Rune Haugseng

Fabian Hebestreit

AND Sil Linskens

Joost Nuiten

TWO-VARIABLE FIBRATIONS, FACTORISATION SYSTEMS AND ∞-CATEGORIES OF SPANS

1. Introduction 1. Introduction 1 2. Adequate triples and ∞-categories of spans 2. Adequate triples and ∞-categories of spans 5 3. Cartesian fibrations and ∞-categories of spans 3. Cartesian fibrations and ∞-categories of spans 13 4. Orthogonal adequate triples and their duals 4. Orthogonal adequate triples and their duals 20 5. Cartesian fibrations between orthogonal adequate triples 5. Cartesian fibrations between orthogonal adequate triples 27 6. Dualisation and straightening of two-variable fibrations 6. Dualisation and straightening of two-variable fibrations 34 7. (Op)lax arrow and twisted arrow ∞-categories

Introduction

Categories of spans typically appear in algebra and geometry as a convenient way to encode coand contravariant functorialities (transfer and restriction) and base change isomorphisms between these. For example, span categories of finite G-sets classically appear in representation theory as the indexing categories of Mackey functors [START_REF] Dress | Notes on the theory of representations of finite groups. Part I: The Burnside ring of a finite group and some AGN-applications[END_REF]]. In the ∞-categorical setting, spans play an even more significant role, because they often provide the only feasible method to organise base change isomorphisms in a homotopy-coherent way. As such, they naturally appear in equivariant homotopy theory (via the theory of spectral Mackey functors [START_REF]Spectral Mackey functors and equivariant algebraic K-theory I[END_REF][START_REF] Nardin | Parametrized higher category theory and higher algebra: Exposé IV -Stability with respect to an orbital ∞-category[END_REF]), in motivic homotopy theory [START_REF] Bachmann | Norms in motivic homotopy theory[END_REF]] and in higher algebra [START_REF] Cranch | Algebraic theories, span diagrams and commutative monoids in homotopy theory[END_REF][START_REF] Harpaz | Ambidexterity and the universality of finite spans[END_REF][START_REF] Elmanto | On distributivity in higher algebra I: The universal property of bispans[END_REF].

In each of these cases, one is interested in diagrams indexed by a certain ∞-category of spans that is constructed informally as follows: from the data of an ∞-category X, together with two classes of maps called ingressive (or forwards, denoted ↣) and egressive (or backwards, denoted ↠), one constructs an ∞-category Span(X) with the same objects as X and with morphisms from x to y given by spans z x y.

Composition proceeds by pullback of spans, which of course requires that pullbacks of ingressive and egressive maps exist and remain ingressive and egressive. From a geometric point of view, one can think of Span(X) as an ∞-category of (combinatorial) bordisms in X op , the two legs of a span giving the two boundary inclusions. As such, span categories also arise in various situations as objects of interest in themselves. Notably, they appear in algebraic K-theory [Ba13 Ba13, [START_REF] Blumberg | A universal characterization of higher algebraic K-theory[END_REF] and its hermitian refinements [START_REF] Schlichting | Hermitian K-theory of exact categories[END_REF][START_REF] Hermitian | Theory of stable ∞-categories II: Cobordism categories and additivitiy[END_REF]] via Quillen's Q-construction [START_REF] Quillen | Higher algebraic K-theory I[END_REF]]; see in particular [RS19 RS19, CDH+20b CDH+20b, [START_REF] Hebestreit | Stable moduli spaces of hermitian forms[END_REF] for works on K-theory where such a geometric perspective on spans is brought to fruition. The above informal description of Span(X) has been substantiated by Barwick, who constructs an explicit functor sending each adequate triple (of an ∞-category X and two classes of maps) to a quasi-category Span(X) [START_REF]Spectral Mackey functors and equivariant algebraic K-theory I[END_REF]]. The first purpose of this text is to study some further abstract properties of this construction of span ∞-categories. For example, we give a description of span ∞-categories by a universal property:

Theorem A. There is an adjoint pair Tw r : Cat AdTrip : Span, where Tw r (A) is the twisted arrow ∞-category of A with the ingressive and egressive morphisms being those which induce equivalences in the target and source component, respectively.

Here we follow the same notational conventions as in [START_REF] Haugseng | Lax monoidal adjunctions, two-variable fibrations and the calculus of mates[END_REF]] and denote the ∞-category of small ∞-categories by Cat (and similarly that of ∞-groupoids by Gpd, etc.). Let us point out that our proof of this result relies on a Segal space construction of span ∞-categories, instead of the quasi-categorical approach of Barwick. Theorem A A also has a 2-categorical upgrade, which identifies the diagram ∞-categories Fun(A, Span(X)) with a span ∞-category of the ∞-category of diagrams Tw r (A) → X. This statement has already been applied in hermitian K-theory, where it forms the basis for the parametrised algebraic surgery of [ [START_REF] Hebestreit | Stable moduli spaces of hermitian forms[END_REF].

In addition, we give a description of the 'universal span category', i.e. the cocartesian fibration classified by the functor Span : AdTrip -→ Cat: Theorem B. The opposite of the lax under-category * // lax AdTrip admits the structure of an adequate triple in which a map * X Y y x f µ is egressive if µ : y → f (x) is egressive, and ingressive if µ is ingressive and f is an equivalence. The natural map p : Span ( * // lax AdTrip) op -→ AdTrip taking the underlying map of adequate triples is then equivalent to the cocartesian fibration classified by Span : AdTrip -→ Cat.

As a consequence of Theorem B B, we obtain a new proof of the main result of [ [START_REF] Barwick | Dualizing cartesian and cocartesian fibrations[END_REF], which asserts that the cocartesian unstraightening of a functor F : B → Cat can be constructed from its cartesian unstraightening via a span construction.

Note that the ∞-category Span(X) has the following features: the egressive arrows from X are reverted, the ingressive arrows from X are kept and each map factors uniquely as a 'revertedegressive', followed by an ingressive map. This suggests that taking span ∞-categories acts as an involution on adequate triples arising from (orthogonal) factorisation systems. The second part of the paper studies such orthogonal adequate triples, whose egressive and ingressive arrows form a factorisation system and in which all commuting squares of the form (1.1)

x ′ x y ′ y are cartesian. Our most significant result about these is the following:

Theorem C. Taking span ∞-categories gives rise to a C 2 -action on the full subcategory of orthogonal adequate triples Span ⊥ : AdTrip ⊥ -→ AdTrip ⊥ .

The simplest example of an orthogonal adequate triple is given by a product A × B, with ingressive maps coming from A and egressive maps coming from B. More generally, every cartesian fibration p : X -→ A defines the structure of an orthogonal adequate triple on X, for which a map is ingressive if it is p-cartesian and egressive if it is fibrewise, i.e. its image under p is invertible. In fact, every orthogonal adequate triple arises in this way: Theorem D. An orthogonal factorisation system arises from a cartesian fibration as the cartesian and fibrewise maps if and only if it forms an orthogonal adequate triple. Moreover, this construction restricts to an equivalence between the ∞-category of orthogonal adequate triples and the ∞-category of cartesian fibrations whose fibres have contractible realisation.

This result extends previous work of Lanari [START_REF] Lanari | Cartesian factorization systems and pointed cartesian fibrations of ∞-categories[END_REF], who established such a correspondence for pointed cartesian fibrations (i.e. those whose base and total ∞-category have a terminal object which is preserved by the fibration) and what he calls cartesian factorisation systems. Using the relation between orthogonal adequate triples and cartesian fibrations, the span duality from Theorem C C also gives an auto-equivalence of the ∞-category of cartesian fibrations. This reduces to the duality functor of Barwick, Glasman and Nardin from [ [START_REF] Barwick | Dualizing cartesian and cocartesian fibrations[END_REF]].

Similarly, one can use Theorem C C to obtain duality functors for fibrations over a product A × B. For example, one can consider maps of orthogonal triples p = (p 1 , p 2 ) : X -→ A × B such that:

(1) the ingressive maps of X are exactly the p-cartesian lifts of arrows in A (2) p 2 is a cocartesian fibration and for any square (1.1 1.1) in X where x → y is p 1 -cocartesian, the base change x ′ -→ y ′ is p 1 -cocartesian as well.

Such maps are called orthofibrations in [HHLN21 HHLN21], or two-sided fibrations in [RV22 RV22, Section 7.1], and can be straightened to functors A op × B -→ Cat, which also correspond to cartesian fibrations over A×B op via unstraightening. Using that Span ⊥ sends A×B (with the adequate triple structure mentioned above) to A × B op , the dualisation of Theorem C C also restricts to an equivalence (1.2) Span ⊥ : Ortho(A, B) ≃ Cart(A × B op ) between orthofibrations and cartesian fibrations, taking opposite categories at the level of fibres.

There are similar dualities for the various other types of two-variable fibrations considered in [HHLN21 HHLN21]; For example, Theorem C C also restricts to a duality between curved orthofibrations over A × B and op-Gray fibrations over A × B op (see [START_REF] Haugseng | Lax monoidal adjunctions, two-variable fibrations and the calculus of mates[END_REF] or Section 6 6 for definitions), which under straightening [START_REF]∞, 2)-categories and the Goodwillie calculus I[END_REF]] correspond to 2-functors out of the Gray tensor product B ⊗ A op . Relying on Toën's equivalence Aut(Cat) ≃ C 2 [To05 To05], we extend another result from [ [START_REF] Barwick | Dualizing cartesian and cocartesian fibrations[END_REF]] and show that any composite of dualisations and (un)straightenings as above is uniquely determined by its action on fibres: Theorem E. The three functors Cat op × Cat op → Cat given by (A, B) -→ Fun(A × B, Gpd), Fun(A × B, Cat) and Fun(A ⊠ B, Cat)

have automorphism groups * , C 2 , and * , respectively. The non-trivial automorphism in the middle case is given by post-composition with (-) op : Cat → Cat. (Here Cat denotes the (∞, 2)-category of ∞-categories, as in [HHLN21 HHLN21].)

For example, this implies that the dualisation equivalence (1.2 1.2) arising from Theorem C C coincides with the dualisation constructed in [HHLN21 HHLN21] using straightening and unstraightening (though this can also be deduced from Theorem B B directly). As an application of this explicit dualisation for two-variable fibrations, we consider the fibrations classified by the enhanced mapping functor Map X : X op × X → Cat associated to an (∞, 2)-category X with underlying ∞-category X. On the one hand, this functor is classified by a cartesian fibration (s, t) : Tw r (X) -→ X × X op from the oplax twisted arrow ∞-category of X, explicitly constructed in [ [START_REF] García | Enhanced twisted arrow categories[END_REF]]. We show (Theorem 7.21 7.21) that dually (and upon taking opposite categories), this functor also classifies the orthofibration (s, t) : Ar opl (X) -→ X × X from the ∞-category underlying the oplax arrow ∞-category, defined using the (in principle unrelated) Gray tensor product of Gagna, Harpaz and Lanari from [ [START_REF] Gagna | Gray tensor products and lax functors of (∞, 2)-categories[END_REF]]. This is a typical example of an identification of dual fibrations which seems difficult to see by passing through a form of straightening and unstraightening (as in [START_REF] Haugseng | Lax monoidal adjunctions, two-variable fibrations and the calculus of mates[END_REF]). In a different direction, Theorem E E shows that straightening a cartesian fibration over A × B to a functor A op × B op -→ Cat is naturally equivalent to straightening it first over A and then over B. This should not be surprising, but deducing it from the definitions does not seem entirely obvious (nevertheless, we provide a second proof as 6.20 6.20). As a consequence, we answer a recent question of Clausen:

Corollary F. The Yoneda embedding C → P(C) canonically extends to a natural transformation of functors Cat -→ Cat from the inclusion to the composite

Cat Fun(-,Gpd) --------→ (Cat R ) op ≃ Cat L ⊆ Cat.
Again, this result is certainly expected, and we were surprised to learn that it is apparently not contained in the literature. Let us point out that, essentially by definition, the Yoneda embedding does define a natural transformation to the functor P : Cat -→ Cat defined using that P(C) is the free cocompletion of C. However, it is not a priori clear that this functor agrees with the one appearing in the corollary in a fashion compatible with the Yoneda embedding (but this follows from the result above as well).

