Dario Beraldo

Ivan Di Liberti

Jeremy Hahn

Andrea Gagna

Dennis Gaitsgory

Owen Gwilliam

Jacob Lurie

Valerio Melani

Sam Raskin

Peter Haine

Mark Macerato

Michele Pernice

I started to learn derived geometry and geometric Langlands for the first time at the beginning of my PhD, mostly during the pandemic. That has not been always easy, and I wish to deeply thank first of all my parents, with whom I lived during most of the time, for their constant and warm support and affection, and also for their help with my English writing. 1 My advisors, Mauro Porta and Gabriele Vezzosi, were the ones who suggested the topic of this thesis to me in the first place and encouraged me to embrace the language of derived algebraic geometry. Geographical distance and the pandemic have prevented me from having a frequent interaction with them during most of my PhD, and our longest time together remains Spring 2019 in Berkeley, from which I will remember Mauro's elven verses from the foundational Noldor poem Higher Topos Theory, and Gabriele's unmistakable smile behind G. Karapet's portrait of a derived Artin stack.

Emanuele Pavia experienced with me "all the sorrows, but also the joy and marvel, of this journey": I owe him empathy, mathematics, many curses (especially in the Ghost type variant), and his company during two chocolate-based months of lockdown in Strasbourg.

of wonder. A special thank to Peter, for sharing their knowledge, enthusiasm, and many thoughts about mathematical life; to Michele for willing to undertake such a thorough yet satisfying journey through derived geometry (and another in a galaxy far far away in search of crêpes); to Marco for the equally fruitful and fun period we spent in Regensburg earlier this year and for staring with me at the frown of the terracotta warrior that awaits outside.

Finally, thanks, for their friendship, to Leonardo Abate, Gioacchino Antonelli, Marco Barbato, Lorenzo Benedini (who was so kind to share his house in Milan with me for five weeks), Andrea Cappelletti, Manuel Crispo, Julian Demeio (to whom I owe one, and he knows why), Fabio Ferri, Alberto Morelli, 1.1 The geometric Satake theorem

Motivation

A classical problem in representation theory is the study of a reductive group G (e.g. GL n , SL n , PGL n ) and its Langlands dual Ǧ (e.g. ǦL n = GL n , ŠL n = PGL n ). A celebrated result in the study of Langlands duality is the Satake theorem, which establishes an isomorphism between the C-algebra of compactly supported G(Z p )-biinvariant functions on G(Q p ), called the Hecke algebra of G, and the (complexified) Grothendieck ring of finite dimensional representations of Ǧ. Ginzburg [START_REF] Ginzburg | Perverse sheaves on a Loop group and Langlands' duality[END_REF] and later Mirkovic and Vilonen [START_REF] Mirkovic | Geometric Langlands duality and representations of algebraic groups over commutative rings[END_REF] provided a "sheaf theoretic" analogue (actually a categorification) of this theorem, called the Geometric Satake Equivalence: here G is a complex reductive group, and the statement has the form of an equivalence of tensor categories between the derived category of equivariant perverse sheaves Perv G O (Gr G ) and the category of finite dimensional representations of Ǧ (see Section 1.1.2 below). The key new object here is the affine Grassmannian Gr G , an infinite dimensional algebro-geometric object with the property that Gr G (C) = G(C((t)))/G(C t ).

The importance of results such as Geometric Satake, Derived Satake and their variants is related to the more general (and partly conjectural) Geometric Langlands Duality, which was introduced by Beilinson and Drinfeld in analogy to the celebrated arithmetic Langlands conjecture (see e.g. [START_REF] Beilinson | Quantization of Hitchin's integrable system and Hecke eigensheaves[END_REF]). One can say that if the Geometric Langlands Duality deals with algebraic and geometric data related to a reductive group and a smooth complex curve X, the Geometric Satake Equivalence is a "specialization" which looks at the same data near a chosen closed point in X; the affine Grassmannian Gr G itself, for example, is related to the local geometry of smooth curves, see Proposition 1.1.5 below. The use of techniques from homotopy theory and derived algebraic geometry in this field has provided many powerful results, and the current and most convincing formulations of the Geometric Langlands Duality are themselves derived in nature. One of the currently most accepted statements for a Geometric Langlands Conjecture is in [START_REF] Arinkin | Singular support of coherent sheaves, and the geometric Langlands conjecture[END_REF].

Intuitively, the statement should satisfy the following requirements.

Conjecture 1.1.1. Let G be a reductive complex group, and X a smooth projective complex curve. Then the Geometric Langlands Duality should at least:

• establish an equivalence between some category of sheaves over the stack Bun G (X) of G-torsors over X and a category of sheaves over the stack of Ǧ-local systems on X.

• agree with the Geometric Satake Equivalence (see below) when specialized "at" any closed point of X.

In order to explain Conjecture 1.1.1 we now present a short overview of the Geometric Satake Equivalence. The expert reader can skip directly to Section 1.3. We refer the reader seeking for a detailed explanation on this matter to the excellent notes [START_REF] Zhu | An introduction to affine Grassmannians and to the geometric Satake equivalence[END_REF], covering all of the next subsection and much more, including background, motivations and further developments.

Statement of the Geometric Satake Theorem

Theorem 1.1.2 (Geometric Satake Equivalence). Fix a reductive algebraic group G over C, and a commutative ring k for which the left-hand-side of the following formula is defined: for example, k could be Q ℓ , Z ℓ , Z/ℓ n Z, F ℓ . There exists a symmetric monoidal structure ⋆ on Perv G O (Gr G ), called convolution, and an equivalence of symmetric monoidal abelian categories

(Perv G O (Gr G , k), ⋆) ≃ (Rep fin ( Ǧ, k), ⊗).
Let us explain the meaning of this statement. Here Ǧ is the Langlands dual of G, obtained by dualising the root datum of the original group G, and the right-hand-side is the abelian category of finite-dimensional R-representations of Ǧ, equipped with a tensor (i.e. symmetric monoidal) structure given by the tensor product of representations. In order to define the left hand side, we need to introduce some further definitions: by G O we mean the representable functor C-algebras → Set, R → Hom(R t , G) (also denoted by G(C t )), by G K we mean the ind-representable functor C-algebras → Set, R → Hom(R((t)), G) (also denoted by G(C((t)))), and by Gr G we mean the affine Grassmannian, that is the stack quotient Gr G = G(C((t)))/G(C t ). Ind-representability of G K (and of Gr by consequence) comes from the fact that there is a natural filtration in finite-dimensional projective schemes Gr ≤N , N ≥ 0, induced by [START_REF] Zhu | An introduction to affine Grassmannians and to the geometric Satake equivalence[END_REF]Theorem 1.1.3]. We will call this filtration the lattice filtration.

Remark 1.1.3.

There is a natural action of G O on Gr G by left multiplication, whose orbits define an algebraic stratification of Gr G over the poset X • (T ) + . When viewed from the point of view of the complex-analytic topology, this stratification satisfies the so-called Whitney conditions (for a proof, see [Mat]). This allows to define the category Perv G O (Gr G , k), namely the abelian category of G O -equivariant perverse sheaves on Gr G with values in k-modules. This category is defined as

colim N Perv G O (Gr ≥N , k)
in the sense of [START_REF] Zhu | An introduction to affine Grassmannians and to the geometric Satake equivalence[END_REF]5.1 and A.1.4].

The ind-scheme Gr G is related to the theory of curves in the following sense: if X is a smooth projective complex curve, the formal neighborhood X x at a given closed point x of X is given by the map ϕ x : Spec C t → X. The inclusion C t ⊂ C((t)) induces a map Spec C((t)) → Spec C t ϕx -→ X which is a model for the punctured formal neighbourhood Xx of x.

Convolution product of equivariant perverse sheaves

Now we explain the tensor structure on both sides. The category Rep( Ǧ, k) is equipped with the standard tensor product of representations; we define now the tensor structure given by convolution product on Perv G O (Gr G ). A more detailed account is given in [Zhu16, Section 1, Section 5. 1, 5.4]. Consider the diagram

G K × Gr G G K × G O Gr G Gr G × Gr G Gr G p q m (1.1.2)
where G K × G O Gr G is the stack quotient of the product G K × Gr G with respect to the "anti-diagonal" left action of G O defined by γ • (g, [h]) = (gγ -1 , [γh]). The map p is the projection to the quotient on the first factor and the identity on the second one, the map q is the projection to the quotient by the "anti-diagonal" action of G O , and the map m is the multiplication map (g, [h]) → [gh]. It is important to remark that this construction, like everything else in this section, does not depend on the chosen

x ∈ X(C), since the formal neighbourhoods of closed points in a smooth projective complex curve are all (noncanonically) isomorphic. Note also that the left multiplication action of G O on G K and on Gr G induces a left action of G O × G O on Gr G × Gr G . It also induces an action of G O on G K × Gr G given by (left multiplication, id) which canonically projects to an action of G O on G K × G O Gr G . Note that p, q and m are equivariant with respect to these actions. Now if A 1 , A 2 are two G O -equivariant perverse sheaves on Gr G , one can define a convolution product

A 1 ⋆ A 2 = m * Ã (1.1.3)
where m * is the derived direct image functor, and à is any perverse sheaf on G K × G O Gr G which is equivariant with respect to the left action of G O and such that q * à = p * (A 1 ⊠ A 2 ). (Of course, the tensor product must be understood as a derived tensor product in the derived category.) Note that such an à exists because q is the projection to the quotient and A 2 is G O -equivariant. This is the tensor structure that we are considering on Perv G O (Gr G ).

Remark 1.1.7. Note that m * carries perverse sheaves to perverse sheaves: indeed, it can be proven that m is ind-proper, i.e. it can be represented by a filtered colimit of proper maps of schemes compatibly with the lattice filtration. By [KW01, Lemma III.7.5], and the definition of Perv G O (Gr G , k) as a direct limit, this ensures that m * carries perverse sheaves to perverse sheaves.

Remark 1.1.8. Note that the convolution product can be described as follows. Consider the diagram of stacks

(G O × G O )\(G K × Gr) G O \(G K ×Gr) G O \Gr × G O \Gr G O \Gr p ∼ m
where all the actions are induced by the left multiplication action of G O on G K . Then:

• the horizontal map is an equivalence;

• a G O -equivariant perverse sheaf on Gr is the same thing as a perverse sheaf on G O \Gr;

• the convolution product is equivalently described (up to shifts and perverse truncations) by

A 1 ⋆ A 2 = m * (p * (A 1 ⊠ A 2 )).
Observations similar to Construction 1.1.6 prove the following: Proposition 1.1.9. We have the following equivalences of schemes or ind-schemes:

• G O ≃ Aut Xx (T G ) • G K (R) ≃ {F ∈ Bun G (X × Spec R), α : F| (X\{x})×Spec R ≃ T G | (X\{x})×Spec R , µ : F| Xx×Spec R ≃ T G | Xx × Spec R} • (G K × Gr G )(R) ≃ {F, α : F| (X\{x})×Spec R ≃ T G | (X\{x})×Spec R , µ : F| Xx×Spec R ≃ T G | Xx×Spec R , G, β : F| (X\{x})×Spec R ≃ T G | (X\{x})×Spec R } • (G K × G O Gr G )(R) ≃ {F, α : F| (X\{x})×Spec R ≃ T G | (X\{x})×Spec R , G, η : F| (X\{x})×Spec R ≃ G (X\{x})×Spec R }.
Let us finally explain the meaning of Conjecture 1.1.1. Thanks to Proposition 1.1.5 and Construction 1.1.6, there is a way to interpret Perv G O (Gr G ) as the "specialization at any point of x" of the "Bun G (X)" side of the Geometric Langlands Conjecture, and Rep( Ǧ) as the specialization of the "local systems" side. In this sense, one wants the Geometric Langlands Conjecture to agree with the Geometric Satake Equivalence.

The spherical Hecke category 1.2.1 Equivariant constructible sheaves

We now review the notion of equivariant constructible sheaves on the affine Grassmannian. Recall that the affine Grassmannian admits a stratification in Schubert cells. Let k be a finite ring. We can consider the ∞-category of sheaves with coefficients in k which are constructible with respect to that stratification. We denote this category by Cons(Gr, S ; k). Here we are not assuming any finite-dimensionality constraint on the stalks nor on the cohomology of our sheaves. We denote by Cons fd (Gr, S ; k) the small subcategory of Cons(Gr, S ; k) spanned by constructible sheaves with finite stalks. The ∞-category Cons(Gr, S ; k) admits a t-structure whose heart is the category of perverse sheaves which are constructible with respect to S . We also consider the category Cons G O (Gr, S ; k) of G O -equivariant constructible sheaves with respect to S , defined as lim . . . Cons(G O × Gr, S ; k) Cons(Gr, S ; k) , (

where the stratification on G O ו • •×. . . G O ×Gr is trivial on the first factors and S on the last one. Note that there exists a notion of category of equivariant constructible sheaves with respect to some stratification (instead of a fixed one, like S in our case). In full generality, let us fix an algebraic group H acting on a scheme X defined over a field K, and let us denote by S the orbit stratification on X. We have a pullback square of triangulated (or dg, or stable ∞-1 ) categories (full faithfulness of the vertical arrows comes from the fact that the transition maps in the colimits are fully faithful).

Cons H (X, S ; k) Cons(X, S ; k) D c,H (X; k) D c (X; k). is not: its essential image only generates the target as a triangulated category ( [Ric]).

On the contrary, the left vertical arrow of (1.2.2) is an equivalence. Indeed, the functor is fully faithful because the transition maps in the diagram of which we take the colimit are, and essentially surjective by the following argument. First of all, we reduce to the finite-dimensional terms of the filtration {Gr N } N ∈N (which can be done by the very definition of category of constructible sheaves on an infinite-dimensional variety). Then we apply the following lemma: Lemma 1.2.1. Let H be an group scheme acting on a finite-dimensional scheme Y , and suppose that the orbits form a stratification S of Y . Then the two categories D c,H (Y ) and Cons H (Y, S ) are equivalent.

Proof. Let us consider an equivariant constructible sheaf F (with respect to some stratification) and the maximal open subset U of Y where the sheaf is locally constant: this is nonempty since we know that F is constructible with respect to some stratification. Then U is H-stable by equivariancy of F and maximality of U , and thus its complementary is, and we can apply Noetherian induction.

Remark 1.2.2. Let Gr an

G (resp. G an O ) be the analytic ind-variety (resp. analytic group) corresponding to Gr G (resp G O ). There is an equivalence between the (triangulated, or dg/∞-) category of algebraic G O -equivariant constructible sheaves over the affine Grassmannian, and that of analytic G an O -equivariant constructible sheaves over Gr an G . Indeed, by [BGH20, Proposition 12.6.4] (in turn building on [Art72, Théorème XVI. 4.1]), there is an equivalence of categories

D alg c (Gr G ) ≃ D an c (Gr an G ),
and the construction which adds the equivariant structure is the same on both sides (the analytification functor commutes with colimits and finite limits).

Derived Satake Theorem and the E 3 -monoidal structure

The Geometric Langlands Conjecture is currently formulated as a "derived" statement (see [START_REF] Arinkin | Singular support of coherent sheaves, and the geometric Langlands conjecture[END_REF].

Bezrukavnikov and Finkelberg [START_REF] Bezrukavnikov | Equivariant Satake category and Kostant-Whittaker reduction[END_REF] have proven the so-called Derived Satake Theorem. There, the abelian category Perv G O (Gr G ; k) is replaced by Sph loc.c. (G) (the small spherical Hecke category), which is a higher category admitting the following presentations:

• as the dg-or ∞-category Cons fd G O (Gr G ; k) of G O -equivariant constructible sheaves on Gr G with finite-dimensional stalks;

• as the dg-or ∞-category D-mod G O (Gr G ; k) G O -equivariant D-modules on Gr G with finitedimensional stalks.

Theorem 1.2.3 (Derived Satake Theorem, [BF07, Theorem 5]). Let G be a reductive group over a field k of characteristic zero. There is an equivalence

Cons fd G O (Gr G , k) ≃ Coh(Spec Sym(ǧ * [1])) Ǧ.
The category Perv G O (Gr G , k) is the heart of a t-structure on Sph loc.c (G), and the Geometric Satake Theorem is indeed recovered from the Derived Satake Theorem by passing to the heart. The same diagram as in (1.1.2) provides the formula for the convolution product of constructible sheaves, but the commutativity of the product is lost. However, in [START_REF] Nocera | A model for E3 the fusion-convolution product of constructible sheaves on the affine Grassmannian[END_REF] we recovered techniques similar to the ones that provide the commutativity of ⋆ in the perverse case, in order to prove a subtler result. Theorem 1.2.4 [START_REF] Nocera | A model for E3 the fusion-convolution product of constructible sheaves on the affine Grassmannian[END_REF]). Let G be a reductive group over C and k be a finite field of coefficients. The ∞-category Sph loc.c (G, k) admits an E 3 -monoidal structure in Cat × ∞ , extending the symmetric monoidal convolution product of perverse sheaves. Theorem 1.2.5. The ∞-category Sph loc.c (G) is equivalent, as an E 3 -∞-category, to the E 2 -center of the derived ∞-category of finite-dimensional representations DRep fd ( Ǧ, k).

Both theorems were originally stated by Gaitsgory and Lurie in unpublished work. The second one follows essentially from the Derived Satake Theorem and work of Ben-Zvi, Francis, Nadler and Preygel [START_REF] Ben-Zvi | Integral transforms and Drinfeld centers in derived algebraic geometry[END_REF], [START_REF] Ben-Zvi | Integral transforms for coherent sheaves[END_REF], and it implies the first one. In Chapter 2, we prove Theorem 1.2.4 independently, building the sought E 3 -monoidal structure in an intrinsic way. This is in the same spirit of the Tannakian reconstruction explained above for the case of perverse sheaves, where the existence of a symmetric monoidal structure on the category is a part of the initial datum, and only a posteriori it is interpreted as the natural tensor product in a category of representations. To be precise, we do not work exactly with the usual small spherical category, but with a big version which is presentable Theorem 2.3.6, and then we deduce the sought result as a corollary Corollary 2.3.7. Remark 1.2.6. It is worth noticing that the heart of an E 3 stable ∞-category C ⊗ with a compatible t-structure is a symmetric monoidal category, whereas for C E 1 -monoidal one only recovers a monoidal category. In other words, an E 3 -monoidal structure for Sph loc.c (G) is the "least level" of commutativity allowing to recover the full symmetric monoidal structure on Perv G O (Gr) in a purely formal way.

Remark 1.2.7.

In recent yet unpublished work [START_REF] Campbell | The derived geometric Satake equivalence via factorization[END_REF], Campbell and Raskin proved that, up to a certain renormalization of both sides, one can put a natural factorizable structure on the RHS of the Derived Satake Equivalence, and that the equivalence can be promoted to a factorizable/E 3 -monoidal equivalence.

Remark 1.2.8. It is worth stressing that the complex topology takes on a prominent role in our proofs. When taking constructible sheaves, we always look at the underlying complex-analytic topological space Gr an G of Gr G , with its complex stratification in Schubert cells. With this topology, Gr an G is homotopy equivalent to Ω 2 B(G an ) (although we do not use this in our paper). This is an equivalent way to derive the E 2 -algebra structure on the spherical category. However, by Remark 1.2.2, the category Sph loc.c is the same both from the algebraic or the complex-analytic point of view and therefore our result works in both settings.

An overview

The main guiding principle of the present dissertation (excluding Chapter 4) is the application of the homotopy theory of stratified spaces (as introduced in [Lur17, Appendix A]) to the study of the affine Grassmannian and the spherical Hecke category. This idea was not part of the project at the beginning, but arose in response to some issues in the description of the tensor structure on the spherical Hecke category; later, it took several other forms. We will now highlight which ones and how they arose.

Before doing this, let us just sketch the structure of the present dissertation. This thesis covers my first three works (two of them with coauthors). Chapter 2 is dedicated to the preprint [START_REF] Nocera | A model for E3 the fusion-convolution product of constructible sheaves on the affine Grassmannian[END_REF], which has been the start of many questions leading to the other works. Chapter 3 contains my work with Marco Volpe [NV21], which deals with problems of more topological nature and exploits phenomena related to the formalism of conically smooth spaces. Chapter 4 contains my work with Michele Pernice on Derived Azumaya Algebras [START_REF] Nocera | The derived Brauer map via homogeneous sheaves[END_REF]. I will explain the connection of this work to my PhD project in Section 1.3.3.

The affine Grassmannian as a Whitney stratified space

As mentioned in Remark 1.1.3, the stratification in Schubert cells of the affine Grassmannian satisfies the Whitney conditions. Now, a stratification can be seen as a continuous map s from the topological space Y to a poset P endowed with the Alexandrov topology (Section 2.A.1). A very important consequence of this property is recorded in the following: Proposition 1.3.1. Let (Y, P, s) be a stratified space satisfying the Whitney conditions. Then the stratification is conical in the sense of [START_REF] Lurie | Higher Algebra[END_REF]Definition A.5.5].

Proof. This is a consequence of [Mat70, Proposition 6.2], as proven in Section 3.2.1.

Theorem 1.3.2 ([Lur17, Theorem A.3.9]). Let (Y, P, s) be a stratified space which is locally of singular shape (see [START_REF] Lurie | Higher Algebra[END_REF]Definition A.4.15]) and conically stratified, and suppose that P satisfies the ascending chain condition. Then there exists an ∞-category Exit(Y, P, s) and an equivalence of the form

Cons(Y, P, s; S) ≃ Fun(Exit(Y, P, s), S)
where S is the ∞-category of spaces.

Corollary 1.3.3. Let k be any ring. In the hypotheses of the previous theorem, there is an equivalence of k-linear stable ∞-categories

Cons(Y, P, s; k) ≃ Fun(Exit(Y, P, s), Mod k )
where Mod k is the derived ∞-category of k-mondules.

Two easy but important consequences of this theorem are the following: Corollary 1.3.4. Let (Y, P, s) be a stratified space which is locally of singular shape and conically stratified.

Then its ∞-category of constructible sheaves is presentable.

Corollary 1.3.5. Suppose that (Y, P, s) → (Z, P, t) is a stratified homotopy equivalence of stratified spaces over the same poset P (i.e. a stratified map inducing equivalences at the level of exit-paths-categories). Then it induces an equivalence of ∞-categories Cons(Y, P, s; S) ≃ Cons(Z, P, t; S).

Another important consequence regards the functoriality and the symmetric monoidality properties of the association (Y, P, s) → Cons(Y, P, s; S). The exact same arguments work if we replace S by the derived ∞-category Mod k , with k a ring. We record all these consequences (proving those which have not been proven before) in Section 2.A.

These facts allow to study the affine Grassmannian and the spherical Hecke category Sph(G) from the viewpoint of stratified homotopy theory. Thanks to this, we may deduce properties about Sph(G) from homotopy-theoretic considerations about the affine Grassmannian. This is the main content of Chapter 2, and leads to prove Theorem 1.2.4 by arguments which are purely intrinsic to the topology of Gr G and its variants, and do not involve anything regarding the spectral side of Geometric Langlands.

The affine Grassmannian as a conically smooth stratified space

A much subtler version of Proposition 1.3.1 can be proven. Indeed, the notion of "conical chart" around a point can be promoted to something more rigid, which represents the analogue in the stratified setting of a smooth differential structure on a topological space. This notion is made rigorous in [AFT17, Section 3], where the Authors define what is called a conically smooth structure on a conically stratified space 2 . Roughly speaking, the idea is the same as that of a differential structure: a conically smooth structure is an equivalence class of atlases, and an atlas is a system of conical charts with a "smooth change of charts" property. A stratified space (Y, P, s) together with a conically smooth structure A is called a conically smooth space. The Authors conjecture that every stratified space satisfying the Whitney conditions (which is conical by Proposition 1.3.1) admits such a conically smooth structure. In Chapter 3 (joint work with Marco Volpe) we prove this conjecture: Theorem 1.3.6 (Theorem 3.2.7). Let (Y, P, s) be a stratified space satisfying the Whitney conditions. Then it admits a canonical conically smooth structure. This result is somehow useful per se, in that it adds a very natural class of examples to the newly introduce theory of conically smooth spaces. However, our middle-term plan together with Marco Volpe is to use this fact to prove additional properties of the affine Grassmannian. Indeed, by Theorem 1.3.6 Gr G admits a canonical conically smooth structure, and so does its Ran version Gr Ran (see Definition 2.1.3,Proposition 2.2.5). Now, [AFT17] introduce the notion of constructible bundle, which is a certain class of maps between conically smooth spaces representing the stratified analogue of the notion of smooth fiber bundle.

Our conjecture together with Marco Volpe is the following: where (H, R, u) is a compact stratified space and I is the unit interval stratified with a closed stratum at 0 and an open stratum at (0, 1], admits a filling.

As a corollary, we would obtain that the map strtop(Gr Ran ) → Ran(X an ) (see Construction 2.2.3) satisfies the exit homotopy lifting property. This is a "folklore" fact, whose proof we have not been able to locate anywhere. This is a crucial step in the proof of Theorem 1.2.4.

Towards an affine Grassmannian for surfaces

A long-term goal in the Geometric Langlands Program is to provide a statement of the Geometric Langlands Conjecture regarding a moduli space of sheaves (or stacks) over a surface S, which should replace Bun G (X) where X is a curve. Such a statement is expected to have connections to representationtheoretic objects such as double Hecke algebras or similar constructions.

A part of this program is to provide statements analogous to the Geometric or Derived Satake Theorem, again following the idea that they should represent the "specialization at a point" of the "global" statement. Therefore, one is naturally led to seek for an analogue of the affine Grassmannian Gr G at the level of surfaces.

For instance, one could fix a complex smooth projective surface S and a point s ∈ S(C), and define

Gr G (S, s)(R) = {F ∈ Bun G (S R ), α : F| (S\{x})×Spec R ∼ -→ T G,(S\{x})×Spec R }.
However, one can prove by means of the Hartogs theorem that this functor is "trivial", because every trivialization away from a closed subset of codimension at least 2 extends to S R , and therefore our moduli space is equivalent to G (seen as the automorphism group of T G,S ).

Alternatively, one could fix an algebraic curve C in S and define

Gr G (S, C)(R) = {F ∈ Bun G (S R ), α : F| (S\C)×Spec R ∼ -→ T G,(S\C)×Spec R }.
This last definition presents an important difference with respect to the setting of curves when it comes to the convolution product. Indeed, one can figure out suitable versions of the convolution diagram, but the analogue of the map m in (1.1.2) is not ind-proper ([Kap00, Proposition 2.2.2]), and therefore we are not granted that the pushforward (or the proper pushforward) along that map takes perverse sheaves to perverse sheaves 3 . However, m ! should preserve constructible sheaves, provided that it carries a sufficiently strong stratified structure. Therefore, although a convolution product of perverse sheaves over Gr G (S, x) or Gr G (S, C) is probably not well-defined, there is a good chance that it is well-defined at the level of some ∞-category of (equivariant) constructible sheaves C.