Remark. Along with [START_REF] Haugseng | Lax monoidal adjunctions, two-variable fibrations and the calculus of mates[END_REF] this article is part of a recombination of our earlier preprints [START_REF] Haugseng | A fibrational mate correspondence for ∞-categories[END_REF]] and [ [START_REF] Hebestreit | Orthofibrations and monoidal adjunctions[END_REF]. In [HHLN21 HHLN21], we described the dualisation of two-variable fibrations, as well as its applications to monoidal and parametrised adjunctions, along the lines of [START_REF] Haugseng | A fibrational mate correspondence for ∞-categories[END_REF]]; These applications were also discussed in terms of the dualisation functor Span ⊥ in [HLN20 HLN20] (in particular Proposition A and Theorem C in there). The results from [HLN20 HLN20] concerning span ∞-categories and the comparison of various straightening functors for two-variable fibrations (in particular Theorem B and Corollary D there) are contained in the present paper.

Organisation. In Section 2 2 we redevelop Barwick's theory of span ∞-categories for adequate triples in terms of complete Segal spaces, and then discuss some further properties of the span construction in Section 3 3. In particular, here we prove Theorem A A as Theorem 2.18 2.18 and deduce Theorem B B as a corollary of Theorem 3.8 3.8.

We then introduce orthogonal adequate triples in Section 4 4 and establish Theorem C C as Theorem 4.12 4.12. The relation between orthogonal adequate triples and fibrations is discussed in Section 5 5, where Theorem D D is proven as Proposition 5.4 5.4. In Section 6 6 we show how Theorem C C induces dualities between various types of two-variable fibrations considered in [ [START_REF] Haugseng | Lax monoidal adjunctions, two-variable fibrations and the calculus of mates[END_REF]], and use this to identify oplax arrow and twisted arrow ∞-categories as duals in Section 7 7. We deduce Corollary F F in the final Section 8 8 as Theorem 8.1 8.1.

Finally, the uniqueness of all these dualities, i.e. Theorem E E, is derived in Appendix A A as Theorem A.1 A.1.

Conventions.

In order to declutter notation we will write Gpd, Cat and Cat 2 for the ∞-categories of ∞-groupoids (or spaces), ∞-categories and (∞, 2)-categories, respectively.

The letter ι will denote the core of an ∞-category, i.e. the ∞-groupoid spanned by its equivalences. By a subcategory of an ∞-category we mean a functor such that the induced morphisms on mapping ∞-groupoids and cores are inclusions of path components. A subcategory is full if the functor furthermore induces equivalences on mapping ∞-groupoids, while it is wide if the functor induces an equivalence on cores. Similarly, a sub-2-category of an (∞, 2)-category is a functor inducing subcategory inclusions on mapping ∞-categories and a monomorphism on underlying ∞-groupoids; we say such a sub-2-category is 1-full if it locally full, i.e. is given by full subcategory inclusions on mapping ∞-categories.

Throughout, we shall use small caps such as Cat to indicate large variants of ∞-categories and boldface such as Cat to denote (∞, 2)-categories. We have also reserved sub-and superscripts on category names to refer to changes on morphisms, e.g. Cart(A) ⊆ Cart opl (A).

We will write Ar(C) for the arrow ∞-category Fun([1], C) of an ∞-category C, and Tw ℓ (C) and Tw r (C) for the two versions of the twisted arrow category, geared so that the combined source-target map defines a left fibraton in the former, and a right fibration in the latter case, see Example 2.8 2.8. Finally, we write Λ 0 [2] for the span category 1 ← 0 → 2 and Λ 2 [2] for the cospan.

Acknowledgments. It is a pleasure to thank Dustin Clausen, Yonatan Harpaz, Gijsbert Scheltus Karel Sebastiaan Heuts, Corina Keller, Achim Krause, Markus Land, Denis Nardin, Thomas Nikolaus, Stefan Schwede, Wolfgang Steimle and Lior Yanovski for several very useful discussions.

During the preparation of this text FH and SL were members of the Hausdorff Center for Mathematics at the University of Bonn funded by the German Research Foundation (DFG) and FH was furthermore a member of the cluster "Mathematics Münster: Dynamics-Geometry-Structure" at the University of Münster under grant nos. EXC 2047 and EXC 2044, respectively. FH would also like to thank the Mittag-Leffler Institute for its hospitality during the research program "Higher algebraic structures in algebra, topology and geometry", supported by the Swedish Research Council (VR) under grant no. 2016-06596. FH and JN were further supported by the European Research Council (ERC) through the grants 'Moduli spaces, Manifolds and Arithmetic', grant no. 682922, and 'Derived Symplectic Geometry and Applications', grant no. 768679, respectively. SL was supported by the DFG Schwerpunktprogramm 1786 'Homotopy Theory and Algebraic Geometry' (project ID SCHW 860/1-1)

Adequate triples and ∞-categories of spans

In this section we shall review the theory of adequate triples and their associated span ∞categories, as developed by Barwick in [START_REF]Spectral Mackey functors and equivariant algebraic K-theory I[END_REF]] under the name effective Burnside ∞-categories. We will use the opportunity to present an alternate viewpoint on the material by translating the assertions along the equivalence between ∞-categories and complete Segal spaces. This will allow for simpler proofs with far less explicit combinatorics, and will form the basis for our analysis of functors to span ∞-categories in the next section.

Since composition in span ∞-categories proceeds by pullback, one uses the following data as input for span ∞-categories: 2.1. Definition. An adequate triple (X, X in , X eg ) consists of an ∞-category X together with two wide subcategories X in and X eg , whose morphisms are called ingressive (or forwards, denoted ↣) and egressive (or backwards, denoted ↠), respectively, such that (1) for any ingressive morphism f : y ↣ x and any egressive morphism g : x ′ ↠ x, there exists a pullback

y ′ x ′ y x, g ′ f ′ g f
(2) and in any such pullback f ′ is again ingressive and g ′ egressive.

Squares whose horizontal arrows are ingressive and whose vertical arrows are egressive are called ambigressive, and ambigressive cartesian if they are furthermore pullback diagrams. A functor

F : (X, X in , X eg ) → (Y, Y in , Y eg )
of adequate triples is given by a functor F : X → Y which preserves ambigressive pullbacks. Therefore we may define AdTrip, the ∞-category of adequate triples, as the subcategory of Fun(Λ 2 [2], Cat) spanned by the adequate triples and those natural transformations whose evaluation at 2 ∈ Λ 2 [2] preserves ambigressive pullback squares.

Note that being a natural transformation boils down to preserving ingressive and egressive maps, since these form subcategories. We shall often drop them from notation to avoid cluttering. 2.2. Notation. Given a morphism p : Y → X of adequate triples we write p in : Y in → X in and p eg : Y eg → X eg for the restriction of p to ingressives and egressives.

Example. Let us record the following immediate examples:

(1) For any wide subcategory T of an ∞-category S, we have the adequate triples (S, T, ιS) and (S, ιS, T ) where the ingressives are maps in T and the egressives are equivalences, and vice versa. In particular, for any ∞-category S we always have the adequate triples (S, S, ιS) and (S, ιS, S).

(2) If S admits all pullbacks, we have the adequate triple (S, S, S) where all maps are both egressive and ingressive. This gives rise to a fully faithful embedding Cat pb → AdTrip of the subcategory of Cat spanned by those ∞-categories which admit pullbacks, and those functors which preserve pullbacks.

(3) If (X, X in , X eg ) is an adequate triple, then X rev = (X, X eg , X in ) is an adequate triple.

2.4. Lemma. The ∞-category of adequate triples admits all limits and one simply computes lim X, X in , X eg ≃ lim X, lim X in , lim X eg for every X : I → AdTrip. Likewise, AdTrip has filtered colimits given by colim X, X in , X eg ≃ colim X, colim X in , colim X eg .

Proof. Since the limit of a natural wide subcategory inclusion remains a wide subcategory inclusion, lim X, lim X in , lim X eg is indeed a triple of a category together with two wide subcategories (and likewise for filtered colimits). Notice that an ambigressive square in lim X is cartesian if and only if its image in each X i is cartesian. Furthermore, a cospan x ↣ x ′ ↞ x ′′ in lim X admits an ambigressive pullback if its images in each X i do (in this case those pullbacks assemble into an object of the limit). This implies that lim X is an adequate triple and that a map Z -→ lim X preserves ambigressive pullback squares if and only each composite Z -→ lim X -→ X i does. This implies that the (pointwise) limit in Fun(Λ 2 [2], Cat) also provides the limit in the subcategory AdTrip → Fun(Λ 2 [2], Cat).

A dual argument applies to the colimit of a filtered diagram. In this case, the image of an ambigressive pullback square in some X i under the map X i -→ colim X i remains an ambigressive pullback (since at the level of mapping spaces, filtered colimits commute with pullbacks). Conversely, any cospan x ↣ x ′ ↞ x ′′ in colim X arises as the image of a cospan in some X i , whose pullback in X i then provides the desired ambigressive pullback in colim X. It follows that colim X is an adequate triple and that a map of triples colim X → Z preserves ambigressive pullback squares if and only if each X i → colim X → Z does. This implies that colim X is also the colimit in the subcategory AdTrip → Fun(Λ 2 [2], Cat). □ 2.5. Lemma. Let X and Y be two adequate triples and let us write Fun AdTrip (X, Y ) for the full subcategory of Fun(X, Y ) spanned by the morphisms of adequate triples. Say a natural transformation τ : F ⇒ G is ingressive if it is pointwise ingressive in Y and for every egressive f : x ↠ y in X the square

F (x) F (y) G(x) G(y) F (f ) τx τy G(f )
is cartesian, and dually for an egressive natural transformation. This endows Fun AdTrip (X, Y ) with the structure of an adequate triple, such that the evaluation map Fun AdTrip (X, Y ) × X → Y exhibits it as the internal mapping object in AdTrip. In particular, AdTrip is cartesian closed.

Consequently, one obtains a natural equivalence of adequate triples

Fun AdTrip (X × Y, Z) ≃ Fun AdTrip (X, Fun AdTrip (Y, Z)).
Proof. The pullback of an ingressive and an egressive natural transformation in Fun AdTrip (X, Y ) exists and is computed pointwise as an ambigressive pullback. One readily verifies that the resulting square of natural transformations is ambigressive, so that Fun AdTrip (X, Y ) is an adequate triple.

To verify the universal property, let Z be a third adequate triple. In suffices to verify that under the equivalence Map Cat (Z × X, Y ) ≃ Map Cat (Z, Fun(X, Y )) a functor F : Z × X -→ Y is a map of adequate triples if and only if the corresponding map F ′ : Z -→ Fun(X, Y ) takes values in Fun AdTrip (X, Y ) and determines a map of adequate triples. To see this, note that any ambigressive square in Z × X is uniquely a composite of four types of squares: (a) {z} × σ with σ an ambigressive square in X, (b) τ × {x} with τ ambigressive and g ×

f : [1] × [1] -→ Z × X with (c)
, p -1 (X eg )),
where a map is ingressive if it is p-cartesian and its image is ingressive in X, and egressive if its image under p is. This structure also upgrades p to a map of adequate triples.

Proof. Consider the left solid cospan in the diagram

y 3 y 2 x 1 × x0 x 2 x 2 y 1 y 0 x 1 x 0 with x 1 ↣ x 0 ↞ x 2 its image in X under p.
By assumption the right diagram can be completed to an ambigressive pullback in X and we let y 3 ↣ y 2 be a p-cartesian lift of x 1 × x0 x 2 . The universal property of p-cartesian maps then implies that there exists a unique map y 3 → y 1 which lives over x 1 × x0 x 2 → x 1 and makes the left square commute. Note that the left square is ambigressive by definition. By unraveling the universal properties on mapping spaces one obtains that it is also a pullback. Finally note that with this description of ambigressive pullbacks in Y , it is clear that p preserves ambigressive pullbacks and is thus a map of adequate triples. □

The case of Proposition 2.6 2.6 where p is a cartesian fibration admits the following generalisation: 2.7. Proposition. Let (X, X in , X eg ) be an adequate triple, F : X op → AdTrip a functor and p : Y -→ X its cartesian unstraightening. Then Y has the structure of an adequate triple, in which a map y 1 → y 0 is egressive if it factors as an egressive morphism in the fibre Y p(y1) ≃ F (p(y 1 )), followed by a p-cartesian morphism with egressive image in X (and likewise for the ingressives). Furthermore, this structure makes p a map of adequate triples.