Another possible version of an affine Grassmannian for surfaces is the following (the additional "g" stays for "gerbes"). Let S be a smooth complex surface and s ∈ S(C). We define

Grg G,S,s (R) = {G a G-gerbe over S R , α : G| (S\{s})×Spec R ∼ -→ BG × (S \ {s}) × Spec R)}.
This definition circulated in the mathematical community years ago, and I am currently studying the properties of such an object in an ongoing project. This motivated by interest in the theory of gerbes. In the case G = GL 1 = G m , the following result of Toën holds: Theorem 1.3.9 ( [START_REF] Toën | Derived Azumaya algebras and generators for twisted derived categories[END_REF], see also Section 1.3.4). If S is a quasicompact quasiseparated scheme, then there is an isomorphism of abelian groups between H 2 (S, G m ) × H 1 (S, Z) and the group dBr Az of derived Azumaya algebras over S up to Morita equivalence. In particular, if S is normal, there is an isomorphism

H 2 (S, G m ) ≃ dBr Az .
This correspondence is useful in proving properties of Grg G,S,s , although Grg Gm,S,s is itself a fairly trivial object. Indeed, Jacob Lurie [Lur] suggested to us a way to prove that a "Beauville-Laszlo theorem for G-gerbes" is true (and thus, that Grg G,S,s is independent of S and s) that reduce to the case when G is abelian and then uses a reformulation of Toën's result in terms of prestable O S -linear prestable presentable ∞-categories (we will talk about this refurmulation in Section 1. 3.4).

Motivated by such applications, Michele Pernice and I studied some further properties of the correspondence established in Theorem 1.3.9. This is the content of Chapter 4, which we summarize in Section 1.3.4.

A final remark: we suspect that techniques analogous to the ones used for the construction of the E 3 -structure on Sph(G) can be used to build an E 5 -structure4 on a suitable category of equivariant constructible sheaves/D-modules on Grg G . A naïve topological motivation comes from the fact that we can define a "topological affine Grassmannian of G-gerbes on a surface" as

Map(S 3 , AutB(G an )/ Map(D 4 , AutB(G an ))
where S 3 is the real 3-sphere, D 4 is the real 4-disk, G an is the complex analytic topological group associated to G, and AutB(G an ) is the 2-group of automorphisms of the trivial topological G an -gerbe. Now this space is homotopy equivalent to

Ω 3 AutB(G an ) ∼ Ω 4 BAutB(G an ),
where BAutB(G an ) is the higher classifying space of AutB(G an ), and on the latter space we have an evident E 4 -structure which is the exact analogue of the E 2 -structure on Ω 2 B(G an ) ∼ Gr an G . Not that the relationship between Grg an G and this topological counterpart is at the moment unclear to us (we do not know an immediate argument to deduce that they are equivalent; the analogous statement for Gr G is already nontrivial), and therefore the existence of the E 4 -structure on Ω 4 BAut(G an ) remains just a heuristic motivation.

The derived Brauer map via twisted sheaves

Let us fix a quasicompact quasiseparated scheme X over some field k of arbitrary characteristic.

In 1966, Grothendieck [START_REF] Grothendieck | Le groupe de Brauer : I. Algèbres d'Azumaya et interprétations diverses[END_REF] introduced the notion of Azumaya algebra over X: this is an étale sheaf of algebras which is locally of the form End(E), the sheaf of endomorphisms of a vector bundle E over X. This is indeed a notion of "local triviality" in the sense of Morita theory: two sheaves of algebras A, A ′ are said to be Morita equivalent if the categories LMod A = {F quasicoherent sheaf over X together with a left action of A} and its counterpart LMod A ′ are (abstractly) equivalent; one can prove that LMod End(E) is equivalent to QCoh(X) via the functor

M → E ∨ ⊗ End(E) M .
The classical Brauer group Br Az (X) of X is the set of Azumaya algebras up to Morita equivalence, with the operation of tensor product of sheaves of algebras. Grothendieck showed that this group injects into H 2 (X, G m ) by using cohomological arguments: essentially, he used the fact that a vector bundle corresponds to a GL n -torsor for some n, and that there exists a short exact sequence of groups

1 → G m → GL n → PGL n → 1.
The image of Br Az (X) inside H 2 (X, G m ) is contained in the torsion subgroup, which is often called the cohomological Brauer group of X. 5One of the developments of Grothendieck's approach to the study of the Brauer group is due to Bertrand Toën and its use of derived algebraic geometry in [START_REF] Toën | Derived Azumaya algebras and generators for twisted derived categories[END_REF]. There, he introduced the notion of derived Azumaya algebra as a natural generalization of the usual notion of Azumaya algebra. Derived Azumaya algebras over X form a dg-category Deraz X . There is a functor Deraz X → Dg c (X), this latter being (in Toën's notation) the dg-category of presentable stable O X -linear dg-categories 6 which are compactly generated, with, as morphisms, functors preserving all colimits. The functor is defined by A → LMod A = {quasicoherent sheaves on X with a left action of A} where all terms have now to be understood in a derived sense. One can prove that this functor sends the tensor product of sheaves of algebras to the tensor product of presentable O X -linear dg-categories, whose unit is QCoh(X). Building on classical Morita theory, Toën defined two derived Azumaya algebras to be Morita equivalent if the dg-categories of left modules are (abstractly) equivalent. This agrees with the fact mentioned above that LMod End(E) ≃ QCoh(X) for any E ∈ Vect n (X).

In [Toë10, Proposition 1.5], Toën characterized the objects in the essential image of the functor A → LMod A as the compactly generated presentable O X -linear dg-categories which are invertible with respect to the tensor product, i.e. those M for which there exists another presentable compactly generated O X -linear dg-category M ∨ and equivalences 1

∼ -→ M ⊗ M ∨ and M ∨ ⊗ M ∼ -→ 1.
The proof of this characterization goes roughly as follows: given a compactly generated invertible dg-category M , one can always suppose that M is generated by some single compact generator E M . Now, compactly generated presentable O X -linear dg-categories satisfy an important descent property [Toë10, Theorem 3.7]; from this and from [Toë10, Proposition 3.6] one can deduce that the End M (E M ) has a natural structure of a quasicoherent sheaf of O X -algebras A, and one can prove that M ≃ LMod A .

Both Antieau-Gepner [START_REF] Antieau | Brauer groups and étale cohomology in derived algebraic geometry[END_REF] and Lurie [START_REF] Lurie | Spectral Algebraic Geometry[END_REF]Chapter 11] resumed Toën's work, using the language of ∞-categories in replacement of that of dg-categories. Lurie also generalized the notion of Azumaya algebra and Brauer group to spectral algebraic spaces, see [START_REF] Lurie | Spectral Algebraic Geometry[END_REF]Section 11.5.3]. He considers the ∞groupoid of compactly generated presentable O X -linear ∞-categories which are invertible with respect to the Lurie tensor product ⊗, and calls it the extended Brauer space Br † X . This terminology is motivated by the fact that the set π 0 Br † X has a natural abelian group structure, and by a result of Toën is isomorphic to H 2 ét (X, G m ) × H 1 ét (X, Z): in particular, it contains the usual cohomological Brauer group of X. At the categorical level, Lurie proves that there is an equivalence of ∞-groupoids between Br † (X) and

Map St k (X, B 2 G m × BZ) (St k
is the ∞-category of stacks over the base field k). In particular, Br † (X) is categorically equivalent to a 2-groupoid.

We can summarize the situation in the following diagram:

Deraz X [Morita -1 ] ≃ ∼ -→ Br † (X) ∼ -→ Map St k (X, B 2 G m × BZ) (1.3.1)
where the left term is the maximal ∞-groupoid in the localization of the ∞-category of derived Azumaya algebras to Morita equivalences. At the level of dg-categories, this chain of equivalences is proven in [Toë10, Corollary 3.8]. At the level of ∞-categories, this is the combination of [Lur18, Proposition 11.5.3.10] and [START_REF] Lurie | Spectral Algebraic Geometry[END_REF]p. 11.5.5.4] Note that, while in the classical case we had an injection Br Az (X) → H 2 (X, G m ), in the derived setting one has a surjection dBr † Az (X) := π 0 (Deraz

X [Morita -1 ]) ≃ → H 2 (X, G m ).
If H 1 (X, Z) = 0 (e.g. when X is a normal scheme), then the surjection becomes an isomorphism of abelian groups.

While the first equivalence in (1.3.1) is completely explicit in the works of Toën and Lurie, the second one leaves a couple of questions open:

• since the space Map(X, B 2 G m × BZ) is the space of pairs (G, P ), where G is a G m -gerbe over X and P is a Z-torsor over X, it is natural to ask what are the gerbe and the torsor naturally associated to an element of Br † (X) according to the above equivalence. This is not explicit in the proofs of Toën and Lurie, which never mention the words "gerbe" and "torsor", but rather computes the homotopy group sheaves of a sheaf of spaces Br † X over X whose global sections are Br † (X).

• conversely: given a pair (G, P ), what is the ∞-category associated to it along the above equivalence?

The goal of Chapter 4 is to give a partial answer to the two questions above. The reason for the word "partial" is that we will neglect the part of the discussion regarding torsors, postponing it to a forthcoming work, and focus only on the relationship between linear ∞-categories/derived Azumaya algebras and G m -gerbes. Our precise result is stated in Theorem 4. The content of this chapter is the paper [START_REF] Nocera | A model for E3 the fusion-convolution product of constructible sheaves on the affine Grassmannian[END_REF]. The aim is to provide an extension of the convolution product of equivariant perverse sheaves on the affine Grassmannian, whose definition will be recalled in the first subsection of this Introduction, to the ∞-category of G O -equivariant constructible sheaves on the affine Grassmannian1 . We will endow this extension with an E 3 -algebra structure in ∞-categories, which is the avatar of Mirkovic and Vilonen's commutativity constraint [MV07, Section 5]. The final result is Theorem 2.3.6. Theorem 2.0.1 (Theorem 2.3.6). If G is a reductive complex group and k is a finite ring of coefficients, there is an object

A ∈ Alg E 3 (Cat × ∞,k
) describing an associative and braided product law on the k-linear ∞-category Cons fd G O (Gr G , k) of G O -equivariant constructible sheaves over the affine Grassmannian (see Section 1.2.1). The restriction of this product law to the abelian category of equivariant perverse sheaves coincides up to shifts with the classical (commutative) convolution product of perverse sheaves [START_REF] Mirkovic | Geometric Langlands duality and representations of algebraic groups over commutative rings[END_REF].

From now on, we will fix the reductive group G and denote the affine Grassmannian associated to G simply by Gr.

The theorem will be proven by steps.

• in Section 2.1, we encode the convolution diagram Remark 1.1.8 in a semisimplicial 2-Segal stack Gr x,• , thus providing an associative algebra object in Corr(StrStk C ) which, after quotienting by the analogue of the G O -action, describes the span given in Remark 1.1.8. This is done in order to express the convolution product, formally, as push-pull of constructible sheaves along this diagram.

• Let Ran(X) be the algebraic Ran space of X, i.e. the prestack parametrising finite sets of points in X (see Definition 2.1.1). The affine Grassmannian Gr admits a variant Gr Ran , called the Ran Grassmannian (see Definition 2.1.3), living over Ran(X), and parametrizing G-torsors on X together with a trivialization away from a finite system of points. This object is a reformulation of the classical Beilinson-Drinfeld Grassmannian, and just like that it allows to define the so-called fusion product of perverse sheaves over the affine Grassmannian (see [MV07, Section 5] for the definition of the fusion product via Beilinson-Drinfeld Grassmannian). This fusion product is built on the fact that the Beilinson-Drinfeld Grassmannian satisfies the so-called factorization property (see [START_REF] Zhu | An introduction to affine Grassmannians and to the geometric Satake equivalence[END_REF]Section 3.1]). This property is formulated in a very convenient way for Gr Ran , taking advantage of the features of the Ran space.

Again in Section 2. • Given a finite-type (stratified) scheme Y over the complex numbers, one can consider its analytification Y an (see [START_REF] Reynaud | Géometrie algébrique et géometrie analytique[END_REF]). This is a complex analytic (stratified) space with an underlying (stratified) topological space which we denote by strtop(Y ). This procedure can be extended to a functor

strtop : PSh(StrSch C ) → Sh(StrTop, S)
, where S is the ∞-category of spaces. In Section 2.2, we consider the analytification of all the constructions performed in Section 2.1. The goal is to exploit the topological and homotopy-theoretic properties of the analytic version of Ran(X), which are reflected all the way up to Gr Ran and to the definition of the fusion product. In particular, these properties allow to apply [Lur17, Theorem 5.5. 4.10] and realize the fusion product as a consequence of the very existence of the map Gr an Ran,k → Ran(X) an for each k. Combining this with the simplicial structure of Gr Ran,• , we obtain that strtop(G O \Gr) carries a natural E 3 -algebra structure in Corr(Sh(StrTop, S)) (by taking quotients in Theorem 2.2.30).

• In Section 2.A we show, building on [Lur17, Appendix A] and many other contributions, that there is a natural "equivariant k-valued constructible sheaves" functor Cons from a certain category of stratified topological spaces with action of a group towards the ∞-category of presentable klinear ∞-categories, which is also symmetric monoidal (Corollary 2.A.11). A very important feature of this functor is that it sends stratified homotopy equivalences to equivalences of presentable ∞-categories. Also, it satiesfies the Beck-Chevalley properties, that allow to take push-pull along correspondences of stratified spaces in a functorial way. In our case, the needed formal properties are contained essentially in the fact that the affine Grassmannian is a stratified ind-scheme and its strata are exactly the orbits with respect to the action of G O . Thanks to all these properties, in Section 2.3 we formally obtain that the category Cons G O (Gr) admits a structure of E 3 -algebra in presentable k-linear ∞-categories (with respect to the Lurie tensor product) inherited from the structure built on strtop(G O \Gr) in the previous point. As a corollary, Cons fd G O (Gr) has an induced E 3 -algebra structure in small k-linear categories (with respect to the usual Cartesian product). Some of the techniques used in the present chapter are already "folklore" in the mathematical community; for example, application of Lurie's [Lur17, Theorem 5.5. 4.10] to the affine Grassmannian appears also in [START_REF] Hahn | Multiplicative structure in the stable splitting of ΩSL n (C)[END_REF], though in that paper the Authors are interested in the (filtered) topological structure of the affine Grassmannian and do not take constructible sheaves. Up to our knowledge, the formalism of constructible sheaves via exit paths and exodromy has never been applied to the study of the affine Grassmannian and the spherical Hecke category. Here we use it in order to take into account the homotopy invariance of the constructible sheaves functor, which is strictly necessary for the application of Lurie's [Lur17, Theorem 5.5.4.10].

Convolution over the Ran space

The aim of this section is to expand the construction of the Ran Grassmannian defined for instance in [Zhu16, Definition 3.3.2] in a way that allows us to define a convolution product of constructible sheaves in the Ran setting.

The presheaves Gr Ran,k

The Ran Grassmannian

Let us recall the definition of the basic objects that come into play. Let Alg C be the category of (discrete) complex algebras and X a smooth projective curve over C. Definition 2.1.1. The algebraic Ran space of X is the presheaf Ran(X) :

Alg C → Set R → {finite subsets of X(R)}.
Remark 2.1.2. Let Fin surj be the category of finite sets with surjective maps between them. Then we have an equivalence of presheaves Ran(X) ≃ colim PSh(X)

I∈Finsurj X I . Definition 2.1.3. The Ran Grassmannian of X is the functor Gr Ran : Alg C → Grpd R → {S ∈ Ran(X)(R), F ∈ Bun G (X R ), α : F| X R \Γ S ∼ -→ T G | X R \Γ S },
where T G is the trivial G-bundle on X R , Γ S is the union of the graphs of the s i inside X R , s i ∈ S, and α is a trivialization, i.e. an isomorphism of principal G-bundles with the trivial G-bundle. This admits a natural forgetting map towards Ran(X).

Gr X = X × Aut(Spec C t ) Gr G ,
where X is the space of formal parameters defined in loc. cit.. This parametrizes

Gr X (R) = {x ∈ X(R), F ∈ Bun G (X R ), α : F| X R \Γx ∼ -→ T G | X R \Γx }.
For each finite set I, there is a multiple version

Gr X I (R) = {(x 1 , . . . , x |I| ) ∈ X(R) I , F ∈ Bun G (X R ), α : F| X R \Γx 1 ∪•••∪Γx |I| ∼ -→ T G | X R \Γx 1 ∪•••∪Γx |I| }.
Then we have Gr Ran = colim I∈Fin surj Gr X I .

Definition 2.1.5. We can define G K,X I , G O,X I in the same way. Let also G K,Ran be the functor

R → {S ∈ Ran(X)(R), F ∈ Bun G (X R ), α : F| X R \Γ S ∼ -→ T G | X R \Γ S , µ : F| (X R ) Γ S ∼ -→ T G | (X R ) Γ S } ≃ colim I G K;X I .
Let G O,Ran be the functor

R → {S ∈ Ran(X)(R), g ∈ Aut (X R ) Γ S (T G )} ≃ colim I G O;X I .
By means of [Zhu16, Proposition 3.1.9] one has that Gr Ran ≃ G K,Ran /G O,Ran in the sense of a quotient in PSh(Stk C ), i.e.

Gr Ran ≃ colim

I G K,X I /G O,X I
where each term is the étale stack quotient in the category of stacks (recall that G K is an ind-scheme and G O is a group scheme) and is equivalent to Gr X I .

Definition 2.1.6. We define Gr Ran,k to be the functor Alg C → Grpd

k-1 times G K,Ran × Ran(X) • • • × Ran(X) G K,Ran × Ran(X) Gr Ran , that is R → {S ∈ Ran(X)(R), F i ∈ Bun G (X R ), α i trivialization of F i outside Γ S , i = 1, . . . , k, µ i trivialization of F i on the formal neighborhood of Γ S , i = 1, . . . , k -1}.
We call r k : Gr Ran,k → Ran(X) the natural forgetting map. Note that Gr Ran,0 = Ran(X), Gr Ran,1 = Gr Ran .

A priori, Gr Ran,k is groupoid-valued, because if R is fixed the F i 's may admit nontrivial automorphisms that preserve the datum of the α i 's and the µ j 's. Actually, this is not the case, just like for the classical affine Grassmannian which is ind-representable: It is worthwhile to remark that the map Gr Ran,k → Ran(X) is ind-representable, although Gr Ran,k itself is not. Definition 2.1.8. Let x ∈ X(C) be a closed point of X. There is a natural map {x} → X → Ran(X), represented by the constant functor R → {x} ∈ Set. Let us denote Gr x,k = Gr Ran,k × Ran(X) {x}. Proposition 2.1.9. Gr k,x is independent from the choice of X and x, and

Gr 1,x ≃ Gr G Gr 2,x ≃ G K × Gr G . Proof. Note first that Gr k,x (R) = (F i ∈ Bun G (X R ), α i : F i | X R \({x}×Spec R) ∼ -→ T G | X R \({x}×Spec R) , µ i : F i | (X R ) ({x}×Spec R) ∼ -→ T G | (X R ) ({x}×Spec R) ) i=1,...,k .
By the Formal Gluing Theorem [START_REF] Hennion | Formal gluing along non-linear flags[END_REF] this can be rewritten as

Gr k,x (R) = (F i ∈ Bun G ( (X R ) {x}×Spec R ), α i : F| (X R ) {x}×Spec R ∼ -→ T G | (X R ) {x}×Spec R , µ i : F| (X R ) ({x}×Spec R) ∼ -→ T G | (X R ) ({x}×Spec R) ) i=1,...,k ≃ ≃ (F i ∈ Bun G ( X {x} × Spec R), α i : F| X{x} ×Spec R ∼ -→ T G | X{x} ×Spec R , µ i : F| X {x} ×Spec R ∼ -→ T G | X {x} ×Spec R ) i=1,...,k ,
but X {x} is independent from the choice of the point x, being (noncanonically) isomorphic to Spec C t (and the same for Xx ). The rest of the statement is clear from the definitions.

The 2-Segal structure

Face maps

We now establish a semisimplicial structure on the collection of the Gr Ran,k .

Construction 2.1.10.

Let ∂ i be the face map from [k -1] to [k] omitting i. We define the corresponding map δ i : Gr Ran,k → Gr Ran,k-1 as follows. A tuple (S, F 1 , α 1 , µ 1 , . . . , F k , α k ) is sent to a tuple (S, F 1 , α 1 , µ 1 , . . . , F i-1 , α i-1 , µ i-1 , Fgl(F i , F i+1 , α i , α i+1 , µ i ), µ ′ i , . . . , F k , α k ),
where:

• Fgl(F i , F i+1 , α i , α i+1 , µ i ) is the pair (F ′ i , α ′ i )
formed as follows: the Formal Gluing Theorem ( [START_REF] Hennion | Formal gluing along non-linear flags[END_REF]) allows us to glue the sheaves

F i | X R \Γ S and F i+1 | (X R ) Γ S along the isomorphism µ -1 i | (X R ) Γ S • α i+1 | (X R ) Γ S
. This is our F ′ i . Also, F ′ i inherits a trivialization over (X R ) Γ S described by

α i | (X R ) Γ S µ -1 i | (X R ) Γ S α i+1 | (X R ) Γ S ,
which is the second datum.

• µ ′ i coincides with µ i+1 via the canonical isomorphism between the glued sheaf and F i+1 over the formal neighbourhood of Γ S . Proof. Let k be fixed. We check the face identities δ i δ j = δ j-1 δ i for i < j. The essential nontrivial case is when k = 3, i = 0, j = 1 or i = 1, j = 2 or i = 2, j = 3. Otherwise the verifications are trivial since, if i < j -1, then the two gluing processes do not interfere with one another. The cases i = 0, j = 1 and i = 2, j = 3 are very simple, because there is only one gluing and one forgetting (F 1 or F 3 respectively). In the remaining case, we must compare

F 1,23 = Fgl(F 1 , Fgl(F 2 , F 3 ) with F 12,3 = Fgl(Fgl(F 1 , F 2 ), F 3 ).
(we omit the S, α i , µ i from the notation for short). We have:

• F 1,23 | X R \Γ S ≃ F 1 | X R \Γ S ≃ F 12 | X R \Γ S ≃ F 12,3 | X R \Γ S • F 1,23 | (X R ) Γ S ≃ F 23 | (X R ) Γ S ≃ F 2 | (X R ) Γ S ≃ F 12 | (X R ) Γ S ≃ F 12,3 | (X R ) Γ S .
This tells us that the two sheaves are the same, and from this it is easy to deduce that the same property holds for the trivializations.

We thus have a semisimplicial structure on Gr Ran,• , together with maps r k : Gr Ran,k → Ran(X) which commute with the face maps by construction.

Verification of the 2-Segal property

The crucial property of the semisimplicial presheaf Gr Ran,• is the following: Proof. Case "0, l ≤ k". With the natural notations appearing in [DK19, Proposition 2.3.2], there is a map Gr Ran,{0,1,...,l} × Gr Ran,{0,l} Gr Ran,{0,l,l+1,...,k} → Gr Ran,{0,...,k} = Gr Ran,k inverse to the natural projection. The map sends

(S, F 1 , α 1 , µ 1 , . . . , F l , α l , F ′ l , α ′ l , µ ′ l , ξ : (F ′ l , α ′ l ) ∼ -→ Fgl({F i , α i , µ j } i=1,...,l,j=1,...,l-1 ), F ′ l+1 , α ′ l+1 , µ ′ l+1 , . . . , F ′ k α ′ k ) → (F 1 , α 1 , µ 1 , . . . , F l , α l , µ ′′ l , F ′ l+1 , α ′ l+1 , µ ′ l+1 , . . . , F ′ k , α ′ k ) where µ ′′ l,l+1 is the trivialization of F l on the formal neighbourhood of x defined as F l | (X R ) Γ S ∼ -→ Fgl({F i , α i , µ j } i=1,...,l,j=1,...,l-1 )| (X R ) Γ S ξ -1 --→ F ′ l | (X R ) Γ S µ ′ l -→ T G (X R ) Γ S .
This map is indeed inverse to the natural map arising from the universal property of the fibered product, thus establishing the 2-Segal property in the case "0, l". The case "l, k" can be tackled in a similar way.

Notation 2.1.13. Given a category or an ∞-category C, we denote by 2-Seg ss (C) the (∞-)category of 2-Segal semisimplicial objects in C.

Action of the arc group in the Ran setting

We now introduce analogues of the "arc group" G O to our global context, namely group functors over Ran(X) denoted by Arc Ran,k , each one acting on Gr Ran,k over Ran(X). This functor is a group functor over Ran(X) under the law

The Ran version

(S, g) • (S, h) = (S, g • h).
There is a semisimplicial group object of (PSh C ) /Ran(X) assigning

[k] → k times G O,Ran × Ran(X) • • • × Ran(X) G O,Ran .
The face maps are described by

δ i : (S, g 1 , . . . , g k ) → (S, g 1 , . . . , i g i g i+1 , . . . , g k ).
Definition 2.1.15. We denote this semisimplicial group functor over Ran(X) by Arc Ran,• . 2Note that Arc Ran,0 = Ran(X), and Arc Ran,1 = G O,Ran . It is also useful to define the version of Arc Ran,k over X I :

Arc X I ,k := k times G O,Ran × Ran(X) • • • × Ran(X) G O,Ran .
As usual, the "Ran" version is the colimit over I ∈ Fin surj of the "X I " versions.

The action on Gr Ran,•

The first observation now is that Arc Ran,k-1 acts on Gr Ran,k on the left over Ran(X) in the following way:

(S, g 2 , . . . , g k ).(S, F 1 , α 1 , µ 1 , . . . , F k , α k ) = = (S, F 1 , α 1 , µ 1 g -1 2 , F 2 , g 2 α 2 , µ 2 g -1 3 , . . . , F k , g k α k )
where g i α i is the modification of α i by g i on (X R ) Γ S which, by the usual "local-global" reformulation, induces a new trivialization of F i outside S.

We denote by:

• Φ k-1,k this action of Arc Ran,k-1 on Gr Ran,k .

• Ξ 1,k the left action of Arc Ran,1 on Gr Ran,k altering the first trivialization α 1 .

• Φ k,k the left action of Arc Ran,k on Gr Ran,k obtained as combination of Ξ 1,k and Φ k-1,k .

Definition 2.1.17. Define Conv Ran,k , the Ran version of the convolution Grassmannian, as the quotient of Gr Ran,k by the left action

Φ k-1,k described above. That is, Conv Ran,k = colim I Arc X I ,k-1 \Gr X I ,k ,
where the terms of the colimits are quotient stacks with respect to the étale topology and Arc X I ,k-1 acts through the pullback to X I of the action Φ k-1,k .

Remark 2.1.18. ConvGr k is a presheaf which can alternatively be described as follows:

R → {S ⊂ X(R), F 1 , G 2 , . . . , G k ∈ Bun G (X R ), α 1 : F 1 | X R \Γ S ∼ -→ T G | X R \Γ S , η 2 : G 2 | X R \Γ S ∼ -→ F 1 | X R \Γ S , . . . , η k : G k | X R \Γ S ∼ -→ G k-1 | X R \Γ S }.
This is proven in [Rei12, Proposition III.1.10, (1)], because if we take m = k then Gr p | ∆ in loc. cit. is the pullback along X n → Ran(X) of the functor described in Remark 2.1.18, and Conv m n is the pullback along X n → Ran(X) of our Conv Ran,m .

The convolution product over Ran(X) Connection with the Mirkovic-Vilonen convolution product

Consider the action of Arc Ran,k on ConvGr k induced by Ξ 1,k , which we still call Ξ 1,k by abuse of notation, and consider also Φ 1,1 as an action of Arc Ran,1 on Gr Ran . Note that Φ ×k 1,1 is an action of Arc where:

• p is the map that forgets all the trivializations µ i .

• q is the projection to the quotient with respect to Φ k-1,k , alternatively described as follows: we keep F 1 and α 1 intact, and define G h by induction as the formal gluing of G h-1 (or

F 1 if h = 1) and F h along µ h-1 and α h : indeed, µ h-1 is a trivialization of G h-1 over the formal neighbourhood of Γ S via the canonical isomorphism between G h-1 and F h-1 on that formal neighbourhood. The isomorphism η h : G h | X R \Γ S ∼ -→ G h-1 | X R \Γ S
is provided canonically by the formal gluing procedure.

• m is the map sending

(S, F 1 , α 1 , G 2 , η 2 , . . . , G k , η k ) → (S, G k , α 1 • η 2 • • • • • η k ).
Remark 2.1.19. Consider the special case k = 2. Note first of all that, since this diagram lives over Ran(X), we can take its fiber at {x} ∈ Ran(X). Under the identifications of Proposition 1.1.9, we obtain the diagram (1.1.2).

We can consider the diagram

Φ k,k \Gr Ran,k Ξ 1,k \ConvGr k Φ ×k 1,1 \Gr × Ran(X) k Ran,1 Φ 1,1 \Gr Ran . ∼ (2.1.2)
The horizontal map is an equivalence since it exhibits its target as the quotient

Ξ 1,k Φ k-1,k \Gr Ran,k ≃ Φ k,k \Gr Ran,k . For k = 2 one obtains: Φ 2,2 \Gr Ran,2 Ξ 1,2 \ConvGr 2 Φ ×2 1,1 \Gr × Ran(X) 2 Ran Φ 1,1 \Gr Ran .
∼ Remark 2.1.20. Take again the fiber of this diagram at the point {x} ∈ Ran(X). This results in a diagram of the form

Φ x,2,2 \Gr x,2 Ξ x,1,2 \ConvGr x,2 Φ ×2 1,1,x \Gr ×2 x,1 Φ 1,1,x \Gr x,1 .
∼ Recall now the identifications of Proposition 1.1.9.

Here, the action

Φ x,1,1 is the usual left-multiplication action of G O over Gr. The action Φ x,2,2 is the action of G O × G O on G K × Gr given by (g 1 , g 2 ).(h, γ) = (g 1 hg -1 2 , g 2 γ). Finally, the action Ξ x,1,2 on ConvGr x,2 is the action of G O on G K × G O Gr given by g[h, γ] = [gh, γ].
Consider now two perverse sheaves F and G on the quotient Φ x,1,1 \Gr x,1 . This is equivalent to the datum of two G O -equivariant perverse sheaves over Gr. We can perform the external product F ⊠ G living over Φ ×2

x,1,1 \Gr ×2 , and then pull it back to Φ x,2,2 \Gr x,2 . Under the equivalence displayed above, this can be interpreted as a Ξ 1,2 -equivariant perverse sheaf over ConvGr 2 . By construction, this is exactly what [START_REF] Mirkovic | Geometric Langlands duality and representations of algebraic groups over commutative rings[END_REF] call F ⊠G (up to shifts). Its pushforward along m is therefore the sheaf

F ⋆ G ∈ Perv G O (Gr).
The same construction can be done with constructible sheaves instead of perverse sheaves. In the rest of the chapter we will describe an algebra structure on the category Cons G O (Gr G ) of equivariant constructible sheaves over the affine Grassmannian. This ∞-category is equivalent to Cons(G O \Gr G ), and from this point of view the product law is exactly the one described here.

Stratifications Stratification of Gr, Gr Ran and Gr Ran,k

Construction 2.1.21. Recall from Remark 1.1.3 that the affine Grassmannian has a stratification in Schubert cells. We have explained in Section 1.2.1 that we are interested in considering constructible sheaves on the affine Grassmannian which are equivariant and constructible with respect to this stratification. While in principle the equivariant structure on the sheaf/D-module is sufficient in that implies the constructibility condition by what observed in Section 1.2.1, it will be very important for us to consider "stratified homotopy equivalences" when in the topological setting (see Definition 2.A.21). Therefore, it is essential to keep track of the stratifications. We will now extend the stratification from Gr to Gr Ran,k . We give/recall the definition of stratified schemes and presheaves in Definition 2.A.2 and Construction 2.A.3. In that formalism, the stratification in Schubert cells can be seen as a continuous map of ind-topological spaces zar(Gr G ) → X • (T ) + where zar(Gr G ) is the Zariski ind-topological space associated to Gr G , and X • (T ) + is the poset of dominant coweights of any maximal torus T ⊂ G. Therefore, the datum (Gr G , S ) may be interpreted as an object of StrPSh C . The global version Gr X admits a stratification described in [START_REF] Zhu | An introduction to affine Grassmannians and to the geometric Satake equivalence[END_REF]eq. 3.1.11], which detects the monodromy of the pair (bundle, trivialization) at the the chosen point. By filtering Gr X by the lattice filtration (see discussion after Theorem 1.1.2) at every point of X, we can exhibit Gr X as a stratified ind-scheme, or more generally a stratified presheaf, whose indexing poset is again X • (T ) + . 3 Notation 2.1.22. From now on, an arrow of the form X → P , where X is a complex presheaf and P is a poset, will denote a geometric morphism zar(X) → P, zar(X) being colim X→X scheme zar(X).

We will now construct maps Gr X I → (X • (T ) + ) I for any I. Recall from [START_REF] Zhu | An introduction to affine Grassmannians and to the geometric Satake equivalence[END_REF] the so-called factorising property of the Beilinson-Drinfeld Grassmannian. For |I| = 2, it says the following:

Proposition 2.1.23 ([Zhu16, Proposition 3.1.13]). There are canonical isomorphisms Gr X ≃ Gr X 2 × X 2 ,∆ X, c : Gr X 2 | X 2 \∆ ≃ (Gr X × Gr X )| X 2 \∆ .
For an arbitrary I, the property is stated in [Zhu16, Theorem 3.2.1]. This property allows us to define a stratification on Gr X I over (X • (T ) + ) I endowed with the lexicographical order:

Definition 2.1.24. For |I| = 2, (Gr X I ) ≤(µ,λ) ⊂ Gr X 2 is defined to be the closure of Gr X,≤µ × Gr X,≤k ⊂ (Gr X × Gr X )| X 2 \∆ ∼ -→ (Gr X 2 )| X 2 \∆ inside Gr X 2 .
For an arbitrary I, the definition uses the small diagonals. This stratification coincides with the partition in Arc X I -orbits of Gr X I . We now consider the map s I :

Gr X I → X • (T ) + given by Gr X I → (X • (T ) + ) I → X • (T ) +
where the first map is the one described above, and the second one is the map

(µ 1 , . . . , µ |I| ) → Σ n i=1 µ i .
The map s I is a stratification. Also, by applying [Zhu16, Proposition 3.1.14], one proves that the diagonal map X J → X I induced by a surjective map of finite sets I → J is stratified with respect to s I , s J . Now, Gr Ran is the colimit colim I∈Fin surj Gr X I , and therefore Gr Ran inherits a map towards X • (T ) + . This stratification coincides with the stratification in Arc Ran -orbits of Gr Ran . Finally, Gr Ran,k admits a map towards Ran(X

• (T ) + ) k , inherited from the bundle map Gr Ran,k → k Gr Ran × Ran(X) • • • × Ran(X) Gr Ran .
Definition 2.1.25. We denote the induced stratification Gr Ran,k → (X • (T ) + ) k by σ k .

Interaction with the semisimplicial structure and with the action of Arc Ran,•

Now we want to study the interaction between the stratifications and the semisimplicial structure. In particular, we want to prove that the simplicial maps agree with the stratifications σ k (and consequently τ k ), thus concluding that Gr Ran,• upgrades to a semisimplicial object in (StrPSh C ) /Ran(X) .

Definition 2.1.26. Consider the semisimplicial group

Cw (I) • (coweights) defined by Cw k = (X • (T ) + ) k δ j : (µ 1 , . . . , µ k ) → (µ 1 , . . . , µ j-1 , µ j + µ j+1 , µ j+2 , . . . , µ k ).
Note that the formal gluing procedure Gr Ran,2 → Gr Ran sends the stratum ((µ, λ), n) to the stratum (µ + λ, n). Indeed, unwinding the definitions and restricting to the case of points of cardinality 1 in Ran(X), formal gluing at a point amounts to multiplying two matrices, one with coweight µ and the other with coweight λ, hence resulting in a matrix of coweight µλ.

Thus if S : X • (T ) + × X • (T ) + → X • (T ) + is the sum map, the diagram Gr Ran,2 Gr Ran,1 Cw 2 = X • (T ) + × X • (T ) + Cw 1 = X • (T ) + τ 2 ∂ 1 τ 1 S commutes.
The case of an arbitrary I uses similar arguments. We can therefore say that for any face map ϕ :

[h] → [k] the induced square Gr Ran,k Gr Ran,h Cw k Cw h τ k Gr Ran (ϕ) τ h Cw(ϕ)
commutes. Note that the top row is a map of presheaves, so the correct interpretation of this diagram is: the diagrams

X I × Ran(X) (Gr Ran,k ) ≤N X I × Ran(X) (Gr Ran,h ) ≤N Cw k Cw h
commute for every I and N , where N refers to the lattice filtration.

Lemma 2.1.27. The strata of the stratification σ k are the orbits of the action of Arc Ran,k on Gr Ran,k over Ran(X).

Proof. We know that the orbits of the Schubert stratification of the affine Grassmannian are the orbits of the action of G O . The same is true at the Ran level. We can thus summarise the content of this whole section as follows, again in the notations of Section 2.A.3. Recall that Ran(X) is seen as a trivially stratified presheaf over Sch C .

Theorem 2.1.29. There exists a functor

ActGr • : ∆ op inj → Act((StrPSh C ) /(Ran(X)) ) [k] → ((Gr Ran,k , σ k ) → (Ran(X)), (Arc Ran,k → Ran(X),
Φ k,k stratified action of Arc Ran,k on Gr Ran,k over Ran(X)), which enjoys the 2-Segal property, and such that:

• ActGr 1 = (Gr Ran → Ran(X), G O,Ran → Ran(X), Φ 1,1 : G O,Ran × Ran(X) Gr Ran → Gr Ran )
• the values of ActGr k object for higher k's describe the Mirkovic-Vilonen convolution diagram and its associativity in the sense of Section 2.1.4.

Fusion over the Ran space 2.2.1 Analytification Topological versions of Gr Ran,k and Arc Ran,k

In order to take into account the topological properties of the affine Grassmannian and of its global variants, we will now analytify the construction performed in the previous section. This will allow us to consider the complex topology naturally induced on the analytic analogue of the prestacks Gr Ran,k by the fact that X is a complex curve, as well as a naturally induced stratification on the resulting complex analytic spaces. In 2.A.1 we describe the stratified analytification functor, which in turn induces a functor This construction admits a relative version over Ran(X) which is not exactly the natural one, because of a change of topology we are going to perform on top(Ran(X)).

strtop : PSh(StrSch C ) → PSh(StrTop) (see
Remark 2.2.2. Let M = top(X), which is a real topological manifold of dimension 2. In [Lur17, Definitions 5.5.1.1, 5.5.1.2] J. Lurie defines the Ran space Ran(M ) of a topological manifold. By definition, there is a map of topological spaces top(Ran(X)) → Ran(M ). Indeed,

top(colim I X I ) ≃ colim I top(X I ) ≃ colim I (topX) I = colim I M I ,
because top is a left Kan extension. Now each term of the colimit is the space of I-indexed collections of points in X(C), and hence it admits a map of sets towards Ran(M ). This is a continuous map: indeed, let f : I → X(C) be a function such that f (I) ∈ Ran({U i }) for some disjoint open sets U i . Then there is an open set V in Map Top (I, top(X)) containing f and such that ∀g ∈ V , g(I) ∈ Ran({U i }): for instance,

V = i {g : I → X | g(f -1 (f (I) ∩ U i )) ⊂ U i } suffices.
This induces a continuous map from colim M I to Ran(M ) by the universal property of the colimit topology, and therefore a continuous map top(Ran(X) an ) → Ran(M ) which is the identity set-theoretically (and thus it is compatible with the stratifications). (2.2.1)

In particular, we obtain a map ρ • : strtop(Gr Let us stress that we are consider Act con (PSh(StrTop) /Ran(M ) ) and not Act con (StrTop) because the whole affine Grassmannian is stratified by a poset which does not satisfy the ascending chain condition, whereas the stratifying posets of the truncations at the level N do.

Preimage functors

For every open U ⊂ Ran(M ), there exists a "preimage space" Gr U,k ∈ StrTop, whose underlying set can be described as {tuples in Gr Ran,k (C) such that S lies in U }.

Formally: Definition 2.2.6. We define functors FactGr k : Open(Ran(M )) → StrTop /Ran(M ) as

Open(Ran(M )) ⊂ Top /Ran(M ) ρ -1 k --→ StrTop /FactGr k → StrTop /Ran(M ) sending U to (U, κ| U ) and finally to (ρ -1 k (U ) → Ran(M ), σ k | ρ -1 k (U )
). This operation is compatible with the semisimplicial structure, and therefore we obtain a functor:

FactGr • : Open(Ran(M )) → 2-Seg ss (StrTop /Ran(M ) ).
We can perform the same restriction construction as above for Arc Ran,k and obtain stratified topological groups

Arc U,k acting on Gr U,k ∈ StrTop /Ran(M ) , functorially in U ∈ Open(Ran(M ))
and k. We denote the functor U → (Arc U,k → Ran(M )) by

FactArc k : Open(Ran(M )) → Grp(StrTop /Ran(M ) ).
Again, this construction is functorial in k.

Fusion

Definition 2.2.8. Let StrTop ⊙ /Ran(M )
be the following symmetric monoidal structure on StrTSpc /(Ran(M ),κ) : if ξ : X → Ran(M ), υ : Y → Ran(M ) are continuous maps, we define ξ ⊙ υ to be the disjoint product

(X × Y) disj = {x ∈ X, y ∈ Y | ξ(x) ∩ υ(y) = ∅}
together with the map towards Ran(M ) induced by the map

union : (Ran(M ) × Ran(M )) disj → Ran(M ) (S, T ) → S ⊔ T.
Recall the definition of the operad Fact(M ) ⊗ from [Lur17, Definition 5.5.4.9]. The aim of this subsection is to extend the FactGr k 's and the FactArc k 's to maps of operads respectively FactGr ⊙ k :

Fact(M ) ⊗ → StrTSpc ⊙ /Ran(M ) and FactArc ⊙ k : Fact(M ) ⊗ → Grp(StrTSpc /Ran(M ) ) ⊙ :
the idea is that the first one should encode the gluing of sheaves trivialised away from disjoints systems of points, and the second one should behave accordingly.

The gluing map

We turn back for a moment to the algebraic side. Definition 2.2.9. Let (Ran(X) × Ran(X)) disj be the subfunctor of Ran(X) × Ran(X) parametrising those S, T ⊂ X(R) for which Γ S ∩ Γ T = ∅. Let also (Gr Ran,k × Gr Ran,k ) disj be the preimage of (Ran(X) × Ran(X)) disj with respect to the map

r k × r k : Gr Ran,k × Gr Ran,k → Ran(X) × Ran(X).
Proposition 2.2.10. There is a map of stratified presheaves χ k : (Gr Ran,k × Gr Ran,k ) disj → Gr Ran,k encoding the gluing of sheaves with trivializations outside disjoint systems of points.

Proof. The map χ k is defined as follows: we start with an object

(S, F 1 , α 1 , µ 1 , . . . , F k , α k ), (T, G 1 , β 1 , ν 1 , . . . , G k , β k ),
where S ∩ T = ∅. We want to obtain a sequence (P, H 1 , γ 1 , ζ 1 , . . . , H k , γ k ). Since the graphs of S and T are disjoint, X R \ Γ S and X R \ Γ T form a Zariski open cover of X R . Therefore, by the descent property of the stack Bun G , every couple F i , G i can be glued by means of α i and β i .

Each of these glued sheaves, that we call H i , inherits a trivialization γ i outside Γ S ∪ Γ T , which is well-defined up to isomorphism (it can be seen both as

α i | X R \(Γ S ∪Γ T ) or as β i | X R \(Γ S ∪Γ T )
). Now set P = S ∪ T (in the usual sense of joining the two collections of points).

It remains to define the glued formal trivializations. However, to define a trivialization ζ i of H i over the formal neighbourhood of Γ P amounts to look for a trivialization of H i on the formal neighbourhood of Γ S ⊔ Γ T . But the first part of this union is contained in X R \ Γ T , where H i is canonically isomorphic to F i by construction; likewise, the first part of the union is contained in X R \ Γ S , where H i is canonically isomorphic to G i by construction. Hence, the originary trivializations µ i and ν i canonically provide the desired datum ζ i , and the construction of the map is complete. Moreover, this map is stratified. Indeed, we have the torsor Gr Ran,k → Gr Ran × Ran(X) • • • × Ran(X) Gr Ran , and the stratification on Gr Ran,k is the pullback of the one on Gr Ran × Ran(X) • • • × Ran(X) Gr Ran . Now for any I, J finite sets, the map (Gr X I ×Gr X J ) disj → Gr X I⊔J is stratified by definition (cfr. Definition 2.1.24). Therefore,

(( k Gr X I × X I • • • × X I Gr X I ) × ( k Gr X J × X J • • • × X J Gr X J )) disj → → k Gr X I⊔J × X I⊔J • • • × X I⊔J Gr X I⊔J
is stratified, and taking the colimit for I ∈ Fin surj , we obtain that

((Gr Ran × Ran(X) • • • × Ran(X) Gr Ran ) × (Gr Ran × Ran(X) • • • × Ran(X) Gr Ran )) disj
is stratified. Finally, since the stratification on Gr Ran,k is induced by the one on Gr Ran × Ran(X) • • • × Ran(X) Gr Ran via the torsor map Gr Ran,k → Gr Ran × Ran(X) • • • × Ran(X) Gr Ran , we can conclude.

Construction of

Gr U,k × Gr V,k strtop((Gr Ran,k × Gr Ran,k ) disj ) Gr Ran,k U × V (Ran(M ) × Ran(M )) disj Ran(M ), π ⊂ union (2.2.2)
where the left hand square is a pullback of topological spaces, and the right top horizontal map is induced by Proposition 2.2.10 by applying strtop. Here we use that the underlying complex-analytical topological space of Ran(X) is -set-theoretically -the space of points of M , and therefore the map

strtop(Ran(X) × Ran(X)) → Ran(M ) × Ran(M ) restricts to a well-defined map strtop((Ran(X) × Ran(X)) disj ) → (Ran(M ) × Ran(M )) disj .
Note also that the bottom composition coincides with U × V → U ⋆ V → Ran(M ), the first map being the one taking unions of systems of points; hence, by the universal property of the fibered product of topological spaces,

(FactGr k )(U ) × (FactGr k )(V ) admits a map towards (FactGr k )(U ⋆ V ) = Gr Ran,k × Ran(M ) (U ⋆ V ), which we call p U,V,k . Of course the triangle Arc U,k × Arc V,k Arc U ⋆V,k Ran(M ) , p U,V,k union•π commutes.
Proposition 2.2.12. Remark 2.2.11 induces well-defined maps of operads

FactGr ⊙ k : Fact(M ) ⊗ → StrTop ⊙
/Ran(M ) encoding the gluing of sheaves trivialised outside disjoint systems of points. That is, we have

• FactGr ⊙ k (U ) is the map (Arc U,k ) → U → Ran(M ) for every U open subset of Ran(M ).
• the image of the morphism

(U, V ) → (U ⋆ V ) for any independent U, V ∈ Open(Ran(M )) is the commuting triangle (FactGr k )(U ) × (FactGr k )(V ) (FactGr k )(U ⋆ V ) Ran(M ) ,
where the top map is the gluing of sheaves trivialised outside disjoints systems of points, and the left map is the map that remembers the two disjoint systems of points and takes their union.

Proof. See Section 2.B.2.

Remark 2.2.13. The constructions performed in the proof of Proposition 2.2.12 are compatible with the face maps of Gr Ran,• . Indeed, for any two independent open subsets U, V ⊂ Ran(M ), the squares

Gr U,k × Gr V,k Gr U ⋆V,k Gr U,k-1 × Gr V,k-1 Gr U ⋆V,k-1
are commutative because the original diagrams at the algebraic level commute. That is,

(Gr Ran,k × Gr Ran,k ) disj Gr Ran,k (Gr Ran,k-1 × Gr Ran,k-1 ) disj Gr Ran,k-1
commutes, since the construction involved in the horizontal maps, as we have seen, does not change the formal trivializations, and, by the independence hypothesis, the non-formal trivializations do not change in the punctured formal neighbourhoods involved in the formal gluing procedure.

Proposition 2.2.14. The maps of operads

FactGr ⊙ k : Fact(M ) ⊗ → StrTop ⊙ /Ran(M ) assemble to a map of operads FactGr ⊙ • : Fact(M ) ⊗ → (2-Seg ss (StrTop)) ⊙ /Ran(M ) .
Proof. Since we have already noticed that the functor strtop preserve finite limits, the condition that FactGr • is 2-Segal can be recovered from the algebraic setting. Now the map

FactGr k (U ) → FactGr {0,...,l} (U ) × FactGr {0,l} (U ) FactGr {l,...,k} (U )
is the pullback of Gr Ran,k → Gr Ran,{0,...,l} × Gr Ran,{0,l} Gr Ran,{0,l,...,k} along U → Ran(M ), hence it is a homeomorphism (and the same holds for the {l, k} case).

Construction of FactArc ⊙

k Construction 2.2.15. We can perform a similar construction for

FactArc k : Open(Ran(M )) → Grp(StrTop /Ran(M ) )
as well. Indeed, we can define

(Arc Ran,k × Arc Ran,k ) disj (R) = {(S, g 1 , . . . , g k ) ∈ FactArc k (R), (T, h 1 , . . . , h k ) ∈ Arc Ran,k (R) | Γ S ∩ Γ T = ∅}
and maps

(Arc Ran,k × Arc Ran,k ) disj (R) → Arc Ran,k (R) ((S, g 1 , . . . , g k ), (T, h 1 , . . . , h k )) → (S ∪ T, g 1 h 1 , . . . , g k h k ),
where g i h i is the automorphism of T G (X R ) Γ S ∪Γ T defined separatedly as g i and h i on the two components, which are disjoint by hypothesis. The rest of the construction is analogous, and provides maps of operads

FactArc ⊙ k : Fact(M ) ⊗ → Grp(StrTop /Ran(M ) ) ⊙
which are, as usual, natural and 2-Segal in k ∈ ∆ op inj .

The factorizing property

Our aim now is to verify the so-called factorization property (see [Lur17, Theorem 5.5.4.10]) for the functors

FactGr ⊙ k : Fact(M ) ⊗ → StrTop ⊙ /Ran(M )
and

FactArc ⊙ k : Fact(M ) ⊗ → Grp(StrTop /Ran(M ) ) ⊙ .
This will immediately imply the property.

Proposition 2.2.16 (Generalised factorization property). If U, V are independent, then the maps

FactGr k (U )× FactGr k (V ) → FactGr k (U ⋆ V ), resp. FactArc k (U ) × FactArc k (V ) → FactArc k (U ⋆ V ), are strat-
ified homeomorphisms over Ran(M ), resp. homeomorphisms of topological groups over Ran(M ).

Proof. Note that the right-hand square in Diagram (2.2.2) is Cartesian. Indeed, let us now prove that its algebraic counterpart

(Gr Ran,k × Gr Ran,k ) disj Gr Ran,k (Ran(X) × Ran(X)) disj Ran(X) is cartesian in PSh C .
The pullback of the cospan computed in PSh C is, abstractly, the functor parametrising tuples of the form (S, T ), (P,

H i , γ i , ζ i ), where (S, T ) ∈ (Ran(X)×Ran(X)) disj (R), P = S∪T, (P, H i , γ i , ζ i ) ∈ Gr Ran,k (R). From this we can uniquely reconstruct a sequence (S, F i , α i , µ i , T, G i , β i , ν i ) in (Gr Ran,k × Gr Ran,k ) disj (R).
To do so, define F i ∈ Bun G (X R ) as the gluing of H i with the trivial G-bundle around T , which comes with a trivialization α i :

F i | X R \Γ S ∼ -→ T G | X R \Γ S .
We also define G i as the gluing of H i with the trivial G-bundle around S, coming with a trivialization β i outside T . As for the formal part of the datum, the ζ i 's automatically restrict to the desired formal neighbourhoods. This construction is inverse to the natural map (Gr

Ran,k × Gr Ran,k ) disj → Gr Ran,k × Ran(X) (Ran(X) × Ran(X)) disj . But now, the diagram strtop(Ran(X) × Ran(X)) disj strtop(Ran(X)) (Ran(M ) × Ran(M )) disj Ran(M )
is again Cartesian, since, set-theoretically, the vertical maps are the identity, and we are just performing a change of topology on the bottom map. Hence the right-hand square in Diagram (2.2.2) is Cartesian, because the functor strtop preserves finite limits. This concludes the proof since the outer square in (2.2.2) is Cartesian, and therefore the natural map

FactGr k (U ) × FactGr k (V ) → FactGr k (U ⋆ V
) is a homeomorphism of topological spaces. Now we turn to FactArc k . It suffices to prove that the square

(Arc Ran,k × Arc Ran,k ) disj Arc Ran,k (Ran(X) × Ran(X)) disj Ran(X)
is Cartesian in PSh C . But this is clear once one considers the map

(Ran(X) × Ran(X)) disj × Ran(X) Arc Ran,k → (Arc Ran,k × Arc Ran,k ) disj
given by

(S, T, {g i ∈ Aut (X R ) Γ S ∪Γ T (T G )}) → ((S, {g i | (X R ) Γ S }), (S, {g i | (X R ) Γ T })).
Since the graphs are disjoint, this map is an equivalence. This concludes the proof.

Local constancy

The aim of this subsection is to prove that the functors FactGr ⊙ k and FactArc ⊙ k satisfy a suitable "local constancy" property, which will be used in the following to apply [Lur17, Theorem 5.5.4.10]. Definition 2.2.17. Let f : (Y, s, P ) → (Y ′ , s ′ , P ′ ) be a map of stratified topological spaces (in the notation of Section 2.A.1), and G a topological group. We say that f is a stratified G-bundle if, locally in the topology strloc (see Section 2.A.1), f coincides with the projection G × Y ′ → Y ′ . Lemma 2.2.18. Let G be a stratified group scheme. If f : S → T is a morphism of stratified schemes which is a stratified G-torsor with respect to the étale topology strét(see again Section 2.A.1), then strtop(f ) :

strtop(S) → strtop(T ) is a principal stratified G-bundle.
Proof. First we perform the proof in the unstratified setting. First of all, to be locally trivial with respect to the étale topology at the algebraic level implies to be locally trivial with respect to the analytic topology at the analytic level (see [Rey71, Section 5] and [Bha, Section 5]). Now the analytic topology on an analytic manifold is the topology whose coverings are jointly surjective families of local homeomorphisms. We want to prove that a trivialising covering for top

FactGr k (Ran({E i })) → FactGr k (Ran({D i })) and FactArc k (Ran({E i })) → FactArc k (Ran({D i }))
are stratified homotopy equivalences over (Ran(X • (T ) + )) k in the sense of Definition 2.A.21.