Proof. Consider a cospan f : y 1 ↣ y 0 ↞ y 2 : g where the left arrow is ingressive and the right arrow is egressive in Y . To show that this admits an ambigressive pullback, note that f decomposes into two egressive maps in Y , one contained in a fibre of p and the other p-cartesian (and likewise for y 2 ↠ y 0 ). By the pasting lemma for pullbacks, it therefore suffice to show the existence of an ambigressive pullback in four special cases. First, suppose that both f and g are contained in a single fibre Y x ≃ F (x). Since the fibre forms an adequate triple, there exists an ambigressive pullback of f and g within Y x . This square remains a pullback square in Y by [Lu09a Lu09a, Proposition 4.3.1.10].

Next, suppose that f is contained in a fibre Y x0 and that g : γ * y 0 → y 0 is a p-cartesian lift of an egressive arrow γ : x 2 ↠ x 0 . Since p is a cartesian fibration, we can form the cartesian square

γ * y 1 y 2 y 1 y 0 cart γ * (f ) cart f
in which the vertical arrows are cartesian. This pullback square is ambigressive: the left vertical map is a p-cartesian lift of the egressive arrow γ in X (hence egressive in Y ) and the top horizontal map is ingressive because the base change functor γ * : F (x 0 ) -→ F (x 2 ) is a map of adequate triples. Of course, the same argument provides an ambigressive pullback of f and g where f is p-cartesian and g is contained in a fibre.

Finally, when f and g are p-cartesian lifts of an ingressive and an egressive arrow in X, an ambigressive pullback exists by the exact same argument as in Proposition 2.6 2.6. □ 2.8. Example. Let (X, X in , X eg ) be an adequate triple and let Tw r (X) be its twisted arrow ∞-category, with the convention that the source and target define a right fibration. In particular, s : Tw r (X) -→ X is a cartesian fibration and Proposition 2.6 2.6 endows Tw r (X) with the structure of an adequate triple, in which a map α -→ β given by

x y w z α f β g
is egressive if f is egressive, and ingressive if f is ingressive and g is an equivalence.

2.9. Example. Let A be an ∞-category, considered as an adequate triple (A, A, ιA) with all morphisms ingressive and only the equivalences egressive. Example 2.8 2.8 endows Tw r (A) with the structure of an adequate triple, where a morphism is ingressive if its image under t : Tw r (A) -→ A op is an equivalence, and egressive if its image under s : Tw r (A) -→ A is an equivalence. This determines a functor Tw r : Cat -→ AdTrip.

For the construction of span ∞-categories this structure will be of particular interest when A is a simplex. For example, Tw r ([2]) is the poset

(0 ≤ 2) (0 ≤ 1) (1 ≤ 2) (0 ≤ 0) (1 ≤ 1) (2 ≤ 2)
where the left pointing arrows are ingressive and the right pointing ones are egressive. More generally, the ambigressive pullbacks in Tw r ([n]) are precisely given by the diagrams

(i ≤ l) (i ≤ j) (k ≤ l) (k ≤ j). for i ≤ k ≤ j ≤ l.
We next set out to construct Barwick's functor Span : AdTrip -→ Cat sending each adequate triple to its ∞-category of spans, described informally in the introduction. We will define Span more precisely as a functor into complete Segal spaces. Recall that such functors can be obtained from the following general procedure: 2.10. Construction. Given a cosimplicial object C : ∆ op → A, we obtain a 'singular complex' functor S C : A → sGpd with values in simplicial ∞-groupoids by currying the composition

∆ op × A C op ×id -----→ A op × A Map A ----→ Gpd, i.e. S C (X) n = Map A (C(n), X). If A is cocomplete,
T 0 T 0 × T 0 T 3 T 1 × T 1 ∆ s (s,s) (d {0,2} ,d {1,3} )
is cartesian. A general cosimplicial object C in A therefore gives rise to a functor A → Cat via the composition of S C with ac. When S C takes values in (complete) Segal spaces we have a good understanding of this functor, since there is then no need to localise; this happens precisely when the cosimplicial object C satisfies the dual version of the Segal and completeness conditions. 2.11. Example. Consider the functor S B : Cat -→ sGpd associated to the cosimplicial object Applying this construction to the functor Tw r : ∆ -→ AdTrip sending each simplex [n] to Tw r ([n]) with the structure from Example 2.9 2.9, we obtain: 2.12. Definition. We define Span : AdTrip → Cat as the composition AdTrip

B : ∆ -→ Cat, [n] -→ [n] ⋆ [n]
S Tw r ---→ sGpd ac -→ Cat.
2.13. Theorem (Barwick). The essential image of S Tw r is contained in the complete Segal spaces. In other words, there is a natural equivalence

Map Cat [n], Span(X) ≃ Map AdTrip (Tw r ([n]), X).
We will give a new proof of this result using complete Segal spaces (where Barwick uses quasicategories). Note that the Segal condition gives us the desired description of Span(X, X in , X eg ): objects are objects of X, morphisms from x to y are spans x z y with the left arrow egressive and the right arrow ingressive, and composition proceeds by pullback of spans. Completeness furthermore implies that ιSpan(X, X in , X eg ) ≃ ι(X) via the degenerate spans consisting of identities. Before we dive into the proof, let us record two immediate properties of the construction, the second of which in particular yields inclusions of subcategories

X eg op Span(X) X in , when applied to (X, ιX, X eg ) -→ X ←-(X, X in , ιX).
2.14. Lemma. For an adequate triple (X, X in , X eg ) with reverse X rev = (X, X eg , X in ) as in Example 2.3 2.3, there is a natural canonical equivalence Span(X) op ≃ Span(X rev ).

Proof Proof. We shall prove the first claim; the second then follows from 2.14 2.14. The triple (B, A, ιB) is evidently adequate (cf. Example 2.3 2.3). Note that the space of functors from an adequate triple (X, X in , X eg ) into (B, A, ιB) is equivalent to the space of functors X → B which invert the edges in X eg and take those of X in into A. Applying this to X = Tw r ([n]) shows that the remaining claim is equivalent to the assertion that the source map

s : Tw r ([n]) -→ [n]
is a localisation (necessarily at those maps whose source component is an equivalence). This is in fact true for all ∞-categories C in place of [n], for example since Tw r (C) → C is a cocartesian fibration with contractible fibres (which are always localisations, see Lemma 5.5 5.5 below). The localisation is particularly simple when C admits a terminal object: In that case the source evaluation Tw r (C) → C is even a Bousfield localisation, with fully faithful left adjoint sending c → (c → * ). □

We now turn to the proof of Theorem 2.13 2.13. Instead of following Barwick's strategy of explicitly filling simplices in a point-set implementation of the above construction, our proof will be a simple adaptation of the argument given in [CDH+20b CDH+20b, Section 2.1] for the case of stable ∞-categories (with all maps ingressive and egressive). In fact, we will prove a slightly stronger statement, which will ultimately allow us to deduce Theorem A A. It uses the following definition: 2.16. Definition. Let A be an ∞-category and X an adequate triple. We write Q A (X) for the ∞-category Fun AdTrip (Tw r (A), X), where Tw r (A) is the adequate triple from Example 2.9 2.9 and denote by Q : Cat op × AdTrip -→ AdTrip the resulting functor.

In particular, each adequate triple X gives rise to a natural simplicial diagram Q • (X) in Cat.

2.17. Lemma. Let X be an adequate triple. Then the simplicial ∞-category Q • (X) satisfies the Segal and completeness conditions, that is the Segal maps

Q n (X) -→ Q 1 (X) × Q0(X) Q 1 (X) • • • × Q0(X) Q 1 (X)
are equivalences and

Q 0 (X) Q 0 (X) × Q 0 (X) Q 3 (X) Q 1 (X) × Q 1 (X) ∆ s (s,s) (d {0,2} ,d {1,3} )
is cartesian.

Note that this immediately implies Theorem 2.13 2.13, as ι : Cat → Gpd preserves limits.

Proof. Let J n ⊆ Tw r ([n]) denote the subposet consisting of those (i ≤ j) with j ≤ i + 1, i.e. the zig-zag along the bottom. Note that J n decomposes as an iterated pushout

J n ≃ Tw r ([1]) ∪ Tw r ([0]) Tw r ([1]) ∪ • • • ∪ Tw r ([0]) Tw r ([1])
along the Segal maps; in fact, the nerve N(J n ) is already the iterated pushout of the nerves N(Tw r ([1])) in simplicial ∞-groupoids (see [START_REF] Haugseng | Iterated spans and classical topological field theories[END_REF]Proposition 5.13] for a similar argument). The iterated pullback appearing in the Segal condition is therefore equivalent to the full subcategory J n (X) of Fun(J n , X) spanned by those functors taking left-pointing edges in J n , i.e. those of the form (i ≤ i + 1) → (i ≤ i), to ingressives and right pointing arrows, namely (i ≤ i + 1) → (i + 1 ≤ i + 1), to egressives. Furthermore, this translates the Segal map to the map Q n (X) → J n (X) induced by the restriction Fun(Tw r ([n]), X) -→ Fun(J n , X).

But from the pointwise formula for Kan extensions, one finds that a diagram Similarly, for completeness we first note that the map P → Q 3 (X) from the pullback in question to the lower left corner is fully faithful since the degeneracy Q 0

F : Tw r ([n]) → X lies in Q n (X) if
(X) 2 → Q 1 (X) 2 is (as | Tw r ([1])| ≃ * ).
We claim that its essential image consists exactly of those diagrams whose edges are all equivalences; since also | Tw([3])| ≃ * these are precisely the constant ones, which gives the result. So consider a diagram

F (0 ≤ 3) F (0 ≤ 2) F (1 ≤ 3) F (0 ≤ 1) F (1 ≤ 2) F (2 ≤ 3) F (0 ≤ 0) F (1 ≤ 1) F (2 ≤ 2) F (3 ≤ 3)
all of whose squares are (ambigressive) cartesian and such that the four compositions

F (0 ≤ 2) -→ F (0 ≤ 0), F (0 ≤ 2) -→ F (2 ≤ 2) F (1 ≤ 3) -→ F (1 ≤ 1), F (1 ≤ 3) -→ F (3 ≤ 3)
are equivalences. Then it first follows that, as pullbacks of equivalences, also F (0 ≤ 3) → F (0 ≤ 1) and F (0 ≤ 3) → F (3 ≤ 3) are equivalences and then by two-out-of-six the entire outer slopes are. But then by commutativity of the larger rectangles also F (0 ≤ 1) → F (1 ≤ 1) and F (2 ≤ 3) → F (2 ≤ 2) are equivalences, and then finally as pullbacks thereof also

F (0 ≤ 2) → F (1 ≤ 2) and F (1 ≤ 3) → F (1 ≤ 2). □
Next we shall describe functors into span ∞-categories. More precisely, we will show that the equivalence between functors [n] -→ Span(X) and maps of adequate triples Tw r ([n]) -→ X from Theorem 2.13 2.13 extends to all ∞-categories: 2.18. Theorem. The functors Tw r : Cat AdTrip : Span form an adjoint pair. In other words, for each ∞-category A and each adequate triple, there is a natural equivalence

Map AdTrip (Tw r (A), X) ≃ Map Cat (A, Span(X)),
where Tw r (A) is as in Example 2.9 2.9.

2.19.

Remark. This description of functors into span categories is for example also used in [ [START_REF] Hebestreit | Stable moduli spaces of hermitian forms[END_REF] to perform parametrised surgery on the cobordism ∞-categories Cob(C, Ϙ) from [CDH+20b CDH+20b], which are hermitian refinements of Span(C).

Recall from Definition 2.16 2.16 that we abbreviate Q A (X) = Fun AdTrip (Tw r (A), X).

The theorem will follow readily from the following: 2.20. Proposition. For each adequate triple X, the functor Q -(X) : Cat op → Cat preserves limits.

We will employ the (opposite of the) following criterion: 2.21. Lemma. Let f : D → E be a functor from a small to a cocomplete category, such that the right adjoint in the induced adjunction

|-| f : P(D)
E : S f is fully faithful. Then for a functor F : E → C to another cocomplete category the following are equivalent:

(1) F preserves colimits,

(2) the natural map

|-| F •f =⇒ F • |-| f is an equivalence, and
(3) (i) the natural map f ! (F • f ) =⇒ F is an equivalence, i.e. F is left Kan extended from its restriction along f , and

(ii) |-| F •f : P(D) → C inverts all maps that |-| f inverts.
Applying this to ∆ ⊂ Cat and taking opposites in particular shows that a functor F : Cat op → C preserves limits if and only if it is right Kan extended from its restriction along ∆ op ⊂ Cat op and the restriction F : For (2) ⇒ (3), (ii) follows immediately. To see (i), note that F agrees with the left Kan extension of

∆ op → C is a complete Segal object in C.
|-| F •f = F • |-| f along |-| f , since |-| f is a localisation [Lu09a Lu09a,
-| F •f ≃ F • |-| f as desired. □
Proof of Proposition 2.20 2.20. Since Q(X) is a complete Segal object in Cat by Lemma 2.17 2.17, we have already shown Item (ii) of Lemma 2.21 2.21(3) (3). So we need only show that Q(X) : Cat op → Cat is right Kan extended from its restriction to ∆ op . By the pointwise formula for Kan extensions this means that the tautological map

Q D (X) -→ lim n∈(∆/D) op Q n (X)
is an equivalence for all D ∈ Cat. The diagram ∆/D → ∆ → Cat, giving rise to the right hand side has colimit D and is a typical example of a functor F : I → Cat such that the natural map

( * ) colim i∈I N(F i ) -→ N(colim i∈I F i )
is an equivalence (with both sides evaluating to ND). We will directly show that Q (-) (X) preserves all limits over diagrams with this property.

To this end note that there are natural equivalences of simplicial ∞-groupoids

N(Tw(C)) ≃ i -→ Map Cat ([i] ⋆ [i] op , C) ≃ i -→ Map sGpd (∆ i ⋆ (∆ i ) op , N(C)) .
In particular, since ∆ i ⋆ (∆ i ) op ≃ ∆ 2i+1 is completely compact in sGpd, we find an equivalence

colim i∈I Tw(F i ) Tw colim i∈I F i ∼
because of ( * * ); let us warn the reader that the above map need not be an equivalence without any assumption on F , as the diagram

[1] 0 ← -[0] 1 -→ [1] with pushout [2]
shows. It then follows that the map Fun(Tw(colim

i F i ), X) -→ lim i∈I Fun(Tw(F i ), X)
is an equivalence.

We have to show that this restricts to an equivalence between the full subcategory Q colim Fi (X) on the left (spanned by morphisms of adequate triples) and lim i∈I Q Fi (X) on the right. To this end, notice that any ambigressive pullback square in Tw r (A) is contained (up to equivalence) in Tw r ([3]) for some map [3] -→ A. Consequently, a functor Tw r (A) → X is a morphism of adequate triples if and only if each restriction

Tw r ([3]) -→ Tw r (A) F -→ X is such. Combining this with the fact that colim i∈I Map Cat ([3], F i ) -→ Map Cat [3], colim i∈I F i is an equivalence whenever F satisfies ( * * ), one indeed finds that Q colim Fi (X) ≃ lim i∈I Q Fi (X). □
Proof of Theorem 2.18 2.18. By Proposition 2.20 2.20 ιQ(X) ≃ Map AdTrip (Tw r (-), X) : Cat op → Gpd preserves limits as does Map Cat (-, Span(X)), so by Lemma 2.21 2.21 they are both right Kan extended from ∆ op ⊂ Cat op . Thus they agree if their restrictions to ∆ op agree, which is the case by Theorem 2.13 2.13. □

In fact, one can upgrade Theorem 2.18 2.18 to describe the full functor ∞-category Fun(K, Span(X)):

2.22.

Corollary. There is a canonical equivalence

Fun(A, Span(X)) ≃ Span(Q A (X)), natural in A ∈ Cat and X ∈ AdTrip.
Proof. The left adjoint Tw r : Cat -→ AdTrip preserves the cartesian product. By adjunction, the right adjoint Span then sends the internal hom Q A (X) = Fun AdTrip (Tw r (A), X) from Lemma 2.5 2.5 to the internal hom Fun(A, Span(X)) in Cat. □ 2.23. Example. Since the functors s : Tw r (A) → A and t : Tw r (A) → A op are localisations, it follows easily from the formula of Corollary 2.22 2.22 that the functors Fun(A, X in ) -→ Fun(A, Span(X)) and Fun(A op , X eg ) op -→ Fun(A, Span(X))

are inclusions of (non-full) subcategories for all A ∈ Cat. Alternatively, this statement can also be reduced to the case A = [0] (which we recorded before Lemma 2.14 2.14) by means of Lemma A.4 A.4. In a different direction note that even if X admits pullbacks and carries the trivial structure of an adequate triple with all maps ingressive and egressive, the same is typically not true for Q A (X) = Fun AdTrip (Tw r (A), X); this phenomenon is the basis for the notions of forwards and backwards surgery in [HS21 HS21].

Cartesian fibrations and ∞-categories of spans

The goal of this section is to describe the cocartesian fibration associated to the functor Span : AdTrip → Cat, which is itself given by a certain span ∞-category of adequate triples. As a direct application we will obtain a new and fairly direct proof of the main result of [ [START_REF] Barwick | Dualizing cartesian and cocartesian fibrations[END_REF], which identifies the dual cartesian fibration of a cocartesian fibration in terms of a span construction. We shall more generally analyse the interaction of span categories and (co)cartesian fibrations in Section 5 5 below.

To understand (co)cartesian edges in span ∞-categories, we start by reproducing a criterion due to Barwick. In fact, we provide a proof different from [START_REF]Spectral Mackey functors and equivariant algebraic K-theory I[END_REF]] and inspired by the proof of the additivity theorem for Grothendieck-Witt theory in [CDH+20b CDH+20b, Section 2.6], see Remark 3.3 3.3 for the details of this connection.

3.1. Theorem (Barwick). Let p : Y -→ X be a morphism in AdTrip and let f be an edge in X in such that the following conditions hold:

(1) Every pullback of f along an egressive edge has a lift in Y in which is simultaneously p-cocartesian and p in -cocartesian, for any choice of lift of its source.

(2) Consider any commutative square

σ in Y x ′ z ′ x z g ′ ϕ ψ g
such that p(σ) is an ambigressive pullback in X, the edge g ′ is ingressive, the morphism ϕ is egressive, and the morphism g is a p-cocartesian and ingressive lift of f . Then g ′ is p-cocartesian if and only if the square is an ambigressive pullback.

Then an edge σ : x → y of Span(Y ), represented by a span u

x y

ϕ ψ such that ψ covers f , is Span(p)-cocartesian if ϕ is p eg -cartesian and ψ is p-cocartesian as indicated.
Proof. Unwinding definitions, we have to show that for any span x ← w → z the solid diagram (ignoring the numbers for a moment)

w u • x y z (1) (2)
ϕ ψ (3) (4)
admits an essentially unique dashed filling lying over a given entirely solid diagram in Span(Y ), such that all left pointing arrows are egressive and all right pointing arrows ingressive, and the top square is cartesian. We then first observe that the second condition on f (the image of ψ in Y ) implies that the assertion that the square is cartesian and • → y is egressive is equivalent to the assertion that the map w → • is p-cocartesian.

We can then fill the diagram step by step, as indicated by the numbers in the above diagram, essentially uniquely each time:

(1) There exists a unique egressive filler because ϕ is p eg -cartesian.

(2) The first assumption on f provides a p-cocartesian edge that is also p in -cocartesian, which is automatically unique.

(3) There exists a unique filler making the middle square commute because w → • is p-cocartesian.

(4) There is a unique ingressive filler because w → • is p in -cocartesian.

□

In the special case where f is an equivalence, the two assumptions are automatic, so we obtain:

3.2. Corollary. If p : Y → X is a morphism in AdTrip then a span in Y of the form u x x ′ , ϕ ∼
represents a Span(p)-cocartesian morphism whenever ϕ is p eg -cartesian.

3.3.

Remark. We feel obliged to point out two oversights in the statement of the above result in [Ba17 Ba17, 12.2 Theorem]:

(i) Barwick requires that the edge ϕ be p-cartesian and not p eg -cartesian. Our proof above hopefully makes it transparent why this is not enough. For an explicit counterexample consider X = Y = [1] 2 with p the identity, where we equip the source with the triple structure with the horizontal maps (and the identities) ingressive and everything but the horizontal maps egressive and the target with the same ingressives but all maps egressive.

(ii) As he is working at the point-set level, Barwick has to show that Span(p) is an inner fibration. To this end he assumes that p is an inner fibration, but in fact requires the stronger assumption that p is an isofibration, as can be seen by lifting a 2-simplex in Span(Y ) of the form w

x x x x x f f
where f is some equivalence and all other maps identities.

In order to carry out iterated span constructions as in [BGN18 BGN18] at the point-set level one therefore needs to check that if p is an isofibration, then so is Span(p). This is indeed the case as a direct consequence of equivalences in span ∞-categories being precisely the spans of equivalences (which itself follows from the completeness assertion for ιQ(X) in Lemma 2.17 2.17, see also [Ba17 Ba17, Proposition 3.4]).

3.4. Remark. Suppose that C is stable and consider the target projection p = t : Ar(C) → C, regarded as a map of adequate triples with all maps both ingressive and egressive. One readily checks that every edge f in C satisfies the assumptions of Theorem 3.1 3.1. It follows that Span(Ar(C)) → Span(C) is a cocartesian fibration for every stable C, and thus by Lemma 2.14 2.14 also a cartesian fibration. Taking classifying spaces preserves pullback squares whose right edge is both a cocartesian and cartesian fibration, as can be shown from Quillen's Theorem B (see [START_REF] Steimle | An additivity theorem for cobordism categories[END_REF][START_REF] Steinebrunner | Locally (co)Cartesian fibrations as realisation fibrations and the classifying space of cospans[END_REF] or the proof of [CDH+20b CDH+20b, Theorem 2.5.1] for an account in the present language). Therefore we conclude that

|Span(C)| c →(c→0) ------→ |Span(Ar(C))| t - → |Span(C)| is a fibre sequence. Since K(C) = Ω|Span(C)|
is one possible definition of the algebraic K-space of C, looping this fibre sequence once recovers Waldhausen's additivity theorem

K(Ar(C)) ≃ K(C) × K(C)
since t is clearly split. In this way, Proposition 3.1 3.1 connects to the discussion of additivity in [CDH+20b CDH+20b, Section 2].

As an application of Theorem 3.1 3.1 we can identify the cocartesian fibration corresponding to the functor Span : AdTrip → Cat. More generally, let X be an ∞-category and consider a diagram of adequate triples F : X op -→ AdTrip. Writing p : Y -→ X for the cartesian fibration classified by the underlying functor of F , we can use Proposition 2.7 2.7 to endow Y with the structure of an adequate triple: 3.5. Notation. By forgetting different pieces of the data, the functor F : X op -→ AdTrip gives three diagrams of ∞-categories F, F in and F eg , which are classified by three cartesian fibrations

p : Y -→ X, p in : Y in -→ X, p eg : Y eg -→ X respectively. We write Y fw , Y fw in , Y eg fw for the pullback of Y, Y in , Y eg along ιX → X. Proposition 2.7 2.7 then shows that p : Y -→ X is part of a map of adequate triples Y = Y, Y fw in , Y eg (X, ιX, X)
which gives rise to a map

(3.6) Span(p) : Span(Y ) → Span(X, ιX, X) ≃ X op .
We will show that this is the cocartesian fibration classifying Span • F . First we note: 3.7. Lemma. The functor Span(p) is a cocartesian fibration.

Proof. Corollary 3.2 3.2 immediately implies that Span(p) is a cocartesian fibration, with cocartesian edges given by spans of the form: y ′ y y, where y → y ′ is a p eg -cartesian edge in Y . □

We will now describe the functor classified by this cocartesian fibration:

3.8. Theorem. Let X be an ∞-category. Then there exists a natural equivalence

α : Un cc (Span • (-)) =⇒ Span(Un ct (-) )
of functors Fun(X op , AdTrip) → Cocart(X op ). In other words, for each F : X op -→ AdTrip, the induced cocartesian fibration (3.6 3.6) is classified by the functor Span • F : X op -→ Cat.