Proof. The factorising property tells us that these maps assume the form n i=1 FactGr k (Ran

(E i )) → n i=1 FactGr k (Ran(D i )) and n i=1 FactArc k (Ran(E i )) → n i=1 FactArc k (Ran(D i ))
respectively, and hence it suffices to perform the checks term by term, i.e., to assume n = 1. We deal first with the case FactGr k (Ran({E i })) → FactGr k (Ran({D i })). Now we observe that we can reduce to the case k = 1: indeed, Gr Ran,k is a stratified Arc Ran,k-1 -torsor over the k-fold product Gr Ran × Ran(X) • • • × Ran(X) Gr Ran , and therefore Gr Ran,k → Gr Ran × Ran(M ) • • • × Ran(M ) Gr Ran is a stratified Arc Ran,k-1 -principal topological bundle, in particular it is a Serre fibration. Thus, if we prove that Gr Ran(E) → Gr Ran(D) is a stratified homotopy equivalence, then the same will be true for

Gr Ran(E) × Ran(E) • • • × Ran(E) Gr Ran(E) → Gr Ran(D) × Ran(D) • • • × Ran(D)

This provides a functor

ActGr × : Fact(M ) ⊗ × E nu 1 → Corr(Act con (PSh(StrTop) /Ran(M ) ))[esh 1 ]) ×,⊙ (2.2.3)
which is a map of operads in both variables with respect to the respective symmetric monoidal structures on the target, has the factorization property in the first variable and sends the usual inclusions of Ran spaces of systems of disks to equivalences in the target category.

Interaction of convolution and fusion over Ran(M )

Construction 2.2.25. Let thus ActGr • (-)

• : Fact(M ) ⊗ × ∆ op inj → PSh(Act((StrTop) /Ran(M ) )) ⊙ denote the functor defined by (U, k) → (FactArc k (U ), FactGr k (U ), Φ k,k (U )).
Recall that this functor satisfies the 2-Segal property in k. If we apply (the semisimplicial variant of) [DK19, Theorem 11.1.6] to ActGr • (-) ⊙ , we obtain a functor

ActGr(-, -) ⊙,× : Fact(M ) ⊗ × E nu 1 → Corr(PSh(Act(Top /Ran(M ) ))) ⊙,×
This functor is lax monoidal in the variable Fact(M ) ⊗ with respect to the structure ⊙ on the target, and in the variable E nu 1 with respect to the structure × on the target.

Remark 2.2.26. Now we make the two algebra structures interact. Consider Gr Ran,1 = Gr Ran : we have two ways of defining an "operation" on it:

• restrict to Gr Ran,1 × Ran(M ) Gr Ran,1 and consider the correspondence

Gr Ran,2
Gr Ran,1 × Ran(M ) Gr Ran,1 Gr Ran,1

• restrict to (Gr Ran,1 × Gr Ran,1 ) disj , or more precisely to Gr U,1 × Gr

The stalk of the factorising cosheaf

We can apply [Lur17, Theorem 5.5. 4.10] to the map of operads (2.2.3), since we have proven in the previous subsections that the hypotheses of the theorem are satisfied. We denote the operads

E ⊗ n , E ⊗ M by E n , E M .
Corollary 2.2.27. The functor ActGr × induces a nonunital E M -algebra object

ActGr × M ∈ Alg nu E M (Alg nu E 1 (PSh(Act(Top)[esh -1 ]) × )).
Fix a point x ∈ M . The main consequence of the above result is that the stalk of ActGr × at the point {x} ∈ Ran(M ) inherits an E nu 2 -algebra structure in 2-Seg ss (PSh(Act(Top)[esh -1 ]) × )). We will now explain how, running through [Lur17, Chapter 5] again.

E ∩ F = ∅, D ⨿ D ∼ -→ E ⨿ F → D} ≃ ≃ A(Ran(E 0 )) ⊗ A(Ran(F 0 ))
for any choice of an embedding

D ⨿ D ∼ -→ E 0 ⨿ F 0 → D.
Proof. We need to prove that the colimit degenerates. Take indeed two couples of disks as in the statement, say E 1 , F 1 and E 2 , F 2 . By local constancy, we can suppose that all four disks are pairwise disjoint. Now we can embed both E 1 and E 2 into some E, and F 1 , F 2 into some F , in such a way that E ∩ F = ∅.

Then we have canonical equivalences

A(Ran(E 1 )) ⊗ A(Ran(F 1 )) ∼ -→ A(Ran(E)) ⊗ A(Ran(F ))
and

A(Ran(E 2 )) ⊗ A(Ran(F 2 )) A -→ (Ran(E)) ⊗ A(Ran(F )).
This discussion implies that the operation µ on A x encoded by Lurie's theorem has the form

A x ⊗ A x ≃ A(Ran(D)) ⊗ A(Ran(D)) ∼ -→ A(Ran(E 0 )) ⊗ A(Ran(F 0 )) ∼ -→ A(Ran(E 0 ) ⋆ Ran(F 0 )) → A(Ran(D)) ≃ A x , (2.2.4)
where:

• the first and last equivalences come from local constancy;

• the second equivalence is induced by the chosen embedding

D ⨿ D ∼ -→ E 0 ⨿ F 0 → D;
• the third equivalence is the factorization property.

The discussion about the stalk leads to the main theorem of this section:

Theorem 2.2.30. The stalk at x ∈ M of the E M -algebra object ActGr M from Corollary 2.2.27 can be naturally viewed as an object of

ActGr × x ∈ Alg nu E 2 (Alg nu E 1 (Corr(Act con StrTop[esh -1 ]) × ))
encoding simultaneously the convolution and fusion procedures on G O \Gr G .

Product of constructible sheaves 2.3.1 Taking constructible sheaves

Definition 2.3.1. We denote by A ⊗,nu the functor

Cons ⊗ corr • ActGr × x : E nu 2 × E nu 1 → Pr L,⊗ k .
By construction, one has that A ⊗,nu (1, 1) = Cons G O (Gr) and more in general

(⟨m⟩, ⟨k⟩) → m Cons G ×k O (G ×k-1 K × Gr) ⊗ • • • ⊗ Cons G ×k O (G ×k-1 K × Gr) .

Units and the main theorem

Remark 2.3.2. Let us inspect the behaviour of the E 2 product. We had a map µ : Hck x × Hck x → Hck x from (2.2.4). By construction, when we apply the functor Cons, this is sent forward with lower shriek functoriality. Therefore, for k = 1, we end up with a map

Cons G O (Gr) × Cons G O (Gr) µ ! -→ Cons G O .
Recovering the original structure of the map µ, µ ! decomposes as

Cons G O (Gr) × Cons G O (Gr) ∼ -→ Cons Arc Ran(D) (Gr Ran(D) ) × Cons Arc Ran(D) (Gr Ran(D) ) ∼ -→ Cons Arc Ran(E) (Gr Ran(E) ) × Cons Arc Ran(F ) (Gr Ran(F ) ) ∼ -→ Cons Arc Ran(E)⋆Ran(F ) (Gr Ran(E)⋆Ran(F ) ) → Cons Arc Ran(D) (Gr Ran(D) ) ∼ -→ j * x Cons G O (Gr),
where j x : {x} → Ran(M ) is the inclusion. The formal gluing property evidently commutes with this map at the various levels, so this construction is natural in k. As usual, let us denote by Hck x,k be the evaluation of Hck ×

x at ⟨1⟩ ∈ E nu 2 , ⟨m⟩ ∈ E nu 1 . We have an induced map * → Hck x,k for every k. Now if µ k : Hck x,k × Hck x,k → Hck x,k is the multiplication in Sh(StrTop), we can consider the composition

Hck x,k she ∼ Hck x,k × * id,u k ---→ Hck x,k × Hck x,k µ k -→ Hck x,k
and we find that this composition is the identity. Therefore, u k is a right quasi-unit, functorially in k. The condition that it is a left-quasi unit is verified analogously.

Remark 2.3.4.

Let us now inspect the behaviour of the E 1 product. Let us fix the E 2 entry equal to ⟨1⟩ for simpplicity. Then the product law is described by the map

A x (⟨1⟩, ⟨2⟩) = Cons G O (Gr) ⊗ Cons G O (Gr) ⊠ -→ Cons G O ×G O (Gr × Gr) Hckx(∂ 2 ×∂ 0 ) * =p * -----------→ Cons G O ×G O (FactGr 2,x ) = Cons(Hck x ) ∼ = Cons(G O \(G K × G O Gr)) FactGr(δ 1 ) * =m * ----------→ Cons G O (Gr) = A x (⟨1⟩, ⟨1⟩).
The "pullback" and "pushforward" steps come from the construction of the functor out of the category of correspondences, which by construction takes a correspondence to the "pullback-pushforward" transform between the categories of constructible sheaves over the bottom vertexes of the correspondence. Note that the most subtle step is the equivalence in the penultimate step. If one is to compute explicitly a product of two constructible sheaves F, G ∈ Cons G O (Gr), one must reconstruct the correct equivariant sheaf over ConvGr x,2 whose pullback to Gr x,2 is p * (F ⊠ G), and then push it forward along m (in the derived sense of course). We stress again that this, when restricted to Perv G O (Gr), is exactly the definition of the convolution product of perverse sheaves from [START_REF] Mirkovic | Geometric Langlands duality and representations of algebraic groups over commutative rings[END_REF] (up to shift and t-structure).

Proposition 2.3.5. The map of operads A ⊗

x :

E 2 × E nu 1 → Pr L,⊗ k can be upgraded to a map of operads A ⊗ x : E 2 × E 1 → Pr L,⊗ k .
Proof. Again, it suffices to exhibit a quasi-unit. In this case, this is represented by the element *

G K × Gr = Gr x,2 G K × G O Gr = ConvGr 2,x
Gr × Gr Gr.

p q m
In our specific case, we are given a diagram

G K × Gr G K × G O Gr * × Gr Gr × Gr Gr, p q m t×id j
where j is the closed embedding (F, α) → (T G , id| X\x , F, α) whose image is canonically identified with Gr. Let F ∈ Cons k,G O (Gr). We want to prove that 1 ⊠F ≃ j * (k ⊠ F ), i.e. that

q * j * (k ⊠ F ) ≃ p * (t × id) * (k ⊠ F ).
Note that because of the consideration about the image of j the support of both sides lies in G O × Gr ⊂ G K × Gr, and this yields a restricted diagram

G O × Gr Gr Gr Gr. q p m ∼ j
This proves the claim. By applying m * we obtain

1 ⋆ F ≃ m * (j * (k ⊠ F )) = k ⊠ F = F since mj = id.
Thanks to these results, our functor A ⊗ x is promoted to a map of operads E 2 × E 1 → Pr L,⊗ k . By the Additivity Theorem ([Lur17, Theorem 5.1.2.2]), we obtain an E 3 -algebra object in Pr L,⊗ k . Summing up:

Theorem 2.3.6 (Main theorem). Let G be a complex reductive group and k be a finite ring of coefficients.

There is an object A ⊗ x ∈ Alg E 3 (Pr L,⊗ k ) describing an associative and braided product law on the ∞-category

Cons G O (Gr G ) of G O -

equivariant constructible sheaves over the affine Grassmannian. The restriction of this product law to the abelian category of equivariant perverse sheaves coincides, up to shifts and perverse truncations, with the classical (commutative) convolution product of perverse sheaves [MV07].

Corollary 2.3.7.

There is an induced

E 3 -monoidal structure in Cat × ∞ on Cons fd G O (Gr).
Proof. The inclusion Pr L k → Cat ∞,k is lax monoidal with respect to the ⊗-structure on the source and the ×-structure on the target. Therefore, Cons G O (Gr) has an induced E 3 -algebra structure in Cat × ∞,k . One can easily check that the convolution product restricts to the small (not presentable) subcategory of finite-dimensional sheaves, and this concludes the proof.

2.A Constructible sheaves on stratified spaces: theoretical complements

2.A.1 Stratified schemes and stratified analytic spaces

Definitions

The following definitions are particular cases of [BGH20, p. 8.2.1] and ff. . Definition 2.A.2. Let StrSch = Sch × Top StrTop, where the map Sch → Top sends a scheme X to its underlying Zariski topological space, and the other map is the evaluation at [0], be the category of stratified schemes, and StrAff its full subcategory of stratified affine schemes.

Analogously, one can define stratified complex schemes StrSch C and stratified complex affine schemes StrAff C . The key point now is that there is an analytification functor an : Aff C → Stn C , the category of Stein analytic spaces. This is defined in [Rey71, Théorème et définition 1.1] (and for earlier notions used there, see also [START_REF] Grothendieck | Sur les faisceaux algébriques et les faisceaux analytiques cohérents[END_REF]p. 6]). In this way we obtain a Stein space, which is a particular kind of complex manifold with a sheaf of holomorphic functions. We can forget the sheaf and the complex structure and recover an underlying Hausdorff topological space (which corresponds to the operation denoted by | -| in [START_REF] Reynaud | Géometrie algébrique et géometrie analytique[END_REF]) thus finally obtaining a functor top : Sch C → Top.

A reference for a thorough treatment of analytification (also at a derived level) is [START_REF] Holstein | Analytification of mapping stacks[END_REF].

Let now StrStn

C = Stn C × Top,ev 0 StrTop.
There is a natural stratified version of the functor top, namely the one that assigns to a stratified affine complex scheme (S, s : zar(S) → P ) the underlying topological space of the associated complex analytic space, with the stratification induced by the map of ringed spaces u : S an → S: Proof. Since StrTop = Fun(∆ 1 , Top) × ev 1 ,Top,Alex Poset (see Section 2.A.1), it suffices to show that:

StrAff C → StrStn C → StrTop (S, s) → (S an , s • u) → (|S an |, s • u).
• the functor ev 0 : Fun(∆ 1 , Top) → Top preserves and reflects pullbacks.

• the functor Alex : Poset → Top preserves and reflects pullbacks;

Now, the first point follows from the fact that limits in categories of functors are computed componentwise. The second point can be verified directly, by means of the following facts:

• the functor preserves binary products. Indeed, given two posets P, Q, then the underlying sets of P × Q and Alex(P ) × Alex(Q) coincide. Now, the product topology on Alex(P ) × Alex(Q) is coarser than the Alexandrov topology Alex(P × Q). Moreover, there is a simple base for the Alexandrov topology of a poset P , namely the one given by "half-lines" P p 0 = {p ∈ P | p ≥ p 0 }. Now, if we choose a point (p 0 , q 0 ) ∈ Alex(P × Q), the set (P × Q) (p 0 ,q 0 ) is a base open set for the topology of Alex(P × Q), but it coincides precisely with P p 0 × Q q 0 . Therefore the Alexandrov topology on the product is coarser that the product topology, and we conclude. Note that this latter part would not be true in the case of an infinite product.

• equalizers are preserved by a simple set-theoretic argument. Therefore, we can conclude that finite limits are preserved.

• Alex is a full functor (by direct verification). Since we have proved that it preserves finite limits, then it reflects them as well.

Topologies

In our setting, there are two specially relevant Grothendieck topologies to consider: the étale topology on the algebraic side and the topology of local homeomorphisms on the topological side (which has however the same sheaves as the topology of open embeddings). We have thus sites Aff C , ét and Top, loc. We can therefore consider the following topoi:

• Sh ét (Sch C );

• PSh(Sh ét (Sch C )), which we interpret as "étale sheaves over the category of complex presheaves";

• Sh loc (Top). This last topos is indeed equivalent to the usual topos of sheaves over the category of topological spaces and open covers. Now, there are analogs of both topologies in the stratified setting. Namely, we can define strét as the topology whose coverings are stratification-preserving étale coverings, and strloc as the topology whose coverings are jointly surjective famiilies of stratification-preserving local homemorphisms. Therefore, we have well-defined stratified analogues:

• Sh strét (StrSch C ) • Sh strét (StrPSh C ) := PSh(Sh strét (StrSch C ))
• Sh strloc (StrTop).

The stratification functor strtop : StrSch C → StrTop sends stratified étale coverings to stratified coverings in the topology of local homeomorphism, and thus induces a functor Sh strét (StrPSh C ) → Sh strloc (StrTop).

2.A.2 Symmetric monoidal structures on the constructible sheaves functor

Constructible sheaves on conically stratified spaces locally of singular shape

Fix a finite ring of coefficients k (this can be extended to the ℓ-adic setting and to more general rings of coefficients, but we will not do this in the present work. For a reference, see [START_REF] Liu | Enhanced six operations and base change theorems for Artin stacks[END_REF]).

Remark 2.A.5. Let (X, s) be a stratified topological space, and k be a torsion ring. Suppose that (X, s) is conically stratified, that X is locally of singular shape and that P , the stratifying poset, satisfies the ascending chain condition (see [Lur17, Definition A.5.5 and A.4.15 resp.]). By [Lur17, Theorem A.9.3] the ∞-category of constructible sheaves on X with respect to s with coefficients in k, denoted by Cons k (X, s), is equivalent to the ∞-category Fun(Exit(X, s), Mod k ).

Here Exit(X, s) is the ∞-category of exit paths on (X, s) (see [Lur17, Definition A.6.2], where it is denoted by Sing A (X), A being the poset associated to the stratification). We will often write Cons instead of Cons k .

We review the definition of conical stratifications in more detail in Definition 3.1.11.

Remark 2.A.6. In recent work by Porta and Teyssier [START_REF] Porta | Topological exodromy with coefficients[END_REF], the hypothesis of "being locally of singular shape" has been removed. For simplicity, we will often work in this higher degree of generality.

Let StrTop con denote the 1-category of stratified topological spaces (X, P, s) such that the stratification is conical, X is locally of singular shape, and P satisfies the ascending chain condition. This category admits finite products because the product of two cones is the cone of the join space. Therefore, there is a well-defined symmetric monoidal Cartesian structure StrTop × con .

Corollary 2.A.7. Let (X, P, s) ∈ StrTop con , and k be a ring. Then Cons k (X, P, s) is a presentable stable k-linear category.

Therefore, the ∞-category Pr L k of presentable stable k-linear ∞-categories will be our usual environment from now on.

Symmetric monoidal structure Lemma 2.A.8. The functor

Exit : StrTop con → Cat ∞ (X, s : X → P ) → Exit(X, s) = Sing P (X)
carries a symmetric monoidal structure when we endow both source and target with the Cartesian symmetric monoidal structure. That is, it extends to a symmetric monoidal functor

StrTop × con → Cat × ∞ .
Proof. Given two stratified topological spaces X, s : X → P, Y, t : Y → Q, in the notations of [Lur17, A.6], consider the commutative diagram of simplicial sets

Sing P ×Q (X × Y ) Sing(X × Y ) Sing(X) × Sing(Y ) N (P × Q) Sing(P × Q) N (P ) × N (Q) Sing(P ) × Sing(Q). ∼ ∼ ∼
The inner diagram is Cartesian by definition. Therefore the outer diagram is Cartesian, and we conclude that Sing P ×Q (X × Y ) is canonically equivalent to Sing P (X) × Sing Q (Y ). Since Sing P (X) models the ∞-category of exit paths of X with respect to s, and similarly for the other spaces, we conclude.

Remark 2.A.9. There exists a functor P ( * ) : Cat × ∞ → Pr L,⊗ sending an ∞-category C to the presheaf ∞-category P(C), a functor F : C → D to the functor F * : P(D) → P(C) given by given by restriction under F . We would like to set a symmetric monoidal structure on this functor. However, this is slightly complicated. Up to our knowledge, the symmetric monoidal structure is well-studied on its "covariant version", in the following sense. Proof. The existence of an oplax-monoidal structure follows from [Lur17, p. 4.8.1]. As for symmetric monoidality, apparently, a detail in the proof of [Lur17, Proposition 4.8.1.15] needs to be fixed: for any pair of ∞-categories C, D, the equivalence P(C) × P(D) follows from the universal property of the tensor product of presentable categories, and not from [Lur17, Corollary 4.8.1.12]. Indeed, for any cocomplete ∞-category E one has:

Cocont(P(C × D), E) ≃ Fun(C × D, E) ≃ Fun(C, Fun(D, E)) ≃ ≃ Fun(C 0 , Cocont(P(D), E)) ≃ Cocont(P(C), Cocont(P(D), E)) ≃ ≃ Bicocont(P(C) × P(D), E).

Corollary 2.A.11. There is a well-defined symmetric monoidal functor

Cons ⊗ (!) : StrTop × con → Pr L,⊗ k (X, s) → Cons k (X, s) f → f formal ! = Lan Exit(f ) .
Proof. The previous constructions provide us with a symmetric monoidal functor

StrTop × Exit(-) ----→ Cat × ∞ op -→ Cat × ∞ P(-) ---→ Pr L,⊗ sending (X, s) → Fun(Exit(X, s), S), f → Lan Exit(f ) .
But now, with the notations of [Lur17, Subsection 1.4.2], for any ∞-category C we have Fun(C, Sp) = Sp(Fun(C, S)).

Then we can apply [Rob14, Remark 4.2.16] and finally [Rob14, Theorem 4.2.5], which establish a symmetric monoidal structure for the functor Sp(-) : Pr L,⊗ → Pr L,⊗ stable . The upgrade from Sp to Mod k is straightforward and produces a last functor Pr L,⊗ stable → Pr L,⊗ k (the ∞-category of presentable stable k-linear categories).

From now on, we will often omit the "linear" part of the matter and prove statements about the functor Cons : StrTop × con → Pr L,⊗ and its variations, because tha passage to the stable k-linear setting is symmetric monoidal. Remark 2.A.12. We denote the functor Lan Exit(f ) by f formal ! because, in general, it does not coincide with the proper pushforward of sheaves. We will see in the next subsection that it does under some hypothesis on f . We will also see that, as a corollary of Corollary 2.A.11, there exists a symmetric monoidal structure on the usual contravariant version

Cons ( * ) : StrTop op con → Pr L,⊗ (X, s) → Cons(X, s) f → f * = -• Exit(f )
as well.

2.A.3 Constructible sheaves and correspondences

First of all, we need to recall some properties of the category of constructible sheaves with respect to an unspecified stratification.

Definition 2.A.13. Let X be a topological space. Then there is a well-defined ∞-category of constructible sheaves with respect to a non-fixed stratification

D c (X) = colim s:X→P stratification Cons(X, s).
where the colimit is taken over the category StrTop × Top {X} of stratifications of X and refinements between them.

Our aim is to prove the following theorem (we will indeed prove a more powerful version, see Theorem 2.A.22).

It sends morphisms in horiz to pullback functors along those morphisms, and morphisms in vert to proper pushforward functors along those morphisms.

Unlike in the stratified case (i.e. the case when the stratification is fixed at the beginnig, treated in the previous subsections), for D c (-) there is a well-defined six functor formalism. In particular Lemma 2.A.15. For any continuous map f : X → Y , there are well defined functors f * , f ! : D c (Y ) → D c (X). Moreover, f * has a right adjoint Rf * and f ! has a left adjoint Rf ! .

From now on, we will write f * for Rf * and f ! for Rf ! . With these notations, the Proper Base Change Theorem (stated e.g. in [Kim15, Theorem 6] at the level of abelian categories, and in [START_REF] Volpe | Six functor formalism for sheaves with non-presentable coefficients[END_REF] at the level of derived ∞-categories) holds: Theorem 2.A.16 (Base Change theorem for constructible sheaves). For any Cartesian diagram of unstratified topological spaces

X ′ X Y ′ Y f ′ g ′ f g (2.A.2)
there is a canonical transformation of functors

D c (X) → D c (Y ′ ) f ′ ! g ′ * → g * f !
which is an equivalence of functors. The same holds for

f ! g * → g ′ * f ! as functors D c (Y ) → D c (X).
We are now ready to prove Theorem 2.A.14. We follow the approach used by D. Gaitsgory and N. Rozenblyum in [GR17, Chapter 5] to construct IndCoh as a functor out of the category of correspondences. As in these works, we need a theory of (∞, 2)-categories. In particular, we need to extend Cat ∞ to an (∞, 2)-category, which we shall denote by Cat 2-cat ∞ : this is done in [START_REF] Gaitsgory | A Study in Derived Algebraic Geometry[END_REF], informally by allowing natural transformations of functors which are not natural equivalences. We proceed by steps. proper,all . We perform the verifications necessary to the application of that theorem:

• The pullback and 2-out-of-3 properties for horiz = all, co-adm = proper, isom, which are immediately verified.

• • Given a Cartesian diagram

X Y Z W ε 0 γ 1 γ 0 ε 1
with ε i ∈ open and γ i ∈ proper, the Beck-Chevalley condition is satisfied (every functor from correspondences must satisfy it; this is the "easy" part of the extension theorems). Hence there is an equivalence

(γ 1 ) * ε * 0 ≃ ε * 1 (γ 0 ) * .
Now we use the other Beck-Chevalley condition, the one introduced and checked in the previous point, using the fact that ε 1 ∈ open. This new set of adjunctions gives us a morphism

(ε 1 ) ! (γ 1 ) * → (γ 0 ) * (ε 0 ) ! .
We require that this morphism is an equivalence. But since γ i ∈ proper, we have that (γ i ) * ≃ (γ i ) ! and we conclude by commutativity of the diagram and functoriality of the proper pushforward.

Note that we have used exactly once that, respectively, for an open embedding f * ≃ f ! and for a proper morphism

f * = f ! .
This completes the proof of Theorem Definition 2.A.21. We define she to be the class of maps in StrTop con which admit a homotopy inverse which is itself stratified. We define esh to be the class of maps f :

H → H ′ , g : Y → Y ′ in Act con (StrTop)
which admit a homotopy inverse f : H ′ → H, ḡ : H ′ → H also living in Act con (StrTop) (i.e. f , ḡ stratified and ḡ f -equivariant). These classes easily extend to the the setting of presheaves.

Theorem 2.A.22. There is a well-defined "equivariant constructible sheaves" functor

Cons ⊗ act,corr : Corr(P(Act con (PSh(StrTop))[esh -1 ])) × → Pr L,⊗ k .

Proof. We argue as in [START_REF] Gaitsgory | A Study in Derived Algebraic Geometry[END_REF]Chapter 5,3.4]. First of all, we have defined above a (symmetric monoidal) functor q : Act con (StrTop) × → Sh(StrTop) × , in turn inducing a symmetric monoidal functor q corr between the categories of correspondences. We now show that there exists a symmetric monoidal functor Corr(Sh(Top)) × → Pr L,⊗ . ----→ Pr L is symmetric monoidal with respect to the Cartesian structure on the source and to the Lurie tensor product on the target. Therefore, to conclude we can apply arguments similar to the ones used in [GR17, Chapter 5, 4.1.5], that is essentially [GR17, Chapter 9, Proposition 3.2.4], whose hypotheses are trivially verified since horiz = all.

By right

In particular, the functor

Cons : StrTop op con → Cat ∞ (X, s) → Cons(X, s) f → f * = -• Exit(f )
is symmetric monoidal, as we had announced in Remark 2.A.12.

2.B Omitted proofs and details

2.B.1 Proof of Proposition 2.1.7

Proof. Fix a discrete complex algebra R, and let ξ = (S, F i , α i , µ i ) i be a vertex of the groupoid Gr Ran,k (R).

We must prove that π 1 (Gr Ran,k (R), ξ) = 0. We know that Ran(X) is a presheaf of sets over complex algebras. Therefore it suffices to prove that for every S ∈ Ran(X)(R), the fiber of Gr Ran,k → Ran(X) at S is discrete.

Consider then an automorphism of a point (S, F i , α i , µ i ): this is a sequence of automorphisms ϕ i for each bundle F i , such that the diagrams

F i | X R \Γ S T G | X R \Γ S F i | X R \Γ S α i ϕ i | X R \Γ S α i
(for i = 1, . . . , k) and

F i | (X R ) Γ S T G | (X R ) Γ S F i | (X R ) Γ S µ i ϕ i | (X R ) Γ S µ i
(for i = 1, . . . , k -1) commute. (Actually, only the commutation of the former set of diagrams is relevant to the proof.)

The first diagram implies that ϕ i is the identity over X R \ Γ S . We want to show that ϕ i is the identity. In order to show this, we consider the relative spectrum Spec

X R (Sym(F i )) of F i , which comes with a map π : Y = Spec X R (Sym(F i )) → X R . An automorphism of F i corresponds to an automorphism f i of Y over X R ,
which in our case is the identity over the preimage of X R \ Γ S inside Y . Let U be the locus {f i = id}. This set is topologically dense, because it contains the preimage of the dense open set X R \ Γ S . We must see that it is schematically dense, that is the restriction map O Y (Y ) → O Y (U ) is injective. If we do so, then ϕ i = id globally. The remaining part of the proof was suggested to us by Angelo Vistoli.