We will use the description of (co)cartesian unstraightening from [GHN17 GHN17, Theorem 7.4]:

3.9. Theorem. For a functor F : B -→ Cat, there is a natural equivalence

Un cc (F ) ≃ colim Tw r (B) (s,t) ---→ B × B op F ×B -/ -----→ Cat .
Dually, for a functor F : B op -→ Cat, there is a natural equivalence

Un ct (F ) ≃ colim Tw r (B op ) (s,t) ---→ B op × B F ×B /- -----→ Cat .
We will apply this in the setting of 3.5 3.5 to compute the cocartesian fibration classified by Span(F ) :

X op -→ Cat. We find that Un cc (Span(F )) is equivalent to the colimit of the diagram Tw r (X op ) X op × X Cat. (s,t) Span(F )×(X op ) -/
Now note that there are natural equivalences (X op ) -/ ≃ (X /-) op ≃ Span(X /-, ι(X /-), X /-) and recall that Span commutes with products of adequate triples to conclude that this colimit is the same as that of the diagram Tw r (X op )

(s,t) ---→ X op × X F ×(X /-,ι(X /-),X /-) --------------→ AdTrip Span ---→ Cat.
We shall prove Theorem 3.8 3.8 by showing that the colimit of the first two functors in this composition is precisely Y = (Y, Y fw in , Y eg ) and that this colimit is preserved by Span. We can directly apply the equivalence of Theorem 3.9 3.9 to conclude that

(3.10) Y ≃ colim F (-) × X /-, Y eg ≃ colim F eg (-) × X /-, and Y in ≃ colim F in (-) × X /-
where all colimits are taken over Tw r (X op ). To exhibit Y fw in = Y in × X ιX as a colimit, we need the following lemma: 3.11. Lemma. Let F : B op -→ Cat be a functor. Then the functor Un ct (F ) × B ιB -→ Un ct (F ) is naturally equivalent to the functor

(3.12) colim Tw r (B op ) F × ι(B /-) -→ colim Tw r (B op ) F × B /-.
Proof. Consider the full subcategory of Fun Fun(B op , Cat), Cat /Un ct spanned by those natural transformations Φ -→ Un ct with the following two properties:

(1) At the terminal diagram, Φ( * ) -→ Un ct ( * ) ≃ B exhibits the inclusion of the core of B.

(2) For each F : B op -→ Cat, the induced square

(3.13) Φ(F ) Un ct (F ) Φ( * ) B.
is cartesian.

This subcategory is contractible and the natural transformation Un ct (-) × B ιB -→ Un ct (-) is (by definition) an object in there. It will therefore suffice to show that the composite natural transformation

Φ(F ) := colim Tw r (B op ) F × ι(B /-) -→ colim Tw r (B op ) F × B /-≃ Un ct (F )
D C Tw r (B op ) ιB Ar(B) s B π q t cst t
as well as the following diagram of ∞-categories By construction, the diagram F ′ only depended on the first factor Tw r (B op ); Consequently, it factors as F ′ = F ′′ p, for some F ′′ : D → Cat. Because p is a localisation it is in particular cofinal, so we conclude that colim C F ′ ≃ colim D F ′′ . Let us first compute the left Kan extension π ! F ′′ of F ′′ along the cocartesian fibration π : D → ιB. We can compute this Kan extension as a colimit over the fibres, which are equivalently the fibres of t : Tw r (B op ) → B. Unravelling the definitions then shows that

F ′ : C Tw r (B op ) B op Cat. q s F Since q : C → Tw r (B op
(π ! F ′′ )(b) = colim B op /b -→ B op F -→ Cat ≃ F (b)
since B op /b has a terminal object. We conclude that there is a natural equivalence

Φ(F ) = colim Tw r (B op ) F × ι(B /-) ≃ colim D F ′′ ≃ colim ιB π ! F ′′ ≃ colim ιB F ≃ Un ct (F ) × B ιB.
In particular, we see that Φ( * ) is an ∞-groupoid, so that (1) (1) Φ( * ) → B is a core inclusion, being a colimit of core inclusions. For (2) (2), note that the vertical maps in (3.13 3.13) are cartesian fibrations, so that it suffices to show that the maps on vertical fibres are equivalences. By naturality of the transformation Φ -→ Un ct in the base B, this map on fibres map is simply the map when B = * .

In this case the map (3.12 3.12) is clearly an equivalence. □ 3.14. Proposition. The diagram

Ψ : Tw r (X op ) (s,t) ---→ X op × X F ×(X /-,ι(X /-),X /-) --------------→ AdTrip
has colimit given by the adequate triple Y = (Y, Y fw in , Y eg ). Proof. The equivalences (3.10 3.10) and Lemma 3.11 3.11 show that the colimit of Ψ in Fun(Λ 2 [2], Cat) is given by the triple Y . To see that this is also a colimit in the subcategory of adequate triples, it suffices to verify that the following are equivalent for a functor ϕ : Y = colim Ψ -→ Z:

(1) ϕ preserves ambigressive pullback squares.

(2) the composite Ψ(f op :

x 1 → x 0 ) → colim Ψ → Z does so for each f op ∈ Tw r (X op ).
Now note that (on underlying ∞-categories) the map Ψ(f op : x 1 → x 0 ) -→ colim Ψ can be identified with the functor λ f : F (x 1 ) × X /x0 -→ Y sending (y, g : x 2 → x 0 ) to g * f * (y). Because g * is a map of adequate triples for every map g in X, λ f is a map of adequate triples, which shows that (1) implies (2).

For the converse, note that every ambigressive pullback square in Y can be obtained as a pasting of two types of squares: (a) the pullback of a fibrewise ingressive map along a p-cartesian lift of some g : x 2 → x 0 and (b) the pullback of an ingressive and egressive map in a single fibre Y x0 ≃ F (x 0 ). It suffices to show that each of these squares is the image of an ambigressive pullback square under some λ f . For (a), we can take the image under λ idx 0 of the square in F (x 0 ) × X /x0 formed by an ingressive arrow in F (x 1 ) and the arrow g → id x0 in X/x 0 . For (b), we simply take the image under λ idx 0 of an ambigressive square in F (x 0 ) × {id x0 }. □

Applying Span to the colimiting cocone of Proposition 3.14 3.14, we therefore obtain a natural cocone of the diagram Tw r (X op )

(s,t) ---→ X op × X F ×(X /-,ι(X /-),X /-) --------------→ AdTrip Span ---→ Cat
whose tip is the ∞-category Span(Y ). This induces a natural functor

α F : Un cc (Span(F )) ≃ colim Tw r (X op ) Span(F ) × (X op ) -/ -→ Span(Y ) = Span Un ct (F ) .
Note that the construction of α F is functorial in the diagram F . Applied to the terminal diagram F = * , it yields the canonical equivalence α * : Span(X, ιX, X) -→ X op from Proposition 2.15 2.15. Using this identification, we then obtain the desired natural transformation α : Un cc (Span•(-)) =⇒ Span(Un ct (-) ) between functors Fun(X op , AdTrip) -→ Cat /X op . 3.15. Lemma. For each F : X op -→ AdTrip, the functor α F preserves cocartesian edges.

Proof. Note that F comes with a natural transformation ιF -→ F , where each ιF (x) comes equipped with the (only possible) adequate triple structure where all morphisms are both ingressive and egressive. Unstraightening, this identifies Un ct (ιF ) with the subcategory of Un ct (F ) on the cartesian arrows, with all arrows egressive and only the equivalences being ingressive.

The map ιF -→ F induces a natural transformation ιSpan(F ) ≃ Span(ιF ) -→ Span(F ), where the first equivalence uses that Span(ιF ) ≃ ι(F ) (Proposition 2.15 2.15). By naturality, we then obtain a commuting diagram Un cc ιSpan(F ) Span(Un ct (ιF ) )

Un cc Span(F ) Span(Un ct (F ) ).

α (ιF ) α F
The result now follows since both vertical arrows can be identified with the inclusion of the wide subcategory of cocartesian arrows: For the left arrow this is evident and for the right arrow, this follows from our description of Un ct (ιF ) and Corollary 3.2 3.2. □ Proof of Theorem 3.8 3.8. We have seen that α constitutes a natural transformation between functors into Cat/X op . By Lemma 3.15 3.15, the value of α at a diagram F is a map between cocartesian fibrations over X op which preserves cocartesian edges. It then suffices to show that α F is an equivalence when restricted to every fibre [Lu09a Lu09a, Corollary 2.4.4.4]. Since Span preserves pullbacks (in particular fibres), we can thus reduce to the case X = * , where the map is equivalent to the identity by inspection. □ 3.16. Remark. Given a functor G : X op -→ Cat, one way to write down the cartesian fibration encoding it is to write down the cocartesian fibration which classifies (-) op • G, and then take its opposite. In the case that G = Span(F ), Lemma 2.14 2.14 shows that G op = Span(F rev ). Therefore the previous result allows us to describe its cocartesian straightening. Finally we can again apply Lemma 2.14 2.14 to describe the opposite of the result. In total one finds that

Span(p) : Span(Y, Y in , Y eg fw ) -→ Span(X, X, ιX) ≃ X
is a cartesian fibration which classifies the functor Span(F ) : X op -→ Cat.

As a special case we recover the key insight of [BGN18 BGN18].

3.17. Corollary. Let p : Y -→ X be a cartesian fibration and define Y ∨ -→ X op to be the functor

Span(p) : Span(Y, Y × X ιX, Y † ) -→ Span(X, ιX, X) ≃ X op
where Y † is the subcategory of p-cartesian morphisms (cf. Proposition 2.6 2.6). This gives rise to an equivalence SD cc : Cart(B) -→ Cocart(B op ) which fits into a commuting triangle

Cart(X) Cocart(X op )
Fun(X op , Cat).

SD cc

Str ct Str cc

Proof. Note that once we have exhibited the triangle as commutative, the two-out-of-three property will immediately imply that SD cc is an equivalence. We will provide a natural equivalence SD cc (Un ct (F )) -→ Un cc (F ) for functors F : X op -→ Cat. To this end, consider the functor δ : Cat -→ AdTrip sending C → (C, C, ιC) and note that Span • δ ≃ id. One then has natural equivalences

SD cc (Un ct (F )) ≃ Span Un ct (δ(F )) ≃ Un cc Span(δ(F )) ≃ Un cc (F )
where the middle equivalence is Theorem 3.8 3.8. □

As another special case, we may consider the universal example.

3.18. Construction. Let us define the lax under-category * // lax AdTrip to be the domain of the cocartesian fibration classifying the functor (-) op : AdTrip -→ Cat. One can identify an object of * // lax AdTrip with a tuple (X, x) of an adequate triple and an object x ∈ X, and a map (X, x) → (Y, y) with a map f : X → Y and a map µ : y → f (x) in Y . We will say that (f, µ) is egressive if µ is egressive, and ingressive if µ is ingressive and f is an equivalence.

T op → AdTrip, t → ((F T ) /t , (F T ) /t , (F T ) /t ),
where the functoriality is given by pullback. Therefore Theorem 3.8 3.8 shows that the functoriality of the T -∞-category Fun × T (A eff (T ), C) is induced by the standard functoriality of Span(-). Similarly, when C = E T , the T -∞-category of T -objects in an ∞-category E, one obtains that the T -∞category of T -commutative monoids is equivalent to the functor Fun × (Span((F T ) /-), E) : T op → Cat.

Orthogonal adequate triples and their duals

Work of Barwick, Glasman and Nardin uses the span category construction from Section 2 2 to produce an equivalence between cartesian fibrations and cocartesian fibrations of ∞-categories [ [START_REF] Barwick | Dualizing cartesian and cocartesian fibrations[END_REF]]. In the previous section we provided an alternative proof of this fact by exhibiting it as a specific example of a general phenomenon: The cocartesian fibration classifying Span • F is given by the span construction applied to a particular adequate triple structure on the cartesian unstraightening of F .

In this section, we will describe a different perspective on this result, which is a priori entirely independent from the discussion of (co)cartesian fibrations: The construction of span ∞-categories upgrades to an automorphism (of order 2) of a certain full subcategory of adequate triples spanned by those triple that we will call orthogonal. In Section 5 5 we will show how this construction interacts with various notions of (cartesian) fibrations between such orthogonal adequate triples. In particular we will extend Lanari's results on the relation between cartesian fibrations and factorisation systems from [La19 La19], and thus recover the main result from [ [START_REF] Barwick | Dualizing cartesian and cocartesian fibrations[END_REF]] in a different way.

To define the notion of an orthogonal adequate triple, let us start by briefly recalling Joyal's notion of a factorisation system on an ∞-category C (also called an orthogonal factorisation system), following [Lu09a Lu09a, Section 5. 4.2. Definition. We call an adequate triple (X, X in , X eg ) orthogonal if every ambigressive square in X is cartesian, and the egressives (↠) and ingressives (↣) are the left and right classes respectively of an orthogonal factorisation system on X. We denote the full subcategory of orthogonal adequate triples by AdTrip ⊥ .

4.3. Remark. Since every ambigressive square in an orthogonal adequate triple is automatically cartesian, it follows that AdTrip ⊥ is a full subcategory of Fun(Λ 2 [2], Cat) (in contrast to AdTrip).

The simplest examples are (A, ιA, A) and (A, A, ιA) for any ∞-category A. We will generate more interesting examples by means of the following observation, which follows directly from the mapping properties of cartesian edges: 4.4. Proposition. If X is an orthogonal adequate triple and p : Y -→ X has all p-cartesian lifts over X in , then the adequate triple Y, Y † , p -1 (X eg )) from Proposition 2.6 2.6 is again orthogonal.

In particular, for a cartesian fibration p : Y → X the triple structure on Y with ingressives the p-cartesian edges and egressives the fibrewise maps is orthogonal; in this way our constructions contain those from [ [START_REF] Barwick | Dualizing cartesian and cocartesian fibrations[END_REF]].

Proof. The fact that any ambigressive square is cartesian follows immediately from the proof of Proposition 2.6 2.6. To show that the ingressives and egressives form a factorisation system on Y , let us start by noting that both classes are stable under retracts (for the cartesian morphisms, this uses that they are characterised by a lifting property relative to the base X). Next consider a commutative square

x y

x ′ y ′ , f g
where f ∈ Y eg and g ∈ Y in . We need to exhibit a unique dotted arrow making the diagram commute. After applying p there exists a unique dotted edge in X making the square commute. Since g is p-cartesian we find that there is a unique dotted edge in Y which makes the square commute. Finally, given an edge f : x → y in Y , one can factor p(f ) = gh with g ∈ X in and h ∈ X eg . Taking a p-cartesian lift g ′ of g, we can factor f = g ′ h ′ with h ′ ∈ p -1 (X eg ), so that Y † and p -1 (X eg ) form a factorisation system. □

4.5. Example. A product A × B of two ∞-
* i ! x -→ x -→ j * j *
x is a fibre sequence for each x ∈ X.

We now claim that X has the structure of an orthogonal adequate triple, in which a map is ingressive or egressive if its image under i ! or j * is an equivalence, respectively. By [CDH+20b CDH+20b, Corollary 2.6.1] j * is a cartesian fibration with an edge in X being j * -cartesian if and only if it is an i ! -equivalence. Thus Proposition 4.4 4.4 (applied with target (U, U, ιU )) yields the structure of an orthogonal triple on X as desired. 4.7. Example. For each ∞-category A, the source projection s : Ar(A) -→ A is a cartesian fibration, so that Ar(A) is part of an adequate triple in which a map is egressive (ingressive) if its image under the source map s : Ar(A) → A (target map t : Ar(A) → A) is an equivalence. For every A → B, the induced functor Ar(A) -→ Ar(B) preserves ingressive and egressive maps, so that we obtain a functor Ar : Cat AdTrip ⊥ .

In fact, Example 4.7 4.7 provides the free orthogonal adequate triple generated by A:

4.8. Proposition. The functor Ar : Cat -→ AdTrip ⊥ is the left adjoint to the forgetful functor AdTrip ⊥ -→ Cat taking underlying ∞-categories. More precisely, the degeneracy map cst : A -→ Ar(A) provides the unit transformation exhibiting Ar as the left adjoint to the forgetful functor.

Proof. It will suffice to define a natural counit map and provide natural homotopies for the triangle identities. Given an adequate triple X, let ϵ X : Ar(X) -→ X be the unique functor that fits into a commuting diagram

Fact(X) Fun([2], X) Ar(X) X. compose ∼ ev1 ϵ X
Here the left vertical functor is the equivalence from Remark 4.1 4.1, a section of which provides a functorial egressive-ingressive factorisation. Since all three solid maps are functorial in the adequate triple X, the map ϵ X is functorial in X. Explicitly, the value of ϵ X on a map µ : f → g in Ar(X) is given by the middle vertical arrow in

• • • • • • f µ0 g µ1
From this we see that ϵ X is a map of adequate triples. Indeed, if µ is ingressive (i.e. µ 1 is an equivalence), then the middle horizontal map is ingressive by the right cancellation property for ingressives [Lu09a Lu09a, Proposition 5.2.8.6 (3)], and likewise in the egressive case where µ 0 is an equivalence. It remains to provide the triangle identities. If X is an adequate triple, then the composite

X Ar(X) X cst ϵ
is evidently naturally equivalent to the identity (using that the functorial factorisation Ar(X) → Fact(X) sends degenerate arrows to constant diagrams). Furthermore, for an ∞-category A, the composite

Ar(A) Ar(Ar(A)) = Fun([1] × [1], A) Ar(A) cst ϵ
sends an arrow f to f ∼ -→ f and then applies the ingressive-egressive factorisation to this natural equivalence. This is again naturally equivalent to the identity. □ 4.9. Proposition. For any adequate triple (X, X in , X eg ), the subcategory inclusions (X eg ) op , X in → Span(X) from Proposition 2.15 2.15 give an orthogonal factorisation system on Span(X). If the triple (X, X in , X eg ) is furthermore orthogonal, then (Span(X), X in , (X eg ) op ) is again an orthogonal adequate triple.

Proof. To prove the first claim, we use the criterion from [Lu09a Lu09a, Proposition 5.2.8.17], which requires us to check that factorisations as in the second item of the definition of an orthogonal factorisation systems are unique. This translates to the statement that the ∞-groupoid of dotted extensions of the solid diagram

• • • • • • ∼ ∼ ∼ ∼
is contractible, which is obvious. Now assume the adequate triple (X, X in , X eg ) is orthogonal. To prove that (Span(X), X in , (X eg ) op ) is adequate, we must show that Span(X) admits ambigressive pullbacks. By Lemma 2.14 2.14, it is equivalent to check that Span(X rev ) admits ambigressive pushouts, which turns out to be notationally slightly more convenient. Let us start by observing that by adjunction a commuting square in Span(X rev ) with horizontal arrows in (X in ) op and vertical arrows in X eg corresponds to a (dotted) diagram ( * )

• • • • • • • • • ∼ ∼ ∼ ∼ ∼ ∼
in X, whose top right and bottom left squares are pullbacks (so that the arrows with the blue labels are equivalences as well). Such a diagram is uniquely determined by its solid part, i.e. the left column and top row. Indeed, we can first fill the top half with equivalences in the vertical direction.

Then we can uniquely factor the composite in the bottom left square as an egressive followed by an ingressive edge (the resulting square is cartesian because X is an orthogonal adequate triple). Finally, we fill the bottom right square with equivalences as indicated.

We can therefore conclude that the triple (Span(X), X in , (X eg ) op ) is both adequate and orthogonal as soon as we can show that any diagram as above defines a pushout square in Span(X rev ). This is, however, rather unpleasant to do directly. We shall instead use the fact that pushout squares in an ∞-category C are precisely the cocartesian edges of the source map s : Ar(C) → C.

Here we regard the diagram ( * ) as a morphism in Ar(Span(X rev )) from the left vertical span to the right vertical span, so that the top horizontal span is the image of this morphism under s.

By Corollary 2.22 2.22 we can identify the functor s : Ar(Span(X rev )) → Span(X rev ) with

Span(Q 1 X rev ) Span(ev (0≤0) )
---------→ Span(X rev ).

Put into this form, we can apply Barwick's criterion 3.1 3.1 for cocartesian edges, or more precisely Corollary 3.2 3.2. It tells us that the diagram ( * ) defines a Span(ev (0≤0) )-cocartesian edge if its left pointing half defines an (ev (0≤0) ) eg -cartesian edge in Q 1 X rev .

To see that it does, recall from Lemma 2.5 2.5 that an edge in Q 1 X rev = Fun AdTrip (Tw r ([1]), X rev ) is egressive if it is pointwise ingressive and its ambigressive square is cartesian. Consider thus the (solid) lifting problem (ignoring the red arrow for a moment)

• • • • • • • • •. ∼ ∼
whose right hand column is the left half of ( * ), and whose lower bent rectangle is a pullback. We have to show that it admits a unique dotted filling (whose lower square is then automatically a pullback by pasting).

Considering the top half of the diagram, there is an essentially unique choice for the upper dotted arrow, because the middle upwards pointing map is an equivalence, and the composite of ingressives it is itself ingressive. By composition we also obtain the red arrow.

The bottom row of the diagram together with the middle left dot, then fit into a diagram

• • • •
Because the egressives and ingressives in X determine an orthogonal factorisation system, this lifting problem has a unique filler. Moreover, because the right class of an orthogonal factorisation system satisfies right cancellation [Lu09a Lu09a, Proposition 5.2.8.6 (3)], this filler is necessarily ingressive. This provides the necessary lift, and concludes the argument. □ 4.10. Definition. We shall refer to (Span(X), X in , (X eg ) op ) as the dual of an orthogonal adequate triple (X, X in , X eg ), and denote by Span ⊥ : AdTrip ⊥ → AdTrip ⊥ the resulting functor.

4.11. Remark. Unravelling the proof above, one arrives at the following description of induced maps out of the ambigressive pushouts in Span(X rev ) from the previous proof: Suppose we are given a commutative square in Span(X rev ) represented by a diagram

• • • • • • • • •. g f α β γ k
whose lower left and upper right squares are cartesian. Factor the composite f g into an ingressive edge f ′ followed by an egressive edge g ′ . From the proof above, we learn that the pushout of the two spans starting in the upper left corner is given by the source of g ′ (or equivalently the target of f ′ ). The induced map out of it unwinds to the span

• • • j k
where j is the solution of the lifting problem

• • • • g ′ α f ′ •β γ j
As a consistency check, note that j is indeed ingressive, as both γ and g ′ are.

This remainder of this section is dedicated to proving the following result: 4.12. Theorem. The functor Span ⊥ is an auto-equivalence of AdTrip ⊥ , which is its own inverse.

Below in Corollary 5.7 5.7 we will upgrade this equivalence to a C 2 -action. In the large our strategy of proof is the same as that of Barwick, Glasman and Nardin in [BGN18 BGN18], who essentially treat the special case corresponding to factorisation systems given by (co)cartesian edges and morphisms lying in a single fibre.

We have to construct a natural equivalence between Span ⊥2 = Span ⊥ • Span ⊥ and the identity of AdTrip ⊥ . To do this, let us start by describing the composite Span ⊥2 more explicitly. 4.13. Observation. Let us write ιQ ♯ : sGpd -→ sGpd for the functor sending a simplicial ∞groupoid S to the simplicial ∞-groupoid given by

ιQ ♯ n (S) = Map sGpd N Tw r ([n]), S .
By the definition of span ∞-categories (see Lemma 2.17 2.17), there is a natural transformation of functors AdTrip -→ sGpd N Span(X) ιQ ♯ (N(X)).

In every simplicial degree, this is given by the inclusion of path components

Map AdTrip (Tw r ([n]), X) ⊆ Map Cat (Tw r ([n]), X).
Applying this reasoning twice, one sees that for X in AdTrip ⊥ , there is a natural map of simplicial ∞-groupoids

N Span ⊥2 (X) ιQ ♯ N(Span ⊥ (X)) ιQ ♯ ιQ ♯ (NX)
which is an inclusion of path components in each degree. To unravel the target of the above map, let us denote by Tw (2) (A) = Tw r (Tw r (A)) the twofold iterated twisted arrow ∞-category of an ∞-category A. We then obtain a natural equivalence

ιQ ♯ n ιQ ♯ (NX) ≃ Map sGpd N Tw r ([n]), ιQ ♯ (NX) ≃ lim [m]∈∆/N(Tw r [n]) Map sGpd N Tw r ([m]), NX ≃ Map sGpd N Tw (2) ([n]) , NX ≃ Map Cat (Tw (2) ([n]), X).
Here the third line uses that N Tw r (-) preserves those colimits of ∞-categories that are preserved by the nerve functor (see the proof of Proposition 2.20 2.20). Summarising, we see that there is a map of simplicial ∞-groupoids, depending functorially on X ∈ AdTrip ⊥ , which is a degreewise inclusion of path components

(4.14) N Span ⊥ • Span ⊥ (X) Map Cat Tw (2) (-), X .
To identify the essential image of (4.14 4.14), let us make the following construction: 4.15. Construction. Note that Tw (2) ([n]) is equivalent to the poset whose objects are tuples abcd

with 0 ≤ a ≤ b ≤ c ≤ d ≤ n, corresponding to a map (a ≤ d) -→ (b ≤ c) in Tw r ([n]
). The partial order is then given by abcd

≤ a ′ b ′ c ′ d ′ when a ≤ a ′ ≤ b ′ ≤ b ≤ c ≤ c ′ ≤ d ′ ≤ d.
We define the following four wide subcategories of Tw (2) ([n]):

(1) Tw (2) ([n]) 1 is the subcategory spanned by the edges abcd → a ′ bcd.