We may suppose that R (and therefore Y and X R ) are Noetherian. Indeed, we can reduce to the affine case and suppose X R = Spec P and Y of the form Spec P [t 1 , . . . , t n ] from the Noether Lemma (observe that Y is finitely presented over X R ). Any global section f of Spec P [t 1 , . . . , t n ] lives in a smaller Noetherian subalgebra P ′ [t 1 , . . . , t n ], because it has a finite number of coefficients in P . Analogously, we can suppose U to be a principal open set of Spec P [t 1 , . . . , t n ] and thus f | U can be seen as a section in some noetherian subalgebra of P [t 1 , . . . , t n ] g , with g a polynomial in P [t 1 , . . . , t n ]. Therefore we conclude that the proof that f is zero can be carried out over a Noetherian scheme.

Let us recall the following facts. • If f : S → T is a flat morphism of noetherian schemes, p ∈ S, then p is associate in S if and only if p is associate in the fiber of f (p) and f (p) is associate in T .

Let now S = Y and T = X R . First of all, if we consider the composition Y red → X R we have that U red = Y red , because two morphisms between separated and reduced schemes coinciding on an open dense set coincide everywhere. Now we note that U contains the generic points of every fiber. Indeed, every f 1 (x) ⊂ Y factors through Y red → Y because the fibers are integral, and hence through

U red = Y red → Y . Now if y is an associate point in Y then it is associate in f -1 (f (y)).
Therefore it is a generic point of f -1 (f (y)), because every fiber of a principal G-bundle is isomorphic to G, which is integral. But U contains all generic points of the fibers, which are their associated points because the fibers are integral. This implies that U is schematically dense.

2.B.2 Proof of Proposition 2.2.12

Proof. Since the inclusions U → V of open sets in Ran(X) induce inclusions (FactGr k ) U → (FactGr k ) V and do not alter the datum of (FactGr k ) U , it suffices to prove that the maps p U,V,k make the diagram

(FactGr k ) U ⋆V × (FactGr k ) W (FactGr k ) U × (FactGr k ) V × (FactGr k ) W (FactGr k ) U ⋆V ⋆W (FactGr k ) U × (FactGr k ) U ⋆V p U ⋆V,W,k id×p V,W,k p U,V,k ×id p U,V ⋆W,k
commute in StrTop. Now this is true because of the following. Define (Ran(X)×Ran(X)×Ran(X)) disj as the subfunctor of Ran(X) × Ran(X) × Ran(X) parametrising those S, T, P ⊂ X(R) whose graphs are pairwise disjoint in X R . Let (Gr Ran,k × Gr Ran,k × Gr Ran,k ) disj be its preimage under r k × r k × r k :

Gr Ran,k × Gr Ran,k × Gr Ran,k → Ran(X) × Ran(X) × Ran(X). Then the diagram

(Gr Ran,k × Gr Ran,k ) disj (Gr Ran,k × Gr Ran,k × Gr Ran,k ) disj Gr Ran,k (Gr Ran,k × Gr Ran,k ) disj χ k id×χ k χ k ×id χ k
commutes because the operation of gluing is associative, as it is easily checked by means of the defining property of the gluing of sheaves. Note also that everything commutes over Ran(M ).

Finally, to prove that the functor defined in Remark 2.2.11 is a map of operads, we use the characterization of inert morphisms in a Cartesian structure provided by [Lur17, Proposition 2.4.1.5]. Note that:

• An inert morphism in Fact(M ) ⊗ is a morphism of the form

(U 1 , . . . , U m ) → (U ϕ -1 (1) , . . . U ϕ -1 (n) )
covering some inert arrow ϕ : ⟨m⟩ → ⟨n⟩ where every i ∈ ⟨n⟩ • has exactly one preimage ϕ -1 (i).

• An inert morphism in StrTop × is a morphism of functors ᾱ between f : P(⟨m⟩ • ) op → Top and g : P(⟨n⟩ • ) op → Top, covering some α : ⟨m⟩ → ⟨n⟩, and such that, for any S ⊂ ⟨n⟩, the map induced by ᾱ from f (α -1 S) → g(S) is an equivalence in StrTop.

By definition, CGr

× k ((U 1 , . . . , U m )) is the functor f assigning T ⊂ ⟨m⟩ • → j∈T (FactGr k ) U j ,
and analogously

CGr × k ((U ϕ -1 (1) , . . . , U ϕ -1 (m) )) is the functor g assigning S ⊂ ⟨n⟩ • → i∈S (FactGr k ) U ϕ -1 (i) .
But now, if α = ϕ and T = ϕ -1 (S), we have the desired equivalence.

2.B.3 Proof of Proposition 2.2.5

Proof. It is sufficient to prove that each FactGr k is conically stratified and locally of singular shape (although as remarked in Remark 2.A.6 this condition is not strictly necessary). Indeed, each (FactGr k ) U , being an open set of FactGr k with the induced stratification, will be conically stratified and locally of singular shape as well. Moreover, it suffices to show that strtop(Gr Ran ) is conically stratified and locally of singular shape. Indeed, this will imply the same property for the k-fold-product of copies of strtop(Gr Ran ) over Ran(M ), and

FactArc k is a principal bundle over this space, with unstratified fiber. This consideration implies the property for FactArc k .

Let us then prove the property for strtop(Gr Ran ). First of all, the Ran Grassmannian is locally of singular shape because of the following argument.

Proposition 2.B.1. Let U : ∆ op → Open(X) be a hypercovering. Then Sing(X) ≃ colim Sing n∈∆ op (U n ).

Proof. We use [START_REF] Lurie | Higher Algebra[END_REF]Theorem A.3.1]. Condition (*) in loc.cit. is satisfied for the following modification of U. Since U is a hypercovering, one can choose for any [n] a covering (U i n ) i of U n , functorially in n. We can thus define a category C → ∆ op as the unstraightening of the functor

∆ op → Set [n] → {(U i n ) i }.
Then there is a functor Ũ : C → Open(X), ([n], U i n ) → U . This functor satisfies (*) in [Lur17, Theorem A.3.1], and therefore colim ([n],U )∈C Sing(U ) ≃ Sing(X). Now note that U is the left Kan extension of

Ũ along C → ∆ op . Therefore, Sing(X) ≃ colim C Sing(U ) ≃ colim ∆ op Sing(U n ).
This allows us to apply the proof of [Lur17, Theorem A.4.14] to any hypercovering of Ran(M ). Now, the space Ran(M ) is of singular shape because is contractible and homotopy equivalences are shape equivalences. Therefore, so are elements of the usual prebase of its topology, namely open sets of the form i Ran(D i ). One can construct a hypercovering of Ran(M ) by means of such open subsets, and therefore we can conclude by applying the modified version of [START_REF] Lurie | Higher Algebra[END_REF]Theorem A.4.14] that we have just proved. It remains to prove that the Ran Grassmannian is conically stratified. Indeed, it is Whitney stratified. The proof of such property has been suggested to us by David Nadler [Nad], and uses very essential properties of the Ran Grassmannian. Consider two strata X and Y of Gr Ran . We want to prove that they satisfy Whitney's conditions A and B; that is:

• for any sequence (x i ) ⊂ X converging to y, such that T x i X tends in the Grassmanian bundle to a subspace τ y of R m , T y Y ⊂ τ y (Whitney's Condition A for X, Y, (x i ), y);5

• when sequences (x i ) ⊂ X and (y i ) ⊂ Y tend to y, the secant lines x i y i tend to a line v, and T x i X tends to some τ y as above, then v ∈ τ y (Whitney's Condition B for X, Y, (x i ), (y i ), y).6

The only case of interest is when X ∩ Y is nonempty. Observe that, when the limit point y ∈ Y appearing in Whitney's conditions is fixed, conditions A and B are local in Y , i.e. we can restrict our stratum Y to an (étale) neighbourdhood U of the projection of y in Ran(M ). Also, both Y and the y i in Condition B live over some common stratum Ran n (M ) of Ran(M ). Using the factorization property, which splits components and tangent spaces, we can suppose that n = 1. Therefore, X projects onto the "cardinality 1" component of Ran(M ), that is M itself. By the locality of conditions A and B explained before, we can suppose that y and the (y i ) involved in Whitney's conditions live over A 1 . There, the total space is Gr A 1 and this is simply the product Gr × A 1 because on the affine line the identification is canonical. From this translational invariance it follows that we can suppose our stratum Y concentrated over a fixed point 0 ∈ A 1 , that is: both y and the y i can be canonically (thus simultaneously) seen inside Gr 0 ⊂ Gr A 1 . Now, by [Kal05, Theorem 2] we know that there exists at least a point y ∈ X ∩ Y such that, for any Condition A or B for some choice of (x i ), (y i )}

(x i ) → y
does not coincide with the whole X ∩ Y . Let π : Gr Ran → Ran(M ) be the natural map. Note that X and

Y are acted upon by RanArc × Ran(M ) π(X) and RanArc × Ran(M ) π(Y ) ≃ G O respectively (recall that Y is a subset of Gr = Gr 0 ⊂ Gr A 1 )
, and this action is transitive on the fibers over any point of Ran(M ). Now, the actions of RanArc and G O take Whitney-regular points with respect to X to Whitney-regular points with respect to X, since it preserves all strata. Therefore, if y is a "regular" point as above, the whole Y is made of regular points, and we conclude.

Background

In this section we briefly review the definition of Whitney stratified space and we recall the basic properties of conically stratified and conically smooth spaces. By doing this, we will also introduce the necessary notations to state our main result, conjectured in [AFT17, Conjecture 1.5.3].

Smooth stratifications of subsets of manifolds (Thom, Mather)

We take the following definitions from [START_REF] Mather | Notes on Topological Stability[END_REF], with minimal changes made in order to connect the classical terminology to the one used in [AFT17].

Definition 3.1.1. Let M be a smooth manifold. A smooth stratification of a subset Z ⊂ M is a partition of Z into smooth submanifolds of M . More generally, if M is a C µ -manifold, then a C µ stratification of a subset Z of M is a partition of Z into C µ submanifolds of M . Remark 3.1.2.
In particular, all strata of a smoothly stratified space Z ⊂ M are locally closed subspaces of Z.

Definition 3.1.3 (Whitney's Condition B in R n ).
Let X, Y be smooth submanifolds of R n , and let y ∈ Y be a point. The pair (X, Y ) is said to satisfy Whitney's Condition B at y if the following holds. Let (x i ) ⊂ X be a sequence converging to y, and (y i ) ⊂ Y be another sequence converging to y. Suppose that T x i X converges to some vector space τ in the r-Grassmannian of R n and that the lines x i y i converge to some line l in the 1-Grassmannian (projective space) of R n . Then l ⊂ τ . • (condition of the frontier) if Y is a stratum of S , consider its closure Ȳ in M . Then we require that ( Ȳ \ Y ) ∩ Z is a union of strata, or equivalently that S ∈ S , S ∩ Ȳ ̸ = ∅ ⇒ S ⊂ Ȳ ;

• (Whitney's condition B) Any pair of strata of S satisfies Whitney's condition B when seen as smooth submanifolds of M .

Given two strata of a Whitney stratification X and Y , we say that X < Y if X ⊂ Ȳ . This is a partial order on S .

Conical and conically smooth stratifications (Lurie, Ayala-Francis-Tanaka)

Definition 3.1.6. Let P be a partially ordered set. The Alexandrov topology on P is defined as follows. A subset

U ⊂ P is open if it is closed upwards: if p ≤ q and p ∈ U then q ∈ U .
With this definition, closed subsets are downward closed subsets and locally closed subsets are "convex" subsets: p ≤ r ≤ q, p, q ∈ U ⇒ r ∈ U . Definition 3.1.7 ([Lur17, Definition A.5.1]). A stratification on a topological space X is a continuous map s : X → P where P is a poset endowed with the Alexandrov topology. The fibers of the points p ∈ P are subspaces of X and are called the strata. We denote the fiber at p by X p and by S the collection of these strata.

In this definition we do not assume any smooth structure, neither on the ambient space nor on the strata. Note that, by continuity of s, the strata are locally closed subsets of X.

Note also that the condition of the frontier in Definition 3.1.5 implies that any Whitney stratified space is stratified in the sense of Lurie's definition: indeed, one obtains a map towards the poset S defined by S < T ⇐⇒ S ⊂ T , which is easily seen to be continuous by the condition of the frontier. Definition 3.1.8. A stratified map between stratified spaces (X, P, s) and (Y, Q, t) is the datum of a continuous map f : X → Y and an order-preserving map ϕ : P → Q making the diagram

X Y P Q f s t ϕ commute.
Definition 3.1.9. Let (Z, P, s) be a stratified topological space. We define C(Z) (as a set) as

Z × [0, 1) {(z, 0) ∼ (z ′ , 0)} .
Its topology and stratified structure are defined in [Lur17, Definition A.5.3]. When Z is compact, then the topology is the quotient topology. Note that the stratification of C(Z) is over P ◁ , the poset obtained by adding a new initial element to P : the stratum over this new point is the vertex of the cone, and the other strata are of the form X × (0, 1), where X is a stratum of Z. A stratified space is conically stratified if it admits a covering by conical charts.

More precisely, the conically stratified spaces we are interested in are the so-called C 0 -stratified spaces defined in [AFT17, Definition 2.1.15]. Here we recall the two important properties of a C 0 -stratified space (X, s : X → P ):

• every stratum X p is a topological manifold;

• there is a basis of the topology of X formed by conical charts

R i × C(Z) → X
where Z is a compact C 0 -stratified space over the relevant P >p . Note that Z will have depth strictly less than X; this observation will be useful in order to make many inductive arguments work.

Hence the definition of [AFT17] may be interpreted as a possible analog of the notion of topological manifold in the stratified setting: charts are continuous maps which establish a stratified homeomorphism between a small open set of the stratified space and some "basic" stratified set.

Following this point of view, one may raise the question of finding an analog of "smooth manifold" (or, more precisely, "smoothly differentiable structure") in the stratified setting. We refer to [AFT17, Definition 3.2.21] for the definition of a conically smooth structure (and to the whole Section 3 there for a complete understanding of the notion), which is a very satisfactory answer to this question. A C 0 -stratified space together with a conically smooth structure is called a conically smooth stratified space.

The definition of conically smooth structure is rather elaborate. As in the case of C 0 -stratified spaces, here we just give a couple of important and enlightening properties of these conically smooth stratified spaces:

• any conically smooth stratified space is a C 0 -stratified space;

• all strata have an induced structure of smooth manifold, like in the case of Whitney stratifications;

• there is a notion of atlas, in the sense of a system of charts whose domains are the so-called basics, i.e. stratified spaces of the form R i × C(Z) where Z is equipped with a conically smooth atlas: indeed, to make this definition rigorous, the authors of [AFT17] employ an inductive argument on the depth, where the case of depth equal to zero corresponds to the usual notion of an atlas for a smooth manifold, and to pass to a successive inductive step they observe that, whenever there is an open stratified embedding R i × C(Z) → M , then depthZ < depthM .

This system admits a notion of "smooth" change of charts, in the sense that charts centered at the same point admit a subchart which maps into both of them in a "rigid" way. We recommend to look at the proof of Theorem 3.2.7 for a more precise explanation of this property.

• the definition of conically smooth space is intrinsic, in the sense that it does not depend on a given embedding of the topological space into some smooth manifold, in contrast to the case of Whitney stratifications (see Definition 3.1.1 and Definition 3.1.5);

• in [AFT17] the authors also introduced a notion of conically smooth maps, which differs substantially from the "naive" requirement of being stratified and smooth along each stratum that one has in the case of Whitney stratifications, and hence define a category Strat of conically smooth stratified spaces. In this setting, they are able to build up a very elegant theory and prove many desirable results such as a functorial resolution of singularities to smooth manifolds with corners and the existence of tubular neighbourhoods of conically smooth submanifolds. These results allow to equip Strat with a Kan-enrichment (and hence, a structure of ∞-category); also, the hom-Kan complex of conically smooth maps between two conically smooth spaces has the "correct" homotopy type (we refer to the introduction to [AFT17] for a more detailed and precise discussion on this topic), allowing to define a notion of tangential structure naturally extending the one of a smooth manifold and to give a very simple description of the exit-path ∞-category of a conically smooth stratified space.

Up to now, the theory of conically smooth spaces has perhaps been in need of a good quantity of explicit examples, specially of topological nature. The following conjecture goes in the direction of providing a very broad class of examples coming from differential geometry and topology. Conjecture 3.1.12 ([AFT17, Conjecture 1.5.3]). Let (M, S ) be a Whitney stratified space. Then it admits a conically smooth structure in the sense of [AFT17].

The rest of the chapter is devoted to the proof of this conjecture (Theorem 3.2.7).

Whitney stratifications are conically smooth

Whitney stratifications are conical

We will need the following lemma, whose proof (to our knowledge) has never been written down. Lemma 3.2.1. Let (M, S ) be a Whitney stratified space, T a smooth unstratified manifold, and let f : M → T be a proper map of topological spaces which is a smooth submersion on the strata. Then for every p ∈ T the fiber of f at p has a natural Whitney stratification inherited from M .

Proof. First of all, by definition of smoothly stratified space we may suppose that M ⊂ S for some manifold S of dimension n. Again by definition the problem is local, and we may then suppose M = R n .

We want to prove Whitney's condition B for any pair of strata of the form X = X ′ ∩ f -1 (p) and

Y = Y ′ ∩f -1 (p)
, where X ′ , Y ′ are strata of M and p ∈ T . To this end, we reformulate the problem in the following way: consider the product M ×T with its structure maps π 1 : M ×T → M, π 2 : M ×T → T , and its naturally induced Whitney stratification. Consider also the following two stratified subspaces of M × T : the graph Γ f and the subspace π -1 2 (p). Note that we can see Γ f as a homeomorphic copy of M inside the product (diffeomorphically on the strata). Having said that, the intersection

Γ f ∩ π -1
2 (p) is exactly the fiber f -1 (p). Consider now strata X, Y in f -1 (p) as above, seen as strata of Γ f . Consider sequences x i ⊂ X, y i ⊂ Y both converging to some y ∈ Y . Let l i be the line between x i and y i , and suppose that l i → l, T x i X → τ . By compactness of the Grassmannians Gr(n, 1) and Gr(n, dim T x i X) (which is independent of i), there exists a subsequence (x i j ) such that T x i j Γ f | X ′ converge to some vector space V ⊃ τ . Since the stratification on M is Whitney, we obtain that l ⊂ V . On the other hand, applying the same argument to π -1 2 (p) (which is again stratified diffeomorphic to M via the map π 1 ), we obtain that, up to extracting another subsequence,

T x i j (π -1 2 (p) ∩ π -1 1 (X)
) converges to some W ⊃ τ . Again, since the stratification on M is Whitney, we obtain that l ⊂ W . Note that the lines l i and l only depend on the points x i , y i and on the embedding of M into some real vector space, and not on the subspace we are working with. Now we would like to show that τ = V ∩ W , and this will follow from a dimension argument that uses the fact that f | X is a smooth submersion onto T .

Note that dim τ = dim T x i X for every i. Moreover, by the submersion hypothesis, this equals

dim X ′ -dim T . Also, dim V = dim T x i j Γ f | X ′ = dim X ′ and dim W = dim T x i j (π -1 2 (p) ∩ π -1 1 (X ′ )) = dim X ′ . To compute dim V ∩ W , it suffices to compute dim(V + W ), which by convergence coincides with dim(T x i j Γ f | X ′ + T x i j (π -1 2 (p) ∩ π -1 1 (X ′ ))). Let V j = T x i j Γ f | X ′ , W j = T x i j (π -1 2 (p) ∩ π -1 1 (X ′ ))).
We have a map of vector spaces

V j ⊕ W j → T x i j X ′ ⊕ T f (x i j ) T sending (v, w) → (w -v, df x i j v).
This map is surjective (since df is) and is zero on the subspace

{(v, v) | v ∈ V j ∩ W j }; hence it induces a surjective map V j + W j → T x i j X ′ ⊕ T f (x i j ) T. It follows that dim(T x i j Γ f | X ′ + T x i j (π -1 2 (p) ∩ π 1 (X ′ ))) ≥ dim X ′ + dim T and therefore dim V ∩ W = dim V + dim W -dim(V + W ) ≤ dim X ′ -dim T = dim τ. Since τ ⊂ V ∩ W
U : U → T is a submersion. Then f | A : A → Y is a locally trivial fibration.
We recommend the reading of Mather's two papers [START_REF] Mather | Notes on Topological Stability[END_REF] and [START_REF] Mather | Stratifications and Mappings[END_REF] to understand the behaviour of Whitney stratified spaces, specially in order to understand the notion of tubular neighbourhood around a stratum, which is the crucial one in order to prove our main result. We refer to [Mat70, Section 6] for a tractation of tubular neighbourhoods. Here we just recall the definition: Definition 3.2.4. Let S be a manifold and X ⊂ M be a submanifold. A tubular neighborhood T of X in M is a triple (E, ε, ϕ), where π : E → X is a vector bundle with an inner product ⟨, ⟩, ε is a positive smooth function on X, and ϕ is a diffeomorphism of B ε = {e ∈ E | ⟨e, e⟩ < ε(π(e))} onto an open subset of S, which commutes with the zero section ζ of E:

B ε X S. ϕ ζ
From [Mat72, Corollary 6.4] we obtain that any stratum W of a Whitney stratified space (M, S ) has a tubular neighbourhood, which we denote by (T W , ε W ); the relationship with the previous notation is the following: T W is ϕ(B ε ) ∩ M (recall that a priori ϕ(B ε ) ⊂ S, the ambient manifold) 1 . We also denote by ρ the tubular (or distance) function

T W → R ≥0 v → ⟨v, v⟩
with the notation as in Definition 3.2.4. Note that ρ(v) < ε(π(v)).

A final important feature of the tubular neighbourhoods of strata constructed in Mather's proof is that they satisfy the so-called "control conditions" or "commutation relations". Namely, consider two strata X < Y of a Whitney stratified space M . Then, if T X and T Y are the tubular neighbourhoods relative to X and Y as constructed by Mather, one has that

π Y π X = π Y ρ X π Y = ρ X .
We explain the situation with an example. Example 3.2.5. Let M be the real plane R 2 and S the stratification given by

X = {(0, 0)} Y = {x = 0} \ {(0, 0)} Z = M \ {x = 0}.
We take R 2 itself as the ambient manifold. Then Mather's construction of the tubular neghbourhoods associated to the strata gives a result like in Fig. 3.1. Here the circle is T X , and the circular segment is a portion of T Y around a point of Y . We can see here that T Y is not a "rectangle" around the vertical line, as one could imagine at first thought, because the control conditions impose that the distance of a point in T W from the origin of the plane is the same as the distance of its "projection" to Y from the origin.

Keeping this example in mind (together with its upper-dimensional variants) for the rest of the tractation may be a great help for the visualization of the arguments used in our proofs. Now we closely review the proof of [START_REF] Mather | Stratifications and Mappings[END_REF]Theorem 8.3], which is essential for the next section. This review is also useful to fix some notations. Note that we will use euclidean disks D n (and not euclidean spaces R n ) as domains of charts for smooth manifolds, because this will turn out to be useful in Section 3.2.2 in order to define some "shrinking" maps in an explicit way.

′ on W such that ε ′ < ε W . Let N be the set {x ∈ T W | ρ W (x) ≤ ε ′ (π W (x))}.
Let also

A = {x ∈ T W | ρ W (x) = ε ′ (π W (x))} and f = π W | A : A → W .
Note that f is a proper stratified submersion, since π W is a proper stratified submersion and for any stratum S of M the differential of π W | S vanishes on the normal to A ∩ S. Hence by Lemma 3.2.1 the restriction of the stratification of M to any fiber of f is again Whitney. Consider the mapping

g : N \ W → W × (0, 1] defined by g(x) = π W (x), ρ W (x) ε ′ (π W (x))
.

The space N \ W inherits from M a Whitney stratification (see Lemma 3.2.2) and, by [Mat70, Lemma 7.3 and above], the map g is a proper stratified submersion. Thus, since A = g -1 (W × {1}), by Lemma 3.2.3 one gets a stratified2 homeomorphism h fitting in the commuting triangle

N \ W A × (0, 1] W × (0, 1] g h f ×id
.

Furthermore, since W = ρ -1 (0) ⊆ N , h extends to a homeomorphism of pairs

(N, W ) (h,id) ---→ (M (f ), W ),
where M (f ) is the mapping cylinder of f (we recall that f :

A → W is the projection (π W | A )). If D i ⊂ R i
is the unit open disk, for any euclidean chart j : D i → W , the pullback of f along j becomes a projection D i × Z → D i . Note that Z is compact by properness of f , and has an induced Whitney stratification being a fiber of f , as we have noticed above. Finally,

M (f ) ≃ M (D i × Z pr 1 --→ D i ) ≃ C(Z) × D i .
From now on the conical charts obtained through the procedure explained in the previous theorem will be referred to as the Thom-Mather charts associated to the Whitney stratified space M . The rest of this chapter will be devoted to prove that these charts constitute a conically smooth structure (atlas) for M , as conjectured in [AFT17, Conjecture 1.5.3 (3)].

Whitney stratifications are conically smooth

We take all the definitions and notations from [AFT17], specially from Section 3. In particular, we recall that the definition of conically smooth structure is given in [AFT17, Definition 3.2.21].

Let now (M, S ) be a Whitney stratified space. Given a chosen system of tubular neighbourhoods around the strata along with their distance and projection functions {ρ X , π X }, we have an induced collection of Thom-Mather charts associated to this choice. Call A this collection. We are now going to prove that this is a conically smooth atlas in the sense of [AFT17, Definition 3.2.10]. We will then prove (Remark 3.2.9) that different choices of systems of tubular neghbourhoods induce equivalent conically smooth atlases, again in the sense of [AFT17, Definition 3. Proof. The proof will proceed by induction on the depth of (M, S ) (see Note 3.1.10). The case of depth 0 is obvious, since any Whitney stratified space over a discrete poset is just a disconnected union of strata which are smooth manifolds. Thus, we may assume that for any Whitney stratified set (M ′ , S ′ ) with depth(M ′ , S ′ ) < depth(M, S ), the Thom-Mather charts induce a conically smooth structure on

(M ′ , S ′ ).
Now we need to show that the Thom-Mather charts induce an atlas of (M, S ) in the sense of [AFT17, Definition 3.2.10]. We know that the charts cover the space M . By Theorem 3. Hence it remains to prove that the "atlas" axiom is satisfied: that is, if m ∈ M is a point, u : R i × C(Z) → M and v : R j × C(W ) → M are Thom-Mather charts with images U and V , such that m ∈ U ∩ V , then there is a commuting diagram

D k × C(T ) D i × C(Z) D j × C(W ) M f g u v (3.2.1)
such that x ∈ Im(uf ) = Im(vg) and that f and g are maps of basics in the sense of [AFT17, Definition 3.2.4].