(2) Tw (2) ([n]) 2 is the subcategory spanned by the edges abcd → ab ′ cd.

(3) Tw (2) ([n]) 3 is the subcategory spanned by the edges abcd → abc ′ d.

(2) ([n]), X) consisting of maps f : Tw (2) ([n]) -→ X that restrict to Tw (2) ([n]) 1 X in Tw (2) ([n]) 2 ιX Tw (2) ([n]) 3 X eg Tw (2) ([n]) 4 ιX. f f f f Proof. A functor f : [n] -→ Span ⊥ • Span ⊥ (X)
corresponds by adjunction to a map Tw r ([n]) -→ Span ⊥ (X) of adequate triples, i.e. a map of

∞-categories f ′ : Tw r ([n]) -→ Span ⊥ (X) such that f ′ (Tw r ([n]) in ) ⊆ Span ⊥ (X) in and f ′ (Tw r ([n]) eg ) ⊆ Span ⊥ (X) eg .
Here we importantly use that every ambigressive square in the target is automatically a pullback, otherwise the functor f ′ would have to furthermore preserve ambigressive pullbacks. By Corollary 2.18 2.18, the map of ∞-categories underlying f ′ corresponds itself to a map Tw r (Tw r ([n])) -→ X of adequate triples, i.e. a map f ′′ : Tw r (Tw r ([n])) -→ X such that

f ′′ Tw r Tw r ([n]) in ⊆ X in and f ′′ Tw r Tw r ([n]) eg ⊆ X eg .
Let us now unravel these conditions using the description of Tw , where the left map is in X eg . Combining all this, the above conditions translate as follows:

(1) Every abcd ≤ a ′ bcd ′ is mapped to X in .

(2) Every abcd ≤ ab ′ c ′ d is mapped to X eg .

(3) Every abcc ≤ a ′ bcc is sent to X in and every abcc ≤ ab ′ cc is sent to an equivalence.

(4) Every aacd ≤ aac ′ d is sent to X eg and every aacd ≤ aacd ′ is sent to an equivalence.

Note that these conditions are certainly implied by the ones from the proposition. Conversely, given these conditions, the ones from the statement follow: for example, the map abcd ≤ abcd ′ fits into a square

aacd aacd ′ abcd abcd ′
The top horizontal arrow is mapped to an equivalence in X by (4), the vertical arrows are sent to maps over X eg by (2). In particular, the bottom arrow maps to X eg , but also to X in by (1). An edge which is in both the left and right part of an orthogonal factorisation system is an equivalence, and therefore abcd ≤ abcd ′ maps to an equivalence. □

Let us write LTw (2) ([n]) for the localisation of Tw (2) ([n]) at the classes of maps (2) (2) and (4) (4) and let LTw (2) ([n]) 1 and LTw (2) ([n]) 3 be its wide subcategories generated by the images of the maps (1) (1) and (3) (3) from Construction 4.15 4.15. Since each map in ∆ induces a map Tw (2) ([m]) -→ Tw (2) ([n]) preserving these four classes of maps, it follows that the triple of ∞-categories

(4.17) LTw (2) ([n]), LTw (2) ([n]) 1 , LTw (2) ([n]) 3 ∈ Fun Λ 2 [2], Cat)
depends functorially on [n]. Proposition 4.16 4.16 implies that the space N(Span ⊥2 (X)) n is naturally equivalent to the space of maps in Fun(Λ 2 [2], Cat)

LTw (2) ([n]), LTw (2) ([n]) 1 , LTw (2) ([n]) 3 -→ X, X in , X eg .
We will now identify the localised triples (4.17 4.17) more explicitly. To do this, consider the composition

[k] × [1] → [k] ⋆ [k] → [k] ⋆ [k] op ⋆ [k] ⋆ [k] op ,
where the first map is the unique natural transformation between the two inclusions

[k] → [k] ⋆ [k]
of the summands into the join. This induces a natural transformation

z : Tw (2) (-) -→ Ar(-)
sending a tuple abcd in Tw (2) ([n]) to ac in Ar([n]). Recall from Example 4.7 4.7 that the arrow category Ar([n]) comes equipped with the structure of an adequate triple in which a map is egressive (ingressive) if its image under the source (target) map is an equivalence. The map z then sends the maps (2) (2) and (4) (4) to equivalences, the maps (1) (1) to ingressive arrows and the maps (3) (3) to egressive arrows. Consequently, we obtain a natural transformation of cosimplicial diagrams in Fun(Λ 2 [2], Cat)

(4.18) LTw (2) (-), LTw (2) (-) 1 , LTw (2) (-) 3 -→ Ar(-).
Restriction along this diagram of pairs then induces a natural map of simplicial ∞-groupoids

ζ : N(X) ≃ Map AdTrip ⊥ Ar(-), X -→ Map Fun(Λ2[2],Cat) LTw (2) (-), X ≃ NSpan ⊥2 (X).
for every X ∈ AdTrip ⊥ . Here the first equivalence follows from the fact that Ar([n]) is the free orthogonal adequate triple generated by [n], by Proposition 4.8 4.8.

4.19. Lemma. For every X ∈ AdTrip ⊥ , the natural transformation ζ is an equivalence of complete Segal spaces.

Proof. Since the domain and the target of ζ are complete Segal spaces, ζ is an equivalence of simplicial ∞-groupoids as soon as it induces an equivalence in simplicial degrees 0 and 1. It therefore suffices to show that the natural transformation (4.18 4.18) is an equivalence of triples of ∞-categories for n = 0 or n = 1.

For n = 0 this is evident since the domain and codomain of (4.18 4.18) are both just a point. For n = 1, at the level of the underlying ∞-categories, we have to verify that the functor z : Tw ) is an equivalence of (adequate) triples, as desired. □

Proof of Theorem 4.12 4.12. Lemma 4.19 4.19 shows that after post-composing with the forgetful functor U : AdTrip → Cat, there is a natural equivalence ζ : U (X) U (Span ⊥2 (X)).

∼

It remains to verify that this equivalence on the underlying ∞-categories also identifies the subcategories of ingressives and egressives. But this is a direct consequence of the naturality of ζ in X ∈ AdTrip ⊥ , since the inclusions Span ⊥2 (X) in → Span ⊥2 (X) ← Span ⊥2 (X) eg arise as the functors underlying the maps of adequate triples Span ⊥2 (X in , X in , ιX) -→ Span ⊥2 (X) ←-Span ⊥2 (X eg , ιX, X eg ) and the same for X in → X ← X eg . □

Cartesian fibrations between orthogonal adequate triples

The purpose of this section is to analyse the interaction between orthogonal adequate triples and cartesian fibrations. In particular, we show that orthogonal adequate triples (X, X in , X eg ) uniquely correspond to cartesian fibrations with contractible fibres, by taking X to X → X[(X eg ) -1 ], generalising results of Lanari [START_REF] Lanari | Cartesian factorization systems and pointed cartesian fibrations of ∞-categories[END_REF]]. More generally, we study various kinds of fibrations between adequate triples that are preserved (or exchanged) by the dualisation equivalence from Theorem 4.12 4.12. When the base X arises from a product of two ∞-categories (Example 4.5 4.5), we can identify these fibrations with the two-variable fibrations studied extensively in [HHLN21 HHLN21]; this specialisation will occupy the next section.

The main notion this section will be concerned with is: 5.1. Definition. We will say that a map p : Y → X of orthogonal adequate triples is an ingressive cartesian fibration if Y admits all p-cartesian lifts over X in and these precisely make up Y in . Given an orthogonal adequate triple X, we will write Cart in (X) ⊆ AdTrip ⊥ /X for the full subcategory on the ingressive cartesian fibrations.

Observation.

Let us make the following remarks:

(1) Let p : Y -→ X be an ingressive cartesian fibration. By definition, Y has all p-cartesian lifts over X in and Y in is the wide subcategory on the p-cartesian lifts of ingressive morphisms. Since Y in and Y eg form a factorisation system, this determines Y eg uniquely and we find that Y arises exactly from the construction in Proposition 4.4 4.4; in particular Y eg = p -1 (X eg ).

(2) For a map of orthogonal adequate triples p : Y → X, an ingressive edge g in Y is p-cartesian if and only if it is p in -cartesian, since lifting an arbitrary map f : x → x ′ in an orthogonal adequate triple along an ingressive g : z → x ′ is equivalent to lifting f ′ : w → x ′ along g where f = f ′ h is the factorisation of f into an egressive followed by an ingressive.

(3) Ingressive cartesian fibrations are stable under pullbacks. In particular, if p : Y -→ X is an ingressive cartesian fibration and x ∈ X, then the fibre over x carries the structure of an orthogonal adequate triple (Y x , ιY x , Y x ).

We start with the following partial converse to Proposition 4.4 4.4: 5.3. Proposition. Let (X, X in , X eg ) be an orthogonal adequate triple. Then the map

p : (X, X in , X eg ) -→ (X[(X eg ) -1 ], X[(X eg ) -1 ], ι(X[(X eg ) -1 ]))
is an ingressive cartesian fibration. In other words, p : X → X[(X eg ) -1 ] is a cartesian fibration, inverts only the egressives in X, and the ingressive maps in X are precisely the p-cartesian edges.

Before we dive into the proof, we spell out some consequences. To this end, let us write Cart for the subcategory of Ar(Cat) spanned by objects the cartesian fibrations and by morphisms those squares

X Y S T p f q
such that f sends p-cartesian edges to q-cartesian edges. By Proposition 5.3 5.3, the assignment X -→ (X → X[(X eg ) -1 ]) extends to a functor

L eg : AdTrip ⊥ -→ Cart,
which is fully faithful by the universal property of a localisation. We claim that the assignment of the orthogonal adequate triple (Y, Y † , p -1 (ιS)) to a cartesian fibration p : Y → S defines a (fully faithful) right adjoint to

L eg : Both Map Cart X → X[(X eg ) -1 ], p and Map AdTrip X, (Y, Y † , p -1 (ιS))
are given by those functors X → Y that take ingressives and egressives to p-cartesian and fibrewise maps, respectively. 5.4. Proposition. The functor L eg restricts to an equivalence between the category AdTrip ⊥ and the full subcategory of Cart spanned by those cartesian fibrations whose fibres all have contractible realisations.

Given the discussion above, this follows immediately from the equivalence between the first and third items in the following: 5.5. Lemma. For a cartesian fibration p : A → B the following conditions are equivalent:

(1) p is a localisation, (2) p is cofinal, and

(3) the realisations of the fibres of p are contractible.

Proof. It is generally true that localisations are cofinal (and coinitial): If p is a localisation it follows straight from the definition of Kan extension as an adjoint to restriction that any functor F : B → C is right (and left) Kan extended from its restriction along p, and thus has the same colimit as F p. This proves (1) ⇒ (2).

To see that (2) ⇔ (3), we observe that the inclusion To see the relationship between this equivalence and ours, note that every fibre X s of a pointed cartesian fibration admits a final object, given by the source x of a cartesian edge x → * , which lies over the unique map s → * in S. Therefore the fibre of pointed cartesian fibrations have contractible realisations, and we conclude that our equivalence extends that of [START_REF] Lanari | Cartesian factorization systems and pointed cartesian fibrations of ∞-categories[END_REF].

p -1 (b) ⊆ b/p, a -→ (a,
In fact, in the case of pointed cartesian fibrations Lemma 5.5 5.5 can be sharpened: It is easy to check that assigning to some s ∈ S the source x of a cartesian lift x → * of s → * defines a right adjoint to p, so that any pointed cartesian fibration is in fact a left Bousfield localisation. As a typical application, one finds that an exact functor between stable ∞-categories is a cartesian fibration if and only if it is a right split Verdier projection (in the language of [CDH+20b CDH+20b, Appendix A]), and both a cartesian and a cocartesian fibration if and only if it participates (as j * ) in a stable recollement as in Example 4.6 4.6. 5.7. Corollary. The self-equivalence Span ⊥ upgrades to a C 2 -action on AdTrip ⊥ .