It is sufficient to consider strata X, Y such that X < Y (that is X is in the closure of Y ) and m ∈ Y . In particular, X will have dimension strictly less than Y . 3In this setting, we may reduce to the case when u is a Thom-Mather chart for X which also contains

{n ∈ V | ρ Y (n) < γ(π Y (n)), π Y (n) ∈ U ∩ Y } ⊆ V ∩ U.
Let γ be defined as follows. Let ε Y : V ∩ Y → R >0 be the radius function asociated to the Thom-Mather chart v. Note that ε Y is equal to the function

y → sup{ρ Y (n) | n ∈ V, π Y (n) = y}
("maximum radius function" for v). Then it makes sense to define

γ = min v(B δ × * ) (γ/ε Y ) > 0.
Now let us consider the self-embedding

D j i - → B δ ⊂ D j
where i is of the form (t 1 , . . . , t j ) → ( t 1 a 1 , . . . ,

t j a j ) (in such a way that m ∈ Im(v • i)). We call ψ : D j × C(W ) i×(•γ) ---→ D j × C(W ).
This construction is a way to "give conical parameters" for a sufficiently small open subspace of v -1 (U ∩V ): the multiplication by γ is the rescalation of the cone coordinate, while i is the rescalation of the "euclidean" coordinate (i.e. the one relative to the D j component). By construction, m ∈ Im(v • ψ) ⊆ U ∩ V . In particular, the image is contained in U \ X.

Lemma 3.2.8. The function ψ is a map of basics.

Proof. We prove that:

• ψ is conically smooth along D j . Indeed, the map on the bottom row of the diagram in [AFT17, Definition 3.1.4] takes the form

(0, 1) × R j × D j × C(W ) → (0, 1) × R j × D j × C(W ) (t, (v 1 , . . . , v j ), (u 1 , . . . , u n ), [s, z]) → t, v 1 a 1 , . . . v j a j , u 1 a 1 , . . . , u j a j , s γ , z . (3.2.2)
As one can see from the formula, this indeed extends to t = 0, and the extension is called Dψ; the differential Dψ of ψ is the restriction of Dψ to t = 0. The same argument works for higher derivatives.

• Dψ is injective on vectors. This is an immediate verification using the formula (3.2.2).

• We have that A ψ -1 ((D j ×C(W ))\D j ) = ψ * ψ -1 ((D j ×C(W ))\D j ) A (D j ×C(W ))\D j . This is proven by looking at the definition of ψ: charts are only rescaled along the cone coordinate, or rescaled and translated in the unstratified part.

Now consider the open subset

D i × Z × (0, 1) ⊂ D i × C(Z)
. By [AFT17, Lemma 3.2.9], basics form a basis for basics, and therefore we may find a map of basics ϕ :

D i ′ × C(Z ′ ) → D i × C(Z) whose image is contained in D i × Z × (0, 1). Therefore we have a diagram D i ′ × Z ′ D j × C(W ) U \ X u ′ v•ψ
But now, depth(U \ X) < depth(U ) ≤ depth(M ), and U \ X with its natural stratification as an open subset of M is Whitney by Lemma 3.2.2 (or also by definition of Thom-Mather chart). Therefore, by induction we may find maps of basics f ′ , g ′ sitting in the diagram

D k × C(T ) D i ′ × Z ′ D j × C(W ) U \ X f ′ g ′ u ′ v•ψ
Let us define f as the composition

D k × C(T ) f ′ -→ D i ′ × C(Z ′ ) ϕ -→ D i × C(Z)
and g as the composition

D k × C(T ) g ′ -→ D j × C(W ) ψ -→ D j × C(W ). Now u • f = u ′ • f ′ = v • ψ • g ′ = v • g.
Since ϕ, ψ, f ′ , g ′ are maps of basics, then also f and g are, and this completes the proof.

Remark 3.2.9. By [Mat70, Proposition 6.1], different choices of Thom-Mather charts induce equivalent conically smooth atlases in the sense of [AFT17, Definition 3.2.10]. Indeed, the construction of a Thom-Mather atlas A depends on the choice of a tubular neighbourhood for each stratum X, along with its distance and projection functions ρ X , π X . Thus, let A, A ′ be two conically smooth atlases induced by different choices of a system of tubular neighbourhoods as above. We want to prove that A ∪ A ′ is again an atlas. The nontrivial part of the verification is the following. Let us fix two strata X < Y , and a point y ∈ Y ; take ϕ X a Thom-Mather chart associated to the A-tubular neighbourhood T X of X, and that ψ ′ Y a Thom-Mather associated to the A ′ -tubular neighbourhood T ′ Y of Y . We want to verify the "atlas condition" (3. Let G be a G m -gerbe over a qcqs scheme X, and χ a character of G m . We define the category of χ-homogeneous sheaves over G, informally, as the full subcategory QCoh χ (G) of QCoh(G) spanned by those sheaves on which ρ = σ χ . Remark 4.1.4. The above definition is a little imprecise, in that it does not specify the equivalences ρ(γ, F) ≃ σ χ (γ, F), γ ∈ G m , F ∈ QCoh(G). A formal definition is given in [BP21, Definition 5.14]. There, the Authors define an idempotent functor (-) χ : QCoh(G) → QCoh(G), taking the "χ-homogeneous component". This functor is t-exact ([BP21, Proof of Lemma 5.17], and comes with canonical maps i χ,F : F χ → F. The category QCoh χ (G) is defined as the full subcategory of QCoh(G) spanned by those F such that i χ (F) is an equivalence. To fix the notations, let us make these constructions explicit. If X is a scheme and G a G m -gerbe, we have the following diagram

G G × BG m G BG m G X u actα q p π π
where p, q are the projections, u is the atlas of the trivial gerbe and act α is the morphism induced by the bending α of G. See [BP21, Section 5] for more specific description of these maps. Given F ∈ QCoh(G) and χ a character of G m , the Authors define

(F) χ := u * (q * q * (act * α (F) ⊗ L ∨ χ ) ⊗ L χ )
and the morphism i χ (F) : (F) χ → F to be the pullback through u of the counit of the adjunction q * ⊣ q * . We will denote by L χ the line bundle over BG m associated to the character χ, while L χ is the pullback of L χ to the trivial gerbe G × G m , i.e. L χ := p * L χ .

[BP21] also prove that there is a decomposition

QCoh(G) ≃ χ∈Hom(Gm,Gm) QCoh χ (G)
building on the fact that F ≃ χ∈Hom(Gm,Gm) F χ . The same result was previously obtained by Lieblich in the setting of abelian categories and by Bergh-Schnürer in the setting of triangulated categories. The relationship between categories of twisted sheaves and Azumaya algebras has been intensively studied, see [De 04], [De ], [START_REF] Lieblich | Twisted sheaves and the period-index problem[END_REF], [START_REF] Hall | Perfect complexes on algebraic stacks[END_REF], [START_REF] Bergh | Decompositions of derived categories of gerbes and of families of Brauer-Severi varieties[END_REF], [START_REF] Binda | GAGA problems for the Brauer group via derived geometry[END_REF]. Given a G m -gerbe G over X, its category of twisted sheaves QCoh id (G) admits a compact generator, whose algebra of endomorphisms is a derived Azumaya algebra A G . In contrast, if we restrict ourselves to the setting of abelian categories and consider abelian categories of twisted sheaves, this reconstruction mechanism does not work anymore. This is one of the reasons of the success of Toën's derived approach.

The construction of the category of twisted sheaves gives rise to a functor Ger Gm (X) → LinCat St (X)

G → QCoh id (G)
taking values in the 2-groupoid Br † (X) of compactly generated invertible categories (see Theorem 4.2.6). To be precise, this was already known, see for instance [START_REF] Hall | Perfect complexes on algebraic stacks[END_REF]Example 9.3]. This functor corresponds to a section of the map Br

† (X) ∼ -→ Map(X, B 2 G m × BZ) → Map(X, B 2 G m
). This section is not fully faithful nor essentially surjective. However, one can observe that QCoh id (G) is not just an ∞-category, but also carries a t-structure which is compatible with filtered colimits, since as recalled in Remark 4.1.4 the functor (-) id is t-exact. This additional datum allows to "correct" the fact that QCoh id (-) is not an equivalence. Indeed, we can change the target from LinCat St (X) to the ∞-category of stable O X -linear presentable categories with a t-structure compatible with filtered colimits. Under the association (C, C ≥0 , C ≤0 ) → C ≥0 , this ∞-category is equivalent to the category LinCat PSt (X) of so-called Grothendieck prestable O X -linear ∞-categories (see Definition 4.1.11). Therefore, we have a functor

Ψ : Ger Gm (X) → LinCat PSt (X) (4.1.1) G → QCoh id (G) ≥0 .

Reminders on stable and prestable linear categories

Definition 4.1.6. Let C be an ∞-category. We will say that C is prestable if the following conditions are satisfied:

• The ∞-category C is pointed and admits finite colimits.

• The suspension functor Σ : C → C is fully faithful.

→ Cat ∞ R → LinCat St (Spec R) R → LinCat PSt (Spec R)
can be right Kan extended to functors

QStk St , QStk PSt : Sch op k → Cat ∞ .
This gives a meaning to the expressions QStk St (X), QStk PSt (X), which can be thought of as "the category of sheaves of QCoh-linear (resp. QCoh ≥0 -linear) (pre)stable categories on X". For any X ∈ Sch k , there are well-defined "global sections functors" This theorem means that, if X is a qcqs scheme over k, every (Grothendieck pre)stable presentable O X -linear ∞-category C has an associated sheaf of ∞-categories on X having C as category of global sections. We will make substantial use of this fact in the present work. The main reason why we will always assume our base scheme X to be qcqs is because it makes this theorem hold.

QStk St (X) → LinCat St (X), QStk PSt (X) → LinCat PSt (X) constructed in [Lur18,
Definition 4.1.16. We denote by Br(X) the maximal ∞-groupoid contained in QStk PSt (X) and generated by ⊗ QCoh(X) ≥0 -invertible objects which are compactly generated categories, and equivalences between them. We denote by Br † (X) the maximal ∞-groupoid contained in QStk St (X) generated by ⊗ QCoh(X) -invertible objects which are compactly generated categories, and equivalences between them. We call dBr(X) := π 0 Br(X) the derived Brauer group of X and dBr † (X) := π 0 (Br † (X)) the extended derived Brauer group of X.

By [Lur18, Theorem 10.3.2.1], to be compactly generated is a property which satisfies descent. The same is true for invertibility, since the global sections functor is a symmetric monoidal equivalence by Theorem 4.1.15. Therefore, Br † (X) is equivalent to the ∞-category of invertible and compactly generated categories in LinCat St , and an analogous statement is true for Br(X). Remark 4.1.17. The stabilization functor mentioned in Remark 4.1.12 restricts to a functor Br(X) → Br † (X), whose homotopy fiber at any object C ∈ Br † (X) is discrete and can be identified with the collection of all t-structures (C ≥0 , C ≤0 ) on C satisfying the following conditions:

• The t-structure (C ≥0 , C ≤0 ) is right complete and compatible with filtered colimits.

• The Grothendieck prestable ∞-category C ≥0 is an ⊗-invertible object of LinCat PSt (X).

• The Grothendieck prestable ∞-category C ≥0 and its ⊗-inverse are compactly generated.

See [Lur18, Remark 11.5.7.3] for further comments.

Consider now the functor Ψ introduced in Eq. (4.1.1). In analogy to the stable setting, we will prove that it factors through Br(X) → LinCat PSt (X) (see Theorem 4.2.6).

The inverse will be described as follows. By Theorem 4.1.15, if X is a quasicompact quasiseparated scheme, and M a prestable presentable ∞-category equipped with an action of QCoh(X) ≥0 , then M is the category of global sections over X of a unique sheaf of prestable QCoh(X) ≥0 -linear categories M.

where G × X G ′ is seen as a G m × G m -gerbe on X.

Proof. First we prove that the external tensor product induces a t-exact equivalence

⊠ : QCoh(G) ⊗ QCoh(X) QCoh(G ′ ) ≃ QCoh(G × X G ′ ). ( 4
(G) ⊗ QCoh(X) QCoh χ ′ (G ′ ) lands in QCoh (χ,χ ′ ) (G × X G ′ ). It is enough to construct an equivalence α(F ⊠ F ′ ) : (F ⊠ F ′ ) (χ,χ ′ ) ≃ (F) χ ⊠ (F ′ ) χ ′
for every F⊠F ′ . With the notation of Remark 4.1.4, we can do this using the following chain of equivalence:

(F ⊠ F ′ ) (χ,χ ′ ) = (p × p ′ ) * (act * (α,α ′ ) (F ⊠ F ′ ) ⊗ L ∨ (χ,χ ′ ) ) ≃ (p × p ′ ) * ((act α × act α ′ ) * (F ⊠ F ′ ) ⊗ (L ∨ χ ⊠ L ∨ χ ′ )) ≃ (p × p ′ ) * ((act * α F ⊗ L ∨ χ ) ⊠ (act * α ′ F ′ ⊗ L ∨ χ ′ )) ≃ (F) χ ⊠ (F ′ ) χ ′ .
A straightforward computation shows that α is in fact a natural transformation and verifies the condition described in Remark 4.2.8. It remains to prove that the restricted functor induces an equivalence, which will automatically be t-exact. But again, it suffices to show this locally, and in the local case this just reduces to the fact that QCoh

χ (U × BG m ) ⊗ QCoh(U ) QCoh χ ′ (U × BG m ) ≃ QCoh(U ) ⊗ QCoh(U ) QCoh(U ) → QCoh(U ) is an equivalence.
ρ : G × X G ′ → G ⋆ G ′ establishes a t-exact equivalence QCoh id (G ⋆ G ′ ) ∼ -→ QCoh (id,id) (G × X G ′ ).
Proof. Note that the character m : G m × G m → G m given by multiplication induces a line bundle L id,id on BG m × BG m . One can prove easily that this line bundle coincides with the external product of L id , the universal line bundle on BG m , with itself. This means that what we denote by L (id,id) ∈ QCoh(G × X G ′ × BG m × BG m ) has the form L id ⊠ L id (again with the usual notations of Remark 4.1.4).

We need to prove that ρ * sends id-twisted sheaves in (id, id)-twisted sheaves. To do this, we will construct an equivalence

α(F) : (ρ * F) (id,id) ≃ ρ * (F id )
for every F in QCoh(G ⋆ G ′ ) and apply Remark 4.2.8. An easy computation shows that α is in fact a natural transformation and it verifies the condition described in Remark 4.2.8. By construction of the ⋆ product, one can prove that the following diagram

G × X G ′ × BG m × BG m G × X G ′ G ⋆ G ′ × BG m G ⋆ G ′ act (α,α ′ ) (ρ,Bm) ρ act αα ′
is commutative, where act (α,α ′ ) is the action map of G × X G ′ defined by the product bending, act αα ′ is the action map of G ⋆ G ′ and Bm is the multiplication map m : G m × G m → G m at the level of classfying stacks. This implies that

act * (α,α ′ ) (ρ * F) = (ρ, Bm) * act * αα ′ (F).
Consider now the following diagram:

G × X G ′ × BG m × BG m G × X G ′ × BG m G × X G ′ G ⋆ G ′ × BG m G ⋆ G ′ (ρ,Bm) q(α,α ′ ) (id,Bm) q (α,α ′ ) (ρ,id) ρ q αα ′
where q(α,α ′ ) , q (α,α ′ ) and q αα ′ are the projections. This is a commutative diagram and the square is a pullback.

Finally, we can compute the (id, id)-twisted part of ρ * F:

(ρ * F) id,id =q (α,α ′ ), * (act * (α,α ′ ) (ρ * F) ⊗ L ∨ id,id ) =q (α,α ′ ), * (id, Bm) * (id, Bm) * (ρ, id) * (act * αα ′ (F)) ⊗ L ∨ id,id ;
notice that since Bm * L id = L id,id we have (ρ, Bm) * L id = L id,id . Furthermore, an easy computation shows that the unit id → (id, Bm) * (id, Bm) * is in fact an isomorphism, because of the explicit description of the decomposition of the stable ∞-category of quasi-coherent sheaves over BG m . These two facts together give us that

(ρ * F) id,id =q (α,α ′ ), * (id, Bm) * (id, Bm) * (ρ, id) * (act * αα ′ (F)) ⊗ L ∨ id,id =q (α,α ′ ), * (id, Bm) * (id, Bm) * (ρ, id) * act * αα ′ (F) ⊗ L ∨ id =q (α,α ′ ), * (ρ, id) * act * αα ′ (F) ⊗ L ∨ id =ρ * q αα ′ , * act * αα ′ (F) ⊗ L ∨ id =ρ * ((F) id ).
To finish the proof, we need to verify that ρ * restricted to id-twisted sheaves is an equivalence with the category of (id, id)-twisted sheaves. Again, this can be checked étale locally, therefore we can reduce to the case G ≃ G ′ ≃ X × BG m where the morphism ρ can be identified with (id X , Bm). We know that for the trivial gerbe we have that QCoh(X) ≃ QCoh id (X × BG m ) where the map is described by F → π * F ⊗ L id , π being is the structural morphism of the (trivial) gerbe. A straightforward computation shows that, using the identification above, the morphism ρ * is the identity of QCoh(X). Finally, t-exactness follows from the fact that ρ is flat (see [AOV08, Theorem A.1]). Remark 4.2.12. One can prove easily that ρ * is an inverse of the morphism ρ * which we have just described. It is still true that send twisted sheaves to twisted sheaves and it has the same functorial property of the pullback, due to the natural adjunction. Furthermore, ρ * is t-exact because ρ is a morphism of gerbes whose image through the banding functor Band(ρ) is a surjective morphism of groups with a linearly reductive group as kernel. This implies that it is the structure morphism of a gerbe banded by a linearly reductive group, therefore ρ * is exact.

Proof of Theorem 4.2.6. First of all, we start with the stable case. We have to lift QCoh id (-) from a morphism of ∞-groupoid to a symmetric monoidal functor, i.e. to extend the action of QCoh id (-) to multilinear maps. Let {G i } i∈I be a sequence of G m -gerbes indexed by a finite set I and H be a G m -gerbe, then we can define a morphism of simplicial set QCoh id (-) :

Mul({G i } i∈I , H) -→ Mul({QCoh id (G i )} i∈I , QCoh id (H))
by the following rule: if f : i∈I G i → H is a multilinear map, we define QCoh id (f ) to be the composi-

tion f * •⊠ n : i∈I QCoh id (G i ) → QCoh id (H) where ⊠ n : i∈I QCoh id (G i ) → QCoh id ( i∈I G i )
is just the n-fold box product. The fact that this association is well-defined follows from Proposition 4.2.9 and Proposition 4.2.11. A priori, QCoh id (-) is a lax-monoidal functor from Ger Gm (X) ⋆ to LinCat St (X) × , where LinCat St (X) × is the symmetric monoidal structure on LinCat St (X) induced by the product of ∞-categories. However, f * and ⊠ n are both QCoh(X)-linear and preserves small colimits. Notice that f * preserves small colimits because it is exact, due to the fact that it is a µ-gerbe, with µ a linearly reductive group (see Remark 4.2.12). This implies that QCoh id (-) can be upgraded to a morphism of ∞-operad if we take the operadic nerve. It remains to prove that it is symmetric monoidal, i.e. to prove that it sends coCartesian morphism to coCartesian morphism. This follows again from the isomorphisms described in Proposition 4.2.11 and Proposition 4.2.9.

The prestable case can be dealt with in the exact same way. Because the equivalences in Proposition 4.2.9 and Proposition 4.2.11 are t-exact, they restrict to the prestable connective part of the ∞-categories. The fact that they remain equivalences can be checked étale locally.

Gerbes of positive trivializations

Definition 4.2.13. Let M ∈ Br(X). We define the functor

Triv ≥0 (M ) : Sch /X → S (S → X) → Equiv QCoh(S) ≥0 (QCoh(S) ≥0 , M(S))
where M is the stack of categories associated to M (see Theorem 4.1.15).

Our aim in this subsection is to prove that for every M ∈ Br(X), the functor Triv ≥0 (M ) has a natural structure of a gerbe over X, and also that the functor Triv ≥0 (-) : Br(X) → Ger Gm (X) can be promoted to a symmetric monoidal functor Br(X) ⊗ → Ger Gm (X) ⋆ . Let us recall the main result about the Brauer space proven in [START_REF] Lurie | Spectral Algebraic Geometry[END_REF] (specialized to the case of qcqs schemes). Theorem 4.2.14 ([Lur18, Theorem 11.5.7.11]). Let X be a qcqs scheme. Then for every u ∈ dBr(X) = π 0 (Br(X)), there exists an étale covering f : U → X such that f * u = 0 in dBr(U ); that is, any representative of f * u is equivalent to QCoh(U ) ≥0 as an ∞-category. Lemma 4.2.15. Let X be a qcqs scheme. Then the stack Equiv QCoh(X) ≥0 (QCoh(X) ≥0 , QCoh(X) ≥0 ) is equivalent to BG m × X.

Proof. Let S → X be a morphism of schemes. A QCoh(S)-linear autoequivalence of QCoh(S) is determined by the image of O S , and therefore amounts to the datum of a line bundle L concentrated in degree 0 (the degree must be nonnegative because we are in the connective setting, and if it were positive then the inverse functor would be given by tensoring by a negatively-graded line bundle, which is impossible).

This implies that the desired moduli space is Pic × X, which is the same as BG m × X.

Proposition 4.2.16. Let X be a qcqs scheme, and M ∈ Br(X). Then Triv ≥0 (M ) has a natural structure of G m -gerbe over X. Furthermore, given an isomorphism f : M → N of prestable invertible categories, then the morphism Triv geq0 (f ) defined by the association ϕ → f • ϕ is a morphism of G m -gerbes.

Proof. To prove that Triv ≥0 (M ) is a G m -gerbe we need to verify that both the structure map and the diagonal of Triv ≥0 (M ) are epimorphisms and to provide a G m -banding. The first two assertions are evident from the fact that Triv ≥0 (M ) is locally of the form X × BG m . Now we provide the banding in the following way. First of all, notice that G m,X can be identified with the automorphism group of the identity endofunctor of QCoh(X) ≥0 , namely O X -linear invertible natural transformations of id QCoh(X) ≥0 . Let I Triv ≥0 (M ) be the inertia stack of Triv ≥0 (M ). We define a functor α M : Triv ≥0 (M ) × G m -→ I Triv ≥0 (M )

as α M (ϕ, λ) := (ϕ, id ϕ * λ) ∈ I Triv ≥0 for every (ϕ, λ) object of Triv ≥0 (M ), * being the horizontal composition of natural transformations. Since being an isomorphism is an étale-local property, we can reduce to the trivial case, for which it is a straightforward computation. The second part of the statement follows from a straightforward computation.

Lemma 4.2.17. The functor Triv ≥0 (-) is symmetric monoidal.

Proof. To upgrade Triv ≥0 to a symmetric monoidal functor, we need to define its action on multilinear maps. Let {M i } i∈I be a sequence of invertible prestable O X -linear ∞-categories indexed by a finite set I and N be another invertible prestable O X -linear category. Then we define Triv ≥0 (-) :

Mul {M i } i∈I , N -→ Mul {Triv ≥0 (M i )} i∈I , Triv ≥0 (N )
in the following way: if f : i∈I M i → N is a morphism in LinCat PSt (X) which preserves small colimits separately in each variable, we have to define the image Triv ≥0 (f ) as a functor Triv ≥0 (f ) :

i∈I Triv ≥0 (M i ) -→ Triv ≥0 (N )
such that Band(Triv ≥0 (f )) is the n-fold multiplication of G m , where n is the cardinality of I. We define it on objects in the following way: if {ϕ i } is an object of i∈I (Triv ≥0 (M i )), then we set

Triv ≥0 (f ) {ϕ i } := f • ϕ i
where ϕ i is just the morphism induced by the universal property of the product.

First of all we need to prove that f • ϕ i is still an equivalence. Let S ∈ Sch X and f : i∈I M i → N morphism in LinCat PSt (X), we can consider the following diagram:

QCoh(S) i∈I M i (S)
i∈I M i (S) N(S)

ϕ i ⊗ϕ i u f (S) f (S) (4.2.2)
where the tensors are in fact relative tensors over QCoh(S). The diagonal map u is the morphism universal between all the QCoh(X)-linear morphisms from i∈I M i (S) which preserve small colimits separately in each variable.Equivalently, one can say that it is a coCartesian morphism in LinCat PSt (S) ⊗ .

Thus, it is enough to prove that both ⊗ϕ i and f are equivalences. Because the source of the functor Triv ≥0 (-) is the ∞-groupoid dBr(S), the morphism f (S) is an equivalence. Furthermore, the morphisms ϕ i are equivalences, therefore it follows from the functoriality of the relative tensor product that ⊗ϕ i is an equivalence. A straightforward but tedious computation shows that it is defined also at the level of 1-morphism and it is in fact a functor. It remains to prove that Band Triv ≥0 (f ) is the n-fold multiplication of G m , where n is the cardinality of I. It is equivalent to prove the commutativity of the following diagram:

i∈I Triv ≥0 (M i ) × G m n Triv ≥0 (N ) × G m i∈I I Triv ≥0 (M i ) I Triv ≥0 (N ) (Triv ≥0 (f ),m n ) α M i α N I Triv ≥0 (f )
where m n is the n-fold multiplicatoon of G m . The notation follows the one in the proof of Lemma 4.2.16. Using diagram 4.2.2 again, we can reduce to the following straightforward statement: let λ 1 , . . . , λ n be QCoh(S)-linear automorphisms of id QCoh(S) , which can be identified with elements of G m (S); then the tensor of the natural transformations coincide with the product as elements of G m , i.e λ 1 ⊗ • • • ⊗ λ n = m n (λ 1 . . . λ n ).

Finally, because we are considering ∞-groupoid, the condition of being stricly monoidal is automotically satisfied.

(2) Dans le cas G = GL n , Gr G est interprétable comme un espace de C t -treillis dans C((t)) n .

(3) Dans le cas général, c'est l'espace des G-torseurs sur le disque complexe formel Spec C t avec une trivialisation sur Spec C((t) !).

Il existe un moyen d'interpréter Perv G O (Gr G ) comme la "spécialisation en tout point de x" du c ôté "Bun G (X)" de la conjecture géométrique de Langlands, et Rep( Ǧ) comme spécialisation du c ôté "systèmes locaux".

Étant donné que la conjecture géométrique de Langlands est actuellement formulée comme une déclaration " dérivée", Bezrukavnikov-Finkelberg [BF08] et Arinkin-Gaitsgory [START_REF] Arinkin | Singular support of coherent sheaves, and the geometric Langlands conjecture[END_REF] ont prouvé la Théorème Dérivée de Satake. Là, la catégorie abélienne Perv G O (Gr G ) est remplacée par la catégorie dite catégorie sphérique de Hecke Sph(G), qui est une catégorie supérieure admettant les présentations suivantes :

- Remarque 1.7. Les dg-categories et les ∞-categories sont des moyens d'encoder l'idée de "catégories avec une notion d'homotopie et d'équivalence d'homotopie" d'une manière particulièrement utile pour traiter les catégories dérivées et l'homotopie la théorie. Dans mes travaux, j'adopte la perspective des ∞categories, qui est systématiquement exposée dans [START_REF] Lurie | Higher Topos Theory[END_REF]. L'une des formulations les plus simples de ce concept est la notion de catégories enrichies dans des espaces topologiques d'ensembles simpliciaux.

Dans ce cadre, des propriétés telles que l'associativité ou la commutativité des structures monoïdales sur les catégories (∞) sont encodées dans certaines "structures de cohérence", qui permettent également des notions intermédiaires d'"associativité plus élevée", appelées E k -structures algébriques, k ∈ N. Par exemple, un loopspace ΩY supporte un produit qui n'est associatif qu'à homotopie près, dans la mesure o ù les associateurs dépendent des paramétrisations de l'intervalle unitaire. C'est ce qu'on appelle une E 1 -algèbre dans les espaces. Cette idée se généralise en Ω k Y, k ≥ 2 : il existe k lois de produit différentes, associatives à homotopie près, et distributives les unes par rapport aux autres, qui coïncident à homotopie près par le théorème d'Eckmann-Hilton. De la même manière, on peut définir des E k -algèbres en ∞-catégories (appelées ∞-catégories E k -monoïdales). Pour les 1-catégories, la situation est très simple. Une E 1 -algèbre en catégories n'est qu'une catégorie monoïdale. Une E 2 -algèbre en catégories est ce qu'on appelle une catégorie monoïdale tressée. Une catégorie ordinaire E k -monoïdale, pour k ≥ 3, n'est qu'une catégorie monoïdale symétrique. Par exemple, Perv G O (Gr G ) avec le produit de convolution est une catégorie E k -monoïdale pour chaque k ≥ 3. Dans le cas des ∞-catégories générales, au contraire, une E k+1 -algèbre dans les ∞-catégories est une notion strictement plus forte qu'une E k -algèbre. En ce sens, mon résultat sur Sph(G) dans [START_REF] Nocera | A model for E3 the fusion-convolution product of constructible sheaves on the affine Grassmannian[END_REF] est l'analogue correct dans le monde dérivé de la commutativité de sur Perv G O (Gr G ) .

Les deux théorèmes Theorème 1.5 et Theorème 1.6 ont été initialement énoncés par Gaitsgory et Lurie (non publié). La seconde découle des résultats de Bezrukavnikov-Finkelberg [BF08], Gaitsgory-Lurie (non publié) et Nadler, et elle implique la première. Dans [START_REF] Nocera | A model for E3 the fusion-convolution product of constructible sheaves on the affine Grassmannian[END_REF], j'ai prouvé Theorème 1.5 indépendamment, en construisant la structure E 3 -monoïdale recherchée de manière intrinsèque. Ceci est dans le même esprit de la reconstruction tannakienne expliquée plus haut pour le cas des faisceaux pervers, o ù l'existence d'une structure monoïdale symétrique sur la catégorie fait partie de la donnée initiale, et ce n'est qu'a posteriori qu'elle est interprétée comme le produit tensoriel naturel dans une catégorie de représentations.

Ma preuve de Theorème 1.5 utilise des outils de la théorie de l'homotopie et de la géométrie algébrique dérivée, en particulier les travaux de Jacob Lurie dans [Lur17] et de Gaitsgory-Rozenblyum dans [START_REF] Gaitsgory | A Study in Derived Algebraic Geometry[END_REF], afin de fournir l'interaction correcte entre la géométrie objets (par exemple la Grassmannienne affine) et des invariants hautement structurés sur eux (par exemple la catégorie des faisceaux constructibles).

FAISCEAUX CONSTRUCTIBLES ET ESPACES CONIQUEMENT LISSES

La nature de ce projet est beaucoup plus topologique. Dans [START_REF] Nocera | A model for E3 the fusion-convolution product of constructible sheaves on the affine Grassmannian[END_REF], je me suis intéressé à comprendre la ∞-catégorie des faisceaux constructibles équivariants sur la Grassmannienne affine par rapport à la stratification dans les cellules de Schubert. En ignorant un instant la structure équivariante, ce sont des faisceaux à valeurs spatiales particulières sur l'espace topologique analytique complexe sous-jacent de Gr G (noté Gr an G ) qui sont localement constants sur les strates de S. 1 La théorie des faisceaux constructibles sur les espaces topologiques et les variétés algébriques a été développée par exemple dans [GM80, Tre09, Lur17, BGH20]. MacPherson a prouvé que sous certaines hypothèses la donnée d'un faisceau constructible sur un espace stratifié (Y, S) est équivalente à une donnée combinatoire, à savoir une représentation de la "catégorie des chemins de sortie" de (Y, mathscrS). Lurie [Lur17, Appendix A] a étendu cette remarque à des hypothèses très générales et à un cadre dérivé, définissant une ∞-category of exit paths Exit(Y, S) et établissant le théorème suivant : De plus, il a un inverse qui est à nouveau monoïdal symétrique, envoyant une ∞-catégorie ∞-linéaire inversible M à sa champ de trivialisations sur X.

LISTE DES PUBLICATIONS

-A model for the E 3 fusion-convolution product of constructible sheaves on the affine Grassmannian (preprint disponible sur https://arxiv.org/abs/2012.08504) -Whitney stratifications are conically smooth (avec Marco Volpe, preprint disponible sur https:// arxiv.org/abs/2105.09243)

( 1 .

 1 2.2)whereD c (X; k) = colim S stratification of X Cons(X, S ; k) and D c,H (X; k) = lim . . . D c (H × X; k) D c (X; k) .Now, the horizontal arrows in (1.2.2) are not equivalences, although they are while restricted to the abelian subcategories of perverse sheaves. Indeed, the forgetful functor Perv G O (Gr, S ; k) → Perv(Gr, S ; k) is an equivalence (see for example [BR18, Section 4.4]), but Cons G O (Gr, S ; k) → Cons(Gr, S ; k)

Remark 2 . 1 . 4 .

 214 Consider the twisted tensor product defined in[START_REF] Zhu | An introduction to affine Grassmannians and to the geometric Satake equivalence[END_REF] (3.1.10)]

Proposition 2 . 1 . 7 .

 217 For any k ≥ 0 the functor Gr Ran,k : Alg C → Grpd factorises through the inclusion Set → Grpd. Proof. See Section 2.B.1.

Proposition 2 . 1 . 11 .

 2111 This construction defines a semisimplicial object Gr Ran,• : ∆ op inj → Fun(Alg C , Set) because the given maps satisfy the simplicial identities.

Proposition 2 . 1 . 12 .

 2112 For any R ∈ Alg C , the semisimplicial set Gr Ran,• (R) enjoys the 2-Segal property, that is the equivalent conditions of [DK19, Proposition 2.3.2].

Remark 2 . 1 . 16 .

 2116 The simplicial object Arc • enjoys the 2-Segal property. The verification is straightforward thanks to the multiplication structure.

Construction 2 . 2 . 3 .

 223 Composition with the map that we have just described yields a functor which we call again strtop : (StrPSh C ) /Ran(X) → PSh(StrTop) /Ran(M ) .

Remark 2 .

 2 2.7. The two functors FactGr k and FactArc k are hypercomplete cosheaves with values in PSh(StrTop /Ran(M ) ).

  FactGr ⊙ k Remark 2.2.11. Consider two independent open subsets U and V of Ran(M ). We have the following diagram

Conjecture 2 . 2 . 20 .

 2220 (f ) with respect to this topology is the same of a classical trivialising open covering. Any open embedding is a local homeomorphism. Conversely, let us suppose that we have a trivialising local homeomorphism, that is a local homeomorphism of topological spaces w : W → top(T ) such that top(S) × top(T ) W is isomorphic to top(G) × W . Now, by definition of local homeomorphism, for every x in the image of w we have that w restricts to an open embedding on some open set U ⊂ W whose image contains x. Moreover,U → W → top(T ) is trivialising a fortiori, that is U × top(T ) top(S) ≃ U × top(G).In conclusion, if we have a jointly surjective family of trivialising local homeomorphisms, the above procedure yields a covering family of trivialising open sets. In the stratified setting, the proof is the same, since the relevant maps are étale-stratified on the algebraic side, and therefore become analytic-stratified on the topological side.Recall the definition of conically smooth space from [AFT17, Section 3] (see Section 3.1.2 for a quick review) and the definition of weakly constructible bundle and constructible bundle from [AFT17, Definition 3.6.1].Lemma 2.2.19. The map Gr Ran → Ran(M ) is a weakly constructible bundle.Proof. Fix a stratum Ran n (M ) in Ran(M ). The restriction of the map Gr Ran → Ran(M ) to that stratum is locally trivial with stratified fiber Gr n , since by the factorization property Gr Ran | Rann(M ) ≃(Gr strtop X ) ×n | (X n ) top disj /Sn, and Gr X ≃ Gr × Aut + (D) X, which is étale-locally trivial over X with fiber Gr. The map Gr Ran → Ran(M ) is a constructible bundle. Conjecture 2.2.21. Let f : (Y, s, P ) → (Y ′ , s ′ , P ′ ) be a constructible bundle of conically smooth spaces. Then stratified homotopies can be lifted along f . Forthcoming work with Mauro Porta, Jean-Baptiste Teyssier and Marco Volpe is devoted to the proof of these two conjectures. Proposition 2.2.22. For every nonempty finite collection of disjoint disks D 1 , . . . , D n ⊆ M containing open subdisks E 1 ⊆ D 1 , . . . , E n ⊆ D n , the maps

  Gr Ran(D) and therefore at the upper level for Gr Ran(E),k → Gr Ran(D),k . But this is true by Lemma 2.2.19. If D is a disk around x in M , we can define Gr D,1 as the preimage along r 1 : Gr Ran → Ran(M ) of the (not open) subset D → Ran(D) ⊂ Ran(M ).Corollary 2.2.23. The natural injective map Gr x,1 → Gr D,1 is a stratified homotopy equivalence over (Ran(X • (T ) + )) k . Notation 2.2.24. From now on, we will denote the relation "being stratified homotopy equivalent" by she ∼ and "being homotopy equivalent as topological groups" by ghe ∼ . We can express this property in a more suitable way by considering the ∞-categorical localization PSh(Act con (StrTop /Ran(M ) )[esh -1 ]) × see Definition 2.A.21. Since she is closed under the symmetric monoidal structure of PSh(StrTop) × , the localization functor extends to a map of operads (Act con PSh(StrTop)) × → Act con PSh(StrTop)[esh -1 ] × and therefore to Corr(Act con PSh(StrTop)) × → Corr(Act con PSh(StrTop)[esh -1 ]) × .

Remark 2 . 2 .

 22 28. By [Lur17, Example 5.4.5.3], a nonunital E M -algebra object A ⊗ in a symmetric monoidal ∞-category C induces a nonunital E n -algebra object in C, where n is the real dimension of the topological manifold M , by taking the stalk at a point x ∈ M . More precisely, there is an object in Alg En (C) whose underlying object is lim {x}∈U ∈Open(Ran(M )) A(U ), which coincides with lim x∈D∈Disk(M ) A(Ran(D)), since the family of Ran spaces of disks around x is final in the family of open neighbourhoods of {x} inside Ran(M ). Now each A ⊗ | Fact(D) ⊗ induces a nonunital E D -algebra by Lurie's theorem [Lur17, p. 5.5.4.10]. But [Lur17, Example 5.4.5.3] tells us that E D -algebras are equivalent to E n algebras. Also, by local constancy (i.e. constructibility), the functor D → A(D) is constant over the family x ∈ D ∈ Disk(M ), and therefore the stalk A x coincides with any of those E n -algebras. This also implies that all stalks at points of M are (noncanonically) isomorphic. 4 Also, the content of [Lur17, Subsection 5.5.4] tells us how the E n -multiplication structure works concretely. Choose a disk D containing x. We interpret this as the only object in the ⟨1⟩-fiber of E D . Recall that a morphism in E D lying over the map ⟨2⟩ → ⟨1⟩ 1, 2 → 1 is the choice of an embedding D ⨿ D → D. Call n D the unique object lying over ⟨n⟩ in E D . Consider the canonical map E D → E M . If A M is the E nu M -algebra object appearing in the conclusion of Lurie's theorem, call A D its restriction to E D . Recall from the proof of Lurie's theorem that A M is obtained by operadic left Kan extension of the restriction A| Disk(M ) along the functor Disk(M ) ⊗ → E M . Then we have that Lemma 2.2.29. A D (1 D ) = A(Ran(D)), and A D (2 D ) = colim{A(Ran(E)) ⊗ A(Ran(F )) | E, F ∈ Disk(M ),

Proposition 2 . 3 . 3 .

 233 Let x ∈ M be a point. Consider the functor A ⊗ x : E nu 2 × E nu 1 → Pr L,⊗ k . Then this can be upgraded to a map of operadsE 2 × E nu 1 → Pr L,⊗ k .Proof. We can apply [Lur17, Theorem 5.4.4.5], whose hypothesis is satisfied since for any ∞-category C the functor C × → Fin * is a coCartesian fibration of ∞-operads ([Lur17, Proposition 2.4.1.5]): therefore, it suffices to exhibit a quasi-unit for any A ⊗ (-, ⟨k⟩), functorial in ⟨k⟩ ∈ E nu 1 . We can consider the map (natural in k) u k : Spec C → Gr x,k represented by the sequence (T G , id| X\{x} , id| Xx , . . . , T G , id X\{x} ) ∈ Gr x,k . Note now that this induces a map of spaces * → Gr x,k .

1 -→

 1 Cons k,G O(Gr). Here 1 is the pushforward along the trivial section t : * → Gr, t( * ) = (T G , id| X\x ), of the constant sheaf with value k. The proof is given in [Rei12, Proposition IV.3.5]. We denote by ⋆ the E 1 -product of equivariant constructible sheaves on Gr described by A x (-). By Remark 2.3.4 for any F ∈ Cons k,G O (Gr) we can compute the product via the convolution diagram

Definition 2 .A. 1 .

 21 Let Top be the 1-category of topological spaces. The category of stratified topological spaces is defined asStrTop C = Fun(∆ 1 , Top) × Top Poset,where the map Fun(∆ 1 , Top) → Top is the evaluation at 1, and Alex : Poset → Top assigns to each poset P its underlying set with the so-called Alexandrov topology (see [BGH20, Definition 1.1.1]).

Construction 2 .A. 3 .Pullbacks of stratified spaces Lemma 2 .A. 4 .

 2324 We can define the category StrPSh C as PSh(StrAff C ). Note that StrTop is cocomplete, because Top, Fun(∆ 1 , Top) and Poset are. By left Kan extension we have a functor strtop : StrPSh C → StrTop. (2.A.1) By construction, this functor preserves small colimits and finite limits (since bothan : Aff C → Stn C and | -| : Stn C → Top preserve finite limits, see [Rey71]). The forgetful functor StrTop → Top preserves and reflects finite limits.

Lemma 2 .

 2 A.10 ([Lur17, Remark 4.8.1.8 and Proposition 4.8.1.15]). There exists a symmetric monoidal functor P (!) : Cat × ∞ → Pr L,⊗ sending an ∞-category C to the ∞-category of S-valued presheaves P(C), a functor F : C → D to the functor P(D) → P(C) given by F ! = Lan F (-).

Theorem 2 .:

 2 A.14. There is an ∞-functor D corr c Corr(Top) → Cat ∞ that coincides with D c when restricted to Top op → Corr(Top).

Step 1 :

 1 (f * , f * ) adjunction and proper base change. The previous discussion tells us that D op c : Top → Cat 2-cat,op ∞ satisfies the left Beck-Chevalley condition [GR17, Chapter 7, 3.1.5] with respect to adm = vert = all, horiz = proper, taking Φ = D c : StrTop op → Cat ∞ . Indeed, for any f ∈ horiz we set Φ ! (f ) = f * , which is right adjoint to f * = Φ(f ). Finally, the base change property for the diagram (2.A.2) completes the proof of the left Beck-Chevalley property. Therefore by [GR17, Chapter 7, Theorem 3.2.2.(a)] the functor Cons op : StrTop → Cat

.

  Every map in adm ∩ co-adm = open ∩ proper should be a monomorphism. But the class open ∩ proper consists of embeddings of unions of connected components. • For any α : X → Y in vert = all, consider the ordinary category Factor(α), whose objects are X X Y, ε γ where ε ∈ adm = open and γ ∈ co-adm = proper, and whose morphisms are commutative diagrams X Consider the (∞, 1)-category N(Factor(α)). We require N(Factor(α)) to be contractible for any α ∈ vert. But this is exactly Nagata's compactification theorem, see [GR17, Chapter 5, Proposition 2.1.6]. • D c | Top should satisfy the right Beck-Chevalley condition with respect to adm = open ⊂ horiz = all. This is true, because for every f ∈ open we have that f * = f ! : now this admits a left adjoint f ! , and the Base Change Theorem holds.

  Kan extension of D c along the Yoneda embedding we obtain a functor P(Top) op → Cat ∞,k . By using the same arguments as in the proof of [GR17, Chapter 5, Theorem 3.4.3], Theorem 2.A.14 provides an extension to Corr(P(Top)). To replace P(Top) with the category of sheaves we use the descent properties of the functor D c (-). We call D c,corr the extension Corr(Sh(Top)) → Cat ∞ . Let Cons act,corr = D c,corr •q corr : Corr(Act con (StrTop)) → Cat ∞,k . Note that if we restrict the functor we just obtained to Act con (Top) → Corr(Act con (Top)) vert , then coincides with the composition Act con (Top) → StrTop con Cons (!) ----→ Pr L . Now: • this proves that Cons act,corr takes values in Pr L k , since it does on objects and on vertical morphism, and for what concerns horizontal morphisms we have that any f * is a left adjoint; • the functor Act con (Top) → StrTop con sends the class esh to stratified homotopy equivalences, and the by the Exodromy Theorem the functor StrTop con Cons (!) ----→ Pr L k sends stratified homotopy equivalences to equivalences of ∞-categories. Therefore, D c,corr sends equivariant stratified homotopy equivalences to equivalences as well, and therefore it factors through the localization; • the functor Act con (Top) → StrTop con is Cartesian symmetric monoidal, and by Corollary 2.A.11 the functor StrTop con Cons (!)

  • ([Mat89, page 181]) If A → B is a flat local homomorphism of local noetherian rings, then depth B = depth A + depth B/mB, where m is the maximal ideal of A.

  as in the hypothesis of Whitney's Condition A, Whitney's Condition A is satisfied, and the same for Whitney's condition B. In other words, the space Sing(X, Y ) = {y ∈ X ∩ Y | y does not satisfy either Whitney's

3 . 2

 32 subsets of manifolds (Thom, Mather) . . . . . . . 77 3.1.2 Conical and conically smooth stratifications (Lurie, Ayala-Francis-Tanaka) . 78 Whitney stratifications are conically smooth . . . . . . . . . . . . . . . . . . . 81 3.2.1 Whitney stratifications are conical . . . . . . . . . . . . . . . . . . . . . . 81 3.2.2 Whitney stratifications are conically smooth . . . . . . . . . . . . . . . . . 86

Definition 3 . 1 . 4 (

 314 Whitney's condition B). Let X, Y be smooth submanifolds of a smooth n-dimensional manifold M , and y ∈ Y . The pair (X, Y ) is said to satisfy Whitney's Condition B at y if there exist a chart of M ϕ : U → R n around y such that (ϕ(U ∩ X), ϕ(U ∩ Y )) satisfies Whitney's Condition B at ϕ(y). Definition 3.1.5 (Whitney stratification). Let M be a smooth manifold of dimension n. A smooth stratification (Z, S ) on a subset Z of M is said to satisfy the Whitney conditions if • (local finiteness) each point has a neighbourhood intersecting only a finite number of strata;

Note 3 . 1 . 1 . 3 . 1 .

 31131 10. A very useful notion relative to stratified spaces (see for example from [AFT17, Definition 2.4.4]) is the notion of depth of a stratified space at a point. For example, let Z be an unstratified space of Lebesgue covering dimension n. Then the depth of the cone C(Z) at the cone point is n + Definition 11 ([Lur17, Definition A.5.5]). Let (X, P, s) be a stratified space, p ∈ P , and x ∈ X p . Let P >p = {q ∈ P | q > p}. A conical chart at x is the datum of a stratified space (Z, P >p , t), an unstratified space Y , and a P -stratified open embedding Y × C(Z) X P whose image hits x. Here the stratification of Y × C(Z) is induced by the stratification of C(Z), namely by the maps Y × C(Z) → C(Z) → P ≥p → P (see Definition 3.1.9).

Figure 3 . 1 :

 31 Figure 3.1: Tubular neighbourhoods in (R 2 , (X, Y, Z)).

  2.10]. Theorem 3.2.7 (Main Theorem). If (M, S ) is a Whitney stratified space, then the Thom-Mather charts exhibit a conically smooth structure on (M, S ).

  2.6, a Thom-Mather chart is in particular an open embedding of the form D i × C(Z) → M where Z has a Whitney stratification S ′ and depth(Z, S ′ ) < depth(M, S ); thus, by the inductive hypothesis, Z is conically smooth and this implies that the Thom-Mather chart is a basic in the sense of [AFT17, Definition 3.2.4].

m

  ∈ Y and v is a Thom Mather chart for a neighbourhood of M in Y , such that v -1 (m) = (0, * ) ( * is the cone point). Consider v -1 (U ∩ Y ) as an open subset of D j × * .This open subset contains some closed ball of radius δ and dimension j centered at 0; denote it by B δ × * . Also, let ρ Y : V → R >0 be the "distance from Y " function associated to the Thom-Mather chart v, and let γ be a positive continuous function on Y (defined at least locally around m) such that there is an inclusion

Proposition 4 . 1 . 2 .

 412 2.1); let T Y be the A-tubular neighbourhood of Y . Now by [Mat70, Proposition 6.1] there is an isotopy between T ′ Y and T Y fixing Y . By pulling back ψ ′ Y to T Y along this isotopy, we obtain an A-Thom Mather chart ψ Y around y; we are now left with two A-charts ϕ X and ψ Y and we finally can apply the fact that A is an atlas. Let now F ∈ QCoh(G). Let I G be the inertia group stack of G over X. Thus F is endowed with a canonical right action by I G , called the inertial action. For an explicit definition, see [BS19, Section 3]. Let G be a G m -gerbe over a qcqs scheme X. The pullback functor QCoh(X) → QCoh(G) establishes an equivalence between QCoh(X) and the full subcategory of QCoh(G) spanned by those sheaves on which the inertial action is trivial. Note that the banding G m × G → I G induces a right action ρ of G m on any sheaf F ∈ QCoh(G), by composing the banding with the inertial action. On the other hand, for any character χ : G m → G m , G m acts on F on the left by scalar multiplication precomposed with χ. Let us call this latter action σ χ . Remark 4.1.3. Let 1 be the trivial character of G m . Proposition 4.1.2 can be restated as: the pullback functor QCoh(X) → QCoh(G) induces an equivalence between QCoh(X) and the full subcategory of QCoh(G) where ρ = σ 1 .

Definition 4 . 1 . 5 .

 415 In the case when χ is the identity character id : G m → G m , QCoh χ (G) is usually called the category of G-twisted sheaves on X.

Remark 4 . 2 . 10 .

 4210 Using the associativity of the box product ⊠, the same proof works in the case of a finite number of G m -gerbes. Proposition 4.2.11. Let G ⋆ G ′ be the G m -gerbe defined in Construction 4.2.1. Then the pullback along

  en tant que dg-ou ∞-category D c,G O (Gr G , k) de G O -réas constructibles équivariants sur Gr G ; -en tant que dg-ou ∞-category D-mod G O (Gr G , k) de G O -équivariants D-modules sur Gr G . J'ai prouvé le premier des deux théorèmes suivants : Theorème 1.5 ([Noc20]). La ∞-category Sph(G) admet un E 3 -structure monoïdale étendant le produit de convolution mono dale symétrique des faisceaux pervers. Theorème 1.6. La ∞-category Sph(G) est équivalente au E 2 -center de la ∞-categorie dérivée des représentations DRep( Ǧ, k).

Theorème 2 . 1 ( 2 . 2 .

 2122 Théorème d'exodromie, [Lur17, Théorème A.9.3]). Soit (Y, S) un espace coniquement stratifié localement de forme singulière. Alors la ∞-catégorie des faisceaux constructibles à valeurs spatiales sur (Y, S) est équivalente à Fun(Exit(Y), Spaces).L'hypothèse d'être "localement de forme singulière" est assez générale, et est satisfaite par la Grassmannienne affine. L'hypothèse que la stratification est conique dit essentiellement que chaque point a un voisinage o ù la stratification ressemble à R n × C(Z), C(Z) étant le c ône topologique d'un espace stratifié Z. On peut interpréter cette condition comme un analogue stratifié de "être une variété topologique".Or, la stratification en cellules de Schubert du Grassmannien affine satisfait les conditions de Whitney, et il est bien connu que Lemma Toute stratification satisfaisant les conditions de Whitney est conique.Cela m'a permis d'appliquer Theorème 2.1 dans [Noc20] et de compléter la preuve de l'existence de la E 3 -structure recherchée sur la catégorie sphérique.Ayala, Francis et Tanaka[AFT17] ont introduit la notion de structure coniquement lisse sur un espace stratifié, qui est interprétable comme un analogue stratifié d'une structure différentiable. Dans[NV21], j'ai prouvé avec Marco Volpe (Ratisbonne) la conjecture suivante contenue dans leur article[AFT17], affinant ainsi Lemma 2.2. Theorème 2.3 (N.-Volpe 2021, [NV21]). Tout espace stratifié satisfaisant les conditions de Whitney [Whi64, Whi65] admet une structure conique lisse. Notre résultat fournit une large classe d'exemples de structures coniquement lisses. En particulier, cela implique que la Grassmannienne affine Gr G admet une structure canonique coniquement lisse.

3 .Conjecture 3 . 1 .

 331 ALG ÈBRES D'AZUMAYA D ÉRIV ÉES ET GERBESLe groupe de Brauer d'un champ a été largement étudié depuis son introduction par Serre et Grothendieck. Dans les mêmes articles, la théorie générale des groupes de Brauer de variétés algébriques est développée, généralisant ainsi le cas des corps : c'est le groupe des algèbres d'Azumaya sur X, à une relation d'équivalence appelée Morita équivalence. Giraud[START_REF] Giraud | Cohomologie non abélienne[END_REF] a montré comment pour toute variété 1. Dans notre cas, nous voulons des faisceaux évalués dans la ∞-catégorie dérivée des Λ-modules pour un anneau de coefficients Λ, mais cela n'est pas pertinent dans la présente section. algébrique S, H 2 (S, G m ) peut être interprété comme un ensemble de G m -gerbes sur S. Il existe une carte des groupes Br(S) → H 2 (S, G m ) impliquant que toute algèbre d'Azumaya sur S donne lieu à un G m -gerbe sur S. Cette application est injective, mais elle prend des valeurs dans le sous-groupe de torsion de H 2 (S, G m ), qui peut en général être un sous-groupe approprié pour les variétés non régulières. En utilisant la théorie des dg/∞-catégories stables présentables, Toën [Toë12] a défini une notion de algèbre d'Azumaya dérivée sur une variété algébrique S et une notion correspondante de Équivalence de Morita étendant l'équivalence classique. Il montre que le groupe dBr(S) des algèbres d'Azumaya dérivées jusqu'à l'équivalence de Morita est isomorphe à H 2 (S, G m ) × H 1 (S, Z) o ù le deuxième facteur est 0 si S est normal. Il y a eu des travaux ultérieurs dans le sens de rendre plus explicite la relation entre G m -gerbes et les algèbres d'Azumaya dérivées. [BP21, Théorème 5.19], suite aux travaux de [Lie04] et [BS19] respectivement au niveau des catégories abéliennes et triangulées, dit que si G est un G m -gerbe sur une base S, alors il y a une action du champ d'inertie I(G/S), induisant une notion de χ-faisceau quasicohérent homogène sur G, pour χ un caractère de G m . Elle dit aussi que la ∞-catégorie QCoh(G) des faisceaux quasi-cohérents se décompose en une somme directe des sous-catégories QCoh χ (G) des faisceaux homogènes. Partant de là, une explicitation de la connexion entre G m -gerbes sur S et les algèbres d'Azumaya dérivées semble possible, selon les lignes suivantes. Mon intérêt pour la théorie des G m -gerbes (et donc des algèbres d'Azumaya) est né du fait que de puissantes modifications de la Grassmannienne affine usuelle Gr G , liée à la dualité conjecturale de Langlands pour les surfaces , sont exprimés en termes de gerbes. Dans l'ouvrage [BP21] apparaît la conjecture suivante : Si Y → S est un G m -gerbe, alors QCoh α (S) ⊗ QCoh β (S) QCoh α+β (S) comme QCoh(S)-linear ∞-categories présentables, pour tout α, β ∈ H 2 (S, G m ).

Corollaire 3 . 2 .Theorème 3 . 3 .

 3233 En particulier, la ∞-category QCoh α (X) est inversible dans Pr L,c.g. X , d'o ù Morita équivalent à une algèbre d'Azumaya dérivée par le travail de Toën. Michele Pernice (Scuola Normale Superiore) et moi avons prouvé cette conjecture. Plus précisément, le théorème principal que nous démontrons est le suivant : Soit S un schéma quasi-séparé quasicompact. Le foncteur Ger G m (S) → {objets inversibles dans Pr L,c.g. X } est monoïdal symétrique par rapport à : -le produit rigidifié de G m -gerbes sur la source (cfr. [BP21]) ; -le produit tenseur de Lurie sur le codomaine.

1.19. Chapter 2 E 3 -monoidal structure on the spherical Hecke category Contents 2.1 Convolution over the Ran space . . . . . . . . . . . . . . . . . . . . . . . . . .

	2.1.1	The presheaves Gr Ran,k . . . . . . . . . . . . . . . . . . . . . . . . . . . .
	2.1.2	The 2-Segal structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
	2.1.3	Action of the arc group in the Ran setting . . . . . . . . . . . . . . . . . .
	2.1.4 The convolution product over Ran(X) . . . . . . . . . . . . . . . . . . .
	2.1.5	Stratifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2 Fusion over the Ran space . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

	2.2.1	Analytification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
	2.2.2 Fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
	2.2.3 The factorizing property . . . . . . . . . . . . . . . . . . . . . . . . . . .
	2.2.4 Local constancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
	2.2.5	Interaction of convolution and fusion over Ran(M ) . . . . . . . . . . . . .
	2.2.6 The stalk of the factorising cosheaf . . . . . . . . . . . . . . . . . . . . . .
	2.

3 Product of constructible sheaves . . . . . . . . . . . . . . . . . . . . . . . . .

	2.3.1	Taking constructible sheaves . . . . . . . . . . . . . . . . . . . . . . . . .
	2.3.2 Units and the main theorem . . . . . . . . . . . . . . . . . . . . . . . . .
	2.

A Constructible sheaves on stratified spaces: theoretical complements . . . . . .

	2.A.1 Stratified schemes and stratified analytic spaces . . . . . . . . . . . . . . . . 59
	2.A.2 Symmetric monoidal structures on the constructible sheaves functor . . . . 61
	2.A.3 Constructible sheaves and correspondences . . . . . . . . . . . . . . . . . 65
	2.B

Omitted proofs and details . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

	2.B.1 Proof of Proposition 2.1.7 . . . . . . . . . . . . . . . . . . . . . . . . . . 70
	2.B.2 Proof of Proposition 2.2.12 . . . . . . . . . . . . . . . . . . . . . . . . . . 72
	2.B.3 Proof of Proposition 2.2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . 74

  1, we give a version of Gr • living over Ran(X). Formally, what we obtain is a semisimplicial 2-Segal (stratified) prestack Gr Ran,• over Ran(X). This is done in order to take into account the factorization structure of Gr Ran and give the setup of the extension of the fusion product to the setting of constructible sheaves. The same is done for G O , thus defining a "global" object G O,Ran ; all the preceding constructions are G O,Ran -equivariant.

of the arc group Construction

2.1.14. Consider the functor G O,Ran : Alg C → Set from Definition 2.1.5. It is immediate to see that this functor takes values in Set just like Gr Ran,k , and admits a map towards Ran(X).

  Ran, Φ k,k on Gr Ran,k , Ξ 1,k on ConvGr k and Φ 1,1 on Gr Ran are compatible with the k-associative convolution diagram

		Gr Ran,k	q	ConvGr k
		p		m	(2.1.1)
	Gr	× Ran(X) k Ran		Gr Ran

× Ran(X) k Ran,1 on Gr × Ran(X) k Ran . The actions Φ ×k 1,1 on Gr × Ran(X) k

  Remark 2.1.28. Both (Gr Ran,• , σ • ) and (Arc Ran,• ) (unstratified) enjoy the 2-Segal property.Proof. We want to use the unstratified version of the same result, proved in Section 2.1.2. In order to do this, it suffices to prove that the functor StrPSh C → PSh C preserves and reflects finite limits. We can reduce this statement to the one that StrSch C → Sch C does. By definition, this follows from Lemma 2.A.4.

  Construction 2.A.3). Recall that this functor preserves finite limits.Hence, if we precompose strtop with Gr Ran,• : ∆ op inj → StrPSh C we obtain a 2-Segal semisimplicial object in stratified spaces. Also, since strtop sends stratified étale coverings to stratified coverings in the topology of local homeomorphisms, it extends to a functor between the categories of sheaves. For simplicity, we set Gr Ran = strtop(Gr Ran ), Gr Ran,k = strtop(Gr Ran,k ).

	Notation 2.2.1.

  Ran,• , σ • ) → Ran(M ), which we consider as an object of 2-Seg ss (PSh(StrTop) /Ran(M ) ) (we abuse of notation by denoting the cardinality stratification again by κ). An important remark: with the notations of Section 2.A.3, the functor ActGr • takes values in Act con (PSh(StrTop) /Ran(M ) ). Every Gr k is a colimit of objects belonging to StrTop con ⊂ StrTop.

	Analogously, the functor ActGr Ran,• : ∆ op ∆ inj → Act(PSh(StrTop) /Ran(M ) ).	op inj → Act((StrPSh C ) /Ran(X) ) induces a functor ActGr • :
	Remark 2.2.4. This is the combination of Lemma 2.1.27 and the following result:
	Proposition 2.2.5. Proof. See Section 2.B.3.

  V,1 for independent open sets U, V ⊂ Ran(M ), and consider the map Gr U,1 × Gr V,1 → Gr U ⋆V,1 induced by Remark 2.2.11. Ran(X) and ⊙. Indeed, there are mapsGr Ran,1 × Ran(M ) Gr Ran,1 → Gr Ran,1 × Gr Ran,1and(Gr Ran,1 × Gr Ran,1 ) disj → Gr Ran,1 × Gr Ran,1which can be encoded as correpondences from Gr Ran,1 ×Gr Ran,1 to Gr Ran,1 × Ran(M ) Gr Ran,1 and (Gr Ran,1 × Gr Ran,1 ) disj respectively. Note that the context of correspondences here is very useful to encode this "restriction" procedure.

	Formally, these "restrictions" are obtained by forgetting both structures to StrTop. Indeed, the forgetful
	functor StrTop /Ran(M ) → StrTop induces a functor

Corr(PSh(StrTop) /Ran(M ) ) → Corr(PSh(StrTop)) which is lax monoidal with respect to both × By this argument, the functor obtained from

ActGr(-, -) ⊙,× : Fact(M ) ⊗ × E nu 1 → Corr(PSh(Act con StrTop /Ran(M ) )) ⊙,×

by composition with Corr(PSh(Act con StrTop /Ran(M ) )) ⊙,× → Corr(PSh(Act con StrTop)) × is a lax monoidal functor in both variables ActGr(-, -) × : Fact(M ) ⊗ × E nu 1 → Corr(PSh(Act con StrTop)) × .

Nagata compactification and proper pushforward Consider

  which is equivalent to the more familiar Corr(StrTop) isom proper;all (horizontal and vertical maps are interchanged while considering the opposite of the correspondence category). We have an (∞, 1)-functor , because we are performing a kind of construction "dual" to that used for IndCoh in[START_REF] Gaitsgory | A Study in Derived Algebraic Geometry[END_REF]).

	2-cat,op ∞ Step 2: We want to apply [GR17, Chapter 7, Theorem 5.2.4] and extend our functor to Corr(StrTop) open extends to a functor (Cons op ) all;proper : Corr(StrTop) proper all;proper → Cat 2-cat,op ∞ Cons proper,all : Corr(StrTop) isom proper,all → Cat ∞ . (2.A.3) all,all ⊃ Corr(StrTop) isom

, or equivalently a functor Cons all;proper : (Corr(StrTop) proper, all;proper ) op → Cat 2-cat ∞ . Now we restrict this functor to the (∞, 1)-category (Corr(StrTop op ) isom all;proper ) op , the classes of morphisms in StrTop given by horiz = all, vert = all, adm = open ⊂ horiz, co-adm = proper∩ ⊂ vert (note that we need adm ⊂ horiz and co-adm ⊂ vert instead of the converse

  All algebraic actions with finitely many orbits induce a Whitney stratification by orbits, and hence their analytic counterpart lies in Act con (Top). Also, by[use], the analytification of algebraic varieties is locally of singular shape. Formally, this means that the functor strtop induces a functor ActStrSch C → Act con StrTop. Act con (StrTop) → StrTop con (which remembers only the stratification in H-orbits) • Act con (StrTop) → Sh loc (Top) sending (Y, H, Φ) to the quotient Y /H defined as the colimit of the usual diagram in the category of sheaves. Note that here the stratification is forgotten. All the preceding constructions can be extended to presheaves, by replacing StrSch C by PSh(StrSch C ) and StrTop by PSh(StrTop).

	Remark 2.A.20. There exist functors
	•
	2.A.14: the application of [GR17, Chapter 7, Theorem 5.2.4]
	provides us with an (∞, 2)-functor from
	Corr(Top) open all,all

to Cat

2-cat

∞ which we restrict to an (∞, 1)-functor from Corr(Top) isom all,all to Cat ∞ .

Equivariant constructible sheaves

Definition 2.A.17. Let ActStrSch C be the category with the following objects {H group scheme over C, (Y, P, s) stratified varieties of finite type over C, Φ action of H over Y, such that P is finite and the strata of (Y, P, s) coincide with the orbits of Φ} and the following morphisms

{f : H → H ′ morphism of group schemes, g : Y → Y ′ f -equivariant morphism of schemes }.

We take the definition of "conically stratified space" Definition 2.A.18. Let Act con StrTop be the category with the following objects {H topological group, (Y, P, s) locally compact conically stratified topological space locally of singular shape, Φ action of H on Y such that the strata of (Y, P, s) coincide with the orbits of Φ, and P satisfies the ascending chain condition} and morphisms analogue to the previous definition. Remark 2.A.19.

  the proof is complete. Unlike the previous lemma, this is just a direct verification allowed by the fact that tangent spaces to open subsets of strata coincide with the tangent spaces to the original strata. One can also apply the more general and very useful argument appearing in[START_REF] Gibson | Topological Stability of Smooth Mappings[END_REF] (1.3), (1.4) and discussion below]. Let f : X → Y be a C 2 mapping, and let A be a closed subset of X which admits a C 2 Whitney stratification S . Suppose f |

	Lemma 3.2.2. Any open subset of a Whitney stratified manifold inherits a natural Whitney stratification
	by restriction.
	Proof. Lemma 3.2.3 (Thom's first isotopy lemma, [Mat72, (8.1)]).

A : A → Y is proper and that for each stratum U of S , f |

  • For every morphism f : Y → ΣZ in C, there exists a pullback and pushout square Grothendieck if it satisfies the equivalent conditions of Proposition 4.1.8. Following [Lur18, Definition C.3.0.5], we denote the ∞-category of Grothendieck presentable ∞-categories (and colimit-preserving functors between them) by Groth ∞ . We also denote the category of presentable stable ∞-categories (and colimit-preserving functors between them) by Pr L St . Groth ∞ inherit a symmetric monoidal structure from Pr L which we denote again by ⊗. Let X be a qcqs scheme. An O X -linear prestable ∞-category is an object of There exists a stabilization functor st X : LinCat PSt (X) → LinCat St (X), induced by the usual stabilization procedure. The category LinCat PSt (X) has a tensor product -⊗ QCoh(X) ≥0 -(which we will abbreviate by ⊗) induced by the Lurie tensor product of presentable ∞-categories. See [Lur18, Theorem C.4.2.1] and [Lur18, Section 10.1.6] for more details. The same is true for LinCat St (X). The stabilization functor LinCat PSt (X) → LinCat St (X) is symmetric monoidal with respect to these structures.LinCat St (X) and LinCat PSt (X) satisfy a very important "descent" property, which is what Gaitsgory [Gai15] calls 1-affineness.

	X 0 Remark 4.1.10. By [Lur17] and [Lur18, Theorem C.4.2.1], both Pr L Y ΣZ. A stable presentable O X -linear ∞-category is an object of LinCat St (X) := Mod QCoh(X) (Pr L,⊗ St ). ∞ ). Remark 4.1.12. Remark 4.1.13. Construction 4.1.14. The functors St and Definition 4.1.11. LinCat PSt (X) := Mod QCoh(X) ≥0 (Groth ⊗ CAlg k

f ′ f Proposition 4.1.7 ([Lur18, Proposition C.1.2.9]). Let C be a presentable ∞-category. Then the following conditions are equivalent: • C is prestable and has finite limits. • There exists a stable ∞-category D equipped with a t-structure (D ≥0 , D ≤0 ) and an equivalence C ≃ D ≥0 . Proposition 4.1.8 ([Lur18, Proposition C.1.4.1]). Let C be a presentable ∞-category. Then the following conditions are equivalent: • C is prestable and filtered colimits in C are left exact. • There exists a stable ∞-category D equipped with a t-structure (D ≥0 , D ≤0 ) compatible with filtered colimits, and an equivalence C ≃ D ≥0 . Definition 4.1.9 ([Lur18, Definition C.1.4.2]

). Let C be a presentable ∞-category. We will say that C is

  discussion before Theorem 10.2.0.1].

	Proof. For the stable part, this is [Lur11, Proposition 6.5]. For the prestable part, this is the combination of
	[Lur18, Theorem 10.2.0.2] and [Lur18, Theorem D.5.3.1]. Symmetric monoidality follows from the fact
	that the inverse of the global sections functor (the "localization functor", see [BP21, Section 2.3]) is strong
	monoidal.
	Theorem 4.1.15. Let X be a qcqs scheme over k. Then the global sections functors
	QStk St (X) → LinCat St (X)
	QStk PSt (X) → LinCat PSt (X)
	are symmetric monoidal equivalences.

  .2.1) By Theorem 4.1.15, every side of the sought equivalence is the ∞-category of global sections of a sheaf in categories over X, which means that the equivalence is étale-local on X. Therefore, by choosing a covering U → X which trivializes both G and G ′ , we can reduce to the caseG = U × BG m → U, G ′ = U × BG m → U ,both maps being the projection to U . But by [BP21, Corollary 5.6], QCoh(U × BG m ) ≃ QCoh(U ) × QCoh(BG m ), and this proves our claim. Finally, t-exactness follows from the Künneth formula. Now we prove that the restriction of the functor (4.2.1) to QCoh χ

About this, I must say that I often wished I knew better the English language, to whose inclination to a certain somberness I owe the tone of this page; I hope that my next things will be, if not more interesting, better written.

In this thesis, we adopt the perspective of ∞-categories, which is systematically exposed in[START_REF] Lurie | Higher Topos Theory[END_REF]. Both dg-categories and ∞-categories are ways to encode the idea of "categories with a notion of homotopy and homotopy equivalence" in a way that is particularly useful to deal with derived categories and homotopy theory. One of the simplest formulations of this concept is the notion of categories enriched in topological spaces or simplicial sets.

To be precise, their notion of "conically stratified space" is slightly different from that of[START_REF] Lurie | Higher Algebra[END_REF], and is defined by induction.

In order for a map to preserve perverse sheaves, one usually assumes that it is semi-small and proper.

Here E1 would stand for an "associativity" coming from some convolution diagram, and E4 would come from the real dimension of S, just like E2 comes from the real dimension of the curve X in our construction.

Other authors, however, use this name for the whole H 2 (X, Gm).

See Section 4.1.2 for a definition of OX -linear category in the ∞-categorical setting.

Our notion of constructible sheaf does not require finitely dimensional stalks. However, the full subcategory obtained by imposing this condition (which is the one usually considered) is closed under the product law that we describe.

We replace the notation "G O " with "Arc" for typographical reasons.

Note that this is true only for points of Ran(M ) coming from single points of M . If we allow the cardinality of the system of points to vary, stalks may take different values. In fact, the factorization property tells us that a system of cardinality m will give the m-ary tensor product in C of the stalk at the single point.

X and Y are said to satisfy Whitney's condition A if this is satisfied for any (xi) ⊂ X tending to y ∈ Y . The space is said to satisfy Whitney's condition A if every pair of strata satisfies it.

Idem.

Also, we usually identify this subspace of M with its preimage in the "abstract" tubular neighbourhood Bε ⊂ E.

With respect to the Whitney stratification induced on A, see Lemma

3.2.1.

One may use Example 3.2.5 as a guiding example, with m a point on {x = 0} \ {(0, 0)}.

In fact, one would need to use the construction of the "opposite of an ∞-operad", see[Bea],[START_REF] Lurie | Higher Algebra[END_REF] Remark

2.4.2.7].

Acknowledgements

Definition 4.1.18. Let X be a quasicompact quasiseparated scheme, and M be an element of Br(X). Then we define the Triv ≥0 (M ) as the sheaf of categories S → Equiv QCoh(S) ≥0 , M(S) .

Finally, observe that both Br(X) and Ger Gm (X) have symmetric monoidal structures: on the first one, we have the tensor product ⊗ inherited by LinCat PSt (X) (Remark 4.1.13), and on the second one we have the rigidified product ⋆ of G m -gerbes. Although the two structures are well-known to experts, we will recall them in Section 4.2.1.

We are now ready to formulate our main theorem.

Theorem 4.1.19. Let X be a qcqs scheme over a field k. Then there is a symmetric monoidal equivalence of 2-groupoids Φ : Br(X) ⊗ ←→ Ger Gm (X) ⋆ : Ψ where

• Ψ(G) = QCoh id (G) ≥0 .

Remark 4.1.20. By taking the π 0 , one obtains an isomorphism of abelian groups dBr(X) ≃ H 2 (X, G m ).

This isomorphism appears also in [START_REF] Lurie | Spectral Algebraic Geometry[END_REF]Example 11.5.7.15], but as mentioned before, it is a consequence of the equivalence of ∞-categories

whose proof however does never refer to the notion of G m -gerbe.

Study of the derived Brauer map

Symmetric monoidal structures

We begin by describing the symmetric monoidal structure on the category of G m -gerbes. Let X be a scheme and G 1 and G 2 be two G m -gerbes on X. One can costruct the product G 1 ⋆ G 2 , which is a G m -gerbe such that its class in cohomology is the product of the classes of G 1 and G 2 (see [START_REF] Binda | GAGA problems for the Brauer group via derived geometry[END_REF]Conjecture 5.23]). Clearly, this is not enough to define a symmetric monoidal structure on the category of G m -gerbes. The idea is to prove that this ⋆ product has a universal property in the ∞-categorical setting, which allows us to define the symmetric monoidal structure on the category of G m -gerbes using the theory of simplicial colored operads and ∞-operads (see Chapter 2 of [START_REF] Lurie | Higher Algebra[END_REF]). Construction 4.2.1. Let AbGer(X) be the (2, 1)-category of abelian gerbes over X and AbGr(X) the (1, 1)-category of sheaves of abelian groups over X. We have the so-called banding functor Band : AbGer(X) -→ AbGr(X), see for example Chapter 3 of [START_REF] Bergh | Decompositions of derived categories of gerbes and of families of Brauer-Severi varieties[END_REF]. It is easy to prove that Band is symmetric monoidal with respect to the two Cartesian symmetric monoidal structures of the source and target, that is it extends to a symmetric monoidal functor Band : AbGer(X) × → AbGr(X) × of colored simplicial operads (and therefore also of ∞-operads).

Recall that, given a morphism of sheaf of groups ϕ : µ → µ ′ and a µ-gerbe G, we can construct a µ ′ -gerbe, denoted by ϕ * G, and a morphism ρ ϕ : G → ϕ * G whose image through the banding functor is exactly ϕ. This pushforward construction is essentially unique and verifies weak funtoriality. This follows from the following result: if G is a gerbe banded by µ, then the induced banding functor

is an equivalence and the pushforward construction is an inverse (see [BS19, Proposition 3.9]). This also implies that Band is a coCartesian fibration. Let Fin * be the Segal category of pointed finite set. Consider now the morphism Fin * → AbGr(X) × induced by the algebra object G m,X in AbGr(X) × . We can consider the following pullback diagram

thus B is a coCartesian fibration. This implies that G is a symmetric monoidal structure over the fiber category

We claim that this symmetric monoidal structure coincides with the ⋆ product of gerbes defined in [START_REF] Bergh | Decompositions of derived categories of gerbes and of families of Brauer-Severi varieties[END_REF]. In fact, following the rigidification procedure in [Ols16, Exercise 12.F] and [AOV08, Appendix A], we can define the ⋆ product as the pushforward of the multiplication map m :

Remark 4.2.2. Let us describe G as a simplicial colored operad. The objects (or colors) of G are G m -gerbes over X. Let {G i } i∈I be a sequence of objects indexed by a finite set I and H another object; we denote by i∈I G i the fiber product of G i over X. The simplicial set of multilinear maps Mul({G i } i∈I , H) is the full subcategory of the 1-groupoid of morphisms Map X ( i∈I G i , H) of gerbes over X such that its image through the banding functor is the n-fold multiplication map of G m , where n is the cardinality of I. Note that because the simplicial sets of multilinear maps are Kan complexes by definition, then they are fibrant simplicial sets. This is why the operadic nerve gives us a symmetric monoidal structure, see [ Next, we will describe the symmetric monoidal structure on Br(X). We know that it is the restriction of the symmetric monoidal structure of LinCat PSt (X), see Remark 4.1.13. Remark 4.2.5. We now describe explicitly the symmetric monoidal structure of LinCat St (X) using the language of simplicial colored operads. The same description will apply to the prestable case. This will come out useful in the rest of the chapter.

The simplicial colored operad LinCat St (X) can be described as follows:

1. the objects (or colors) are stable presentable ∞-categories which are modules over QCoh(X) ( see [Lur17, Section 4.5] for the precise definition of module over an algebra object);

2. given {M i } i∈I a sequence of objects indexed by a finite set I and N another object, the simplicial set of multilinear maps Mul({M i } i∈I , N ) is the mapping space Fun L ( i∈I M i , N ) of functors of stable presentable categories which are QCoh(X)-linear and preserve small colimits separately in each variable.

Derived categories of twisted sheaves

Recall from Section 4.1.1 the definition of QCoh id (G) and QCoh id (G) ≥0 . Our aim in this subsection is to prove the following statement: Theorem 4.2.6. Let X be a qcqs scheme. The functors

and QCoh id (-) ≥0 : Ger Gm (X) ⋆ → LinCat PSt (X) ⊗ carry a symmetric monoidal structure with respect to the ⋆-symmetric monoidal structure on the left hand side and to the ⊗-symmetric monoidal structures on the right hand sides. In particular, since every G m -gerbe is ⋆-invertible and Ger Gm (X) is a 2-groupoid, QCoh id (-) takes values in Br † (X) and QCoh id (-) ≥0 takes values in Br(X).

Remark 4.2.7. Notice that we need to explicit how QCoh id (-) (or QCoh id (-) ≥0 ) acts on the mapping spaces as well. We decided to stick with the covariant notation, i.e we define the image of a morphism of

We omit the proof of the fact that f * sends twisted sheaves to twisted sheaves, which follows from a straightforward computation.

Although there are no real issues with the contravariant notation (considering pullbacks instead of pushforwards), there are some technical complications when one wants to prove that a contravariant functor is symmetric monoidal. 1 Note that, if f is any morphism of gerbes, then, f * = (f * ) -1 .

We will prove Theorem 4.2.6 in two different steps.

) is a morphism between the two diagrams and therefore there exists a unique (up to homotopy) morphism C 0 1 → C 0 2 compatible with all the data. If G is a G m -gerbe and χ is a character of G m , the endofunctor (-) χ of QCoh(G) and the natural transformation i χ introduced in Remark 4.1.4 form a diagram as above. In this situation, QCoh χ (G) is exactly the equalizer. Proposition 4.2.9. Let X be a quasicompact quasiseparated scheme over a field k. Let G, G ′ → X be two G m -gerbes over X, and χ, χ ′ two characters of G m . The external tensor product establishes a t-exact equivalence

Proof of the equivalence

The goal of this subsection is to prove that the constructions

establish a (symmetric monoidal) categorical equivalence between the 2-groupoids Br(X) and Ger Gm (X), thus proving Theorem 4.1.19. Proof. We want to prove that for any M, M ′ ∈ Br(X) the map

is a homotopy equivalence of spaces (more precisely, an equivalence of 1-groupoids). First of all, by using the grouplike monoid structure of Br(X) together with Lemma 4.2.17, we can reduce to the case

If M does not lie in the connected component of 1, then what we have to check is that Map Ger Gm (X) (Triv ≥0 (M ), Triv ≥0 (1)) = ∅.

But if this space contained an object, then in particular we would have an equivalence at the level of global sections between Equiv QCoh(X) ≥0 (M, 1) and Equiv QCoh(X) ≥0 (1, 1). But the first space is empty by hypothesis, while the second is not. On the other hand, if M lies in the connected component of 1, by functoriality of Triv ≥0 (-) we can suppose that M = 1. In this case, we have to prove that the map

is an equivalence of groupoids. But the first space is the groupoid Pic(X), the latter is the space of maps X → BG m , and the composite map is the one sending a line bundle over X to the map X → BG m that classifies it. Proposition 4.2.19. Let X be a qcqs scheme. Then there is a natural equivalence of functors

Proof. Let G be a G m -gerbe over X. Let us observe that for any S → X we have

Indeed, there is an evident map of stacks over X F : Equiv Ger Gm (X × BG m , G) → G and up to passing to a suitable étale covering of X, the map becomes an equivalence: in fact the choice of an equivalence ϕ : S × BG m → S × BG m of gerbes over S amounts to the choice of a map S → BG m , because ϕ must be a map over S (hence pr BGm • ϕ = pr BGm ) and it must respect the banding . Therefore, F is an equivalence over X, and this endows Equiv Ger Gm (X × BG m , G) with a natural structure of a G m -gerbe over X.

The construction ϕ → ϕ * thus provides a morphism of stacks

Now since the pushforward ϕ * of an equivalence ϕ of stacks is t-exact, the construction ϕ → ϕ * factors through the stabilization map Equiv QCoh id QCoh id (X×BG m ) ≥0 , QCoh id (G) → Equiv QCoh(X) QCoh id (X×BG m ), QCoh id (G) .

Note that the stabilization map makes sense in the id-twisted context since the stabilization functor has descent (it is the limit of iterated loops), and thus we can reduce to the case when G is trivial.

We have therefore obtained a map of stacks over X G → Equiv QCoh(X) ≥0 QCoh(X) ≥0 , QCoh id (G) ≥0 = Triv ≥0 (QCoh id (G) ≥0 ).

But now, Triv ≥0 (QCoh id (G)) is a G m -gerbe over X (note that this was not true before passing to the connective setting), and therefore to prove that the map is an equivalence it suffices to prove that it agrees with the bandings, that is, that it is a map of G m -gerbes. This follows from unwinding the definitions.

We are now ready to prove our main result.

Proof of Theorem 4.1.19: Section 4.2.2 and Section 4.2.3 tell us that the two functors are symmetric monoidal and take values in the sought ∞-categories. The fact that they form an equivalence follows from Proposition 4.2.18 and Proposition 4.2.19.