Proof. According to Corollary 3.17 3.17 above, the embedding L eg : AdTrip ⊥ → Cart from Proposition 5.4 5.4 translates Span ⊥ to the functor that takes a cartesian fibration to its fibrewise opposite (defined via straightening, postcomposing with (-) op : Cat → Cat, and then unstraightening). Since Aut(Cat) ≃ C 2 by Toën's theorem [To05 To05], this latter operation defines a C 2 -action, hence so does the former. □

It would be interesting to provide a more direct construction of the coherences for this C 2 -action that does not rely on the (un)straightening equivalence, but we have not pursued this.

As another consequence, we obtain the following description of free cartesian fibrations, a generalisation of which was already proven by different means in [GHN17 GHN17]: 5.8. Corollary. Consider the functor Cart -→ Cat sending a cartesian fibration p : Y → X to its domain Y . This functor admits a left adjoint sending A to the cartesian fibration s : Ar(A) → A.

N rel (X)(n) Fun([n], X)
for the subcategory of all functors [n] -→ X and natural transformations which are pointwise egressive. The core of this ∞-category is simply Map Cat ([n], X). Taking geometric realisations, we then obtain a map of simplicial ∞-groupoids

N(X) = Map Cat (-, X) N rel (X)
such that the induced map on associated ∞-categories is the localisation

X -→ X[(X eg ) -1 ].
We are now going to give a smaller description of N rel (X) , which will show in particular that it already satisfies the Segal condition (so that the associated ∞-category is just its completion). To this end, consider the full subcategories N rel in (X)(n) ⊆ N rel (X)(n) whose objects are functors [n] -→ X with values in X in . For each α ∈ N rel (X)(n), there exists an initial α in N rel in (X)(n) equipped with a map from α, given by the unique factorisation (starting at the end)

α(0) α(1) . . . α(n -1) α(n) α(0) α(1) . . . α(n -1) α(n).
It follows that each inclusion N rel in (X)(n) → N rel (X)(n) admits a left adjoint. We thus obtain a diagram of simplicial categories (5.9)

N(X in ) N rel in (X) N rel in (X) N(X) N rel (X) N rel (X) ∼
where the objects on the left and on the right are simplicial ∞-groupoids and the right vertical map is an equivalence, since it is given in each simplicial degree by the geometric realisation of a right adjoint functor. The simplicial ∞-category N rel in (X) satisfies the Segal conditions: The equivalence Fun(

[n], X) ≃ Fun([1], X) × X • • • × X Fun([1
], X) identifies the subcategories from N rel in (X) on both sides. Furthermore, the 'target' functor (5.10)

t : N rel in (X)([1]) N rel in (X)({1}) = X eg
is both a left fibration (since each ↣↠ fits into a unique ambigressive square) and a right fibration (since ingressive maps can be pulled back along egressive maps), and hence a Kan fibration (i.e. it can be presented by a Kan fibration between quasicategories). This implies that pullbacks along t induce pullbacks after taking realisations, so that the equivalent simplicial ∞-groupoids on the right in (5.9 5.9) also satisfy the Segal conditions.

We will use this description of X[(X eg ) -1 ] in terms of N rel in (X) to identify its over-categories. To this end, recall that for any Segal ∞-groupoid S and s ∈ S(0), the simplicial ∞-groupoid

(5.11) S/s (n) = S([n + 1]) × S({n+1}) {s}
is again a Segal ∞-groupoid, such that the associated ∞-category ac(S/s) ≃ ac(S)/s is a model for the over-category of s [CDH+20b CDH+20b, Lemma 2.4.7]. Let us now take an object x ∈ X and consider the maps of simplicial ∞-groupoids induced by the top row in (5.9 5.9)

(5.12) N(X in )/x N rel in (X)/x N rel in (X) /x.
Here the middle term is given by first taking (5.11 5.11) at the level of simplicial ∞-categories, and then taking realisations. Unraveling the definitions, the simplicial ∞-category N rel in (X)/x is given in degree n by the ∞-category with objects α(0

) ↣ • • • ↣ α(n) ↣ x and morphisms α(0) α(1) . . . α(n) x β(0) β(1) . . . β(n) x.
Since the ingressives and egressives form a factorisation system, all vertical maps are equivalences. Consequently, the first map in (5.12 5.12) is an equivalence (even before taking geometric realisations).

In addition, since the target map (5.10 5.10) is a Kan fibration, so is the map N rel in (X)([n + 1]) -→ N rel in (X)({n + 1}) (being a composite of its base changes). It follows that the pullbacks (5.11 5.11) are preserved under taking classifying spaces, so that the second map in (5.12 5.12) is an equivalence.

Since N rel in (X) has associated ∞-category X[(X eg ) -1 ], the associated ∞-category of N rel in (X) /x is X[(X eg ) -1 ]/p(x).
All in all, we have therefore found that for each object x ∈ X, the composite functor

X in /x X/x X[(X eg ) -1 ]/p(x) ι p
is an equivalence. Under this equivalence, the functor p is identified with the functor X/x -→ X in /x sending each y → x to the ingressive part y ′ ↣ x of its functorial egressive-ingressive factorisation. This functor admits a fully faithful right adjoint, whose essential image is given by the ingressive maps to x. By [AF20 AF20, Lemma 2.16], this implies that p is a cartesian fibration and that an arrow in X is p-cartesian if and only if it is ingressive.

In particular, we can apply Proposition 4.4 4.4 to obtain an orthogonal triple structure on X whose ingressives are the p-cartesian arrows and whose egressives are the maps that are inverted by p. Since the first class coincides with X in , the second class coincides with X eg . □

We now turn to more restrictive types of fibrations between orthogonal triples: 5.13. Definition. A ingressive cartesian fibration p : Y -→ X is said to be (1) an op-Gray fibration if p eg : Y eg -→ X eg is a cartesian fibration, and

(2) a curved orthofibration if p eg : Y eg -→ X eg is a cocartesian fibration.

For X an orthogonal adequate triple, we will write CrvOrtho(X), OpGray(X) for the subcategories of Cart in (X) spanned by the curved orthofibrations and op-Gray fibrations, and maps between them preserving (in addition) p eg -(co)cartesian morphisms.

5.14. Remark. To avoid confusion we would like to explicitly mention that while arrows with the decoration ↠ denoted cartesian edges in [HHLN21 HHLN21], with our conventions it is exactly the ingressive edges in Y which are all p-cartesian and that the egressive edges are almost never all p-cartesian.

The naming schema above is inspired by the definitions of [HHLN21 HHLN21

]. We will show in Section 6 6 that in the case X = (A × B, A × ιB, ιA × B), the definitions above reduce to the corresponding objects of [ [START_REF] Haugseng | Lax monoidal adjunctions, two-variable fibrations and the calculus of mates[END_REF].

Note that Cart in (A, A, ιA) is simply the ∞-category of cartesian fibrations over A, as are the two subcategories CrvOrtho and OpGray in this case. On the other hand, Cart(A, ιA, A) is just Cat/A, while OpGray(A, ιA, A) = Cart(A) and CrvOrtho(A, ιA, A) = Cocart(A).

One can informally think about an op-Gray fibration over X as encoding a lax functor from X to Cat, which is strong on 2-simplices in X in or X eg , as well as 2-simplices of the form x 0 ↠ x 1 ↣ x 2 . This is substantiated (using the scaled straightening construction of Lurie [START_REF]∞, 2)-categories and the Goodwillie calculus I[END_REF]) by the following observation: 5.15. Lemma. An op-Gray fibration p : Y -→ X is in particular a locally cartesian fibration. Furthermore, for each 2-simplex σ : [2] -→ X arising from the composition of an ingressive and an egressive map

x 0 ↠ x 1 ↣ x 2 , the restriction σ * (p) : σ * Y -→ [2] is a cartesian fibration.
Proof. Note that the second part implies that p is a locally cartesian fibration since every morphism in X fits into a 2-simplex σ as indicated. To see that σ * (p) is a cartesian fibration, one simply notes that it admits enough cartesian lifts of x 1 ↣ x 2 (since p does) and locally cartesian lifts over x 0 ↠ x 1 . □ 5.16. Lemma. Let p : Y -→ X be an op-Gray fibration between orthogonal adequate triples. Then the following are equivalent:

(1) The underlying functor p : Y -→ X is a cartesian fibration.

(2) For every ambigressive square

(5.17)

y 00 y 01 y 10 y 11 f f ′ in which f ′ is p eg -cartesian, f is p eg -cartesian as well.
Proof. Assuming (1) (1), the uniqueness of cartesian lifts implies that an egressive morphism in Y is p eg -cartesian if and only if it is p-cartesian. Except for f ′ , all arrows in the square (5.17 5.17) are therefore p-cartesian, which implies that f ′ is p-cartesian as well.

For the converse, Lemma 5.15 5.15 already asserts that p is a locally cartesian fibration. It then suffices to verify that locally p-cartesian morphisms are closed under composition. To this end, let f and g be two composable locally p-cartesian morphisms in Y and consider the diagram

• • • • • • f g
Here the two triangles factor f and g into an egressive followed by an ingressive morphism; Lemma 5.15 5.15 shows that the resulting maps are all locally p-cartesian. The top right square is obtained by factoring the down-right composite into an egressive, followed by an ingressive map. Condition (2) (2) then implies that all individual arrows depicted in the above picture are locally p-cartesian (and the horizontal ones are p-cartesian). Since p eg was a cartesian fibration, the top composite is then locally p-cartesian, and since the right vertical composite is p-cartesian, the composite gf is locally p-cartesian as well. □ 5.18. Definition. We will say that a map p : Y -→ X of orthogonal adequate triples is a cartesian fibration if it satisfies the equivalent conditions of Lemma 5.16 5.16. Dually, a curved orthofibration is called an orthofibration if for each ambigressive square (5.17 5.17) in which f is p eg -cocartesian, f ′ is p eg -cocartesian as well.

Let us write Cart(X) ⊆ OpGray(X) and Ortho(X) ⊆ CrvOrtho(X) for the full subcategories on cartesian and orthofibrations; this notation is justified by the following observation: 5.19. Observation. Note that the datum of a cartesian fibration of orthogonal adequate triples p : Y -→ X is equivalent to that of a cartesian fibration between the underlying ∞-categories. In addition, a map in Cart(X) is required to preserve cartesian lifts over ingressive and egressive maps in X; since every map in X factors into an egressive map followed by an ingressive one, it follows that the maps in Cart(X) preserve all cartesian arrows. In other words, Cart(X) is simply equivalent to the ∞-category of cartesian fibrations over the underlying ∞-category of X.

The main result about these various types of fibration is that they behave well under dualisation: 5.20. Theorem. Let X be an orthogonal adequate triple. Then the natural equivalence

Span ⊥ : AdTrip ⊥ /X -→ AdTrip ⊥ /Span ⊥ (X) restricts to natural equivalences Cart in (X) ≃ Cart in (Span ⊥ (X)) OpGray(X) ≃ CrvOrtho(Span ⊥ (X)) Cart(X) ≃ Ortho(Span ⊥ (X)).
Furthermore, for each fibration p : Y -→ X, the dual fibration Span ⊥ (Y ) -→ Span ⊥ (X) has fibres given by the opposites of the fibres of p.

In the simple case X = (A, A, ιA) this result specialises to the statement that taking span categories for the adequate triple structure given by the cartesian and fibrewise maps provides an equivalence Span :

Cart(A) = Cart in (X) → Cart in (Span ⊥ (X)) = Cart(A).
We already encountered this equivalence, first established by Barwick, Glasman and Nardin [BGN18 BGN18, Theorem 1.7], in Corollary 3.17 3.17. In the case X = (A, ιA, A), on the other hand, the result simply specialises to the equivalence

Cat/A = Cart in (X) → Cart in (Span ⊥ (X)) = Cat/A op ,
given by taking opposites. In the case of a general X, our equivalence combines these two extremes, as we shall discuss in the case of two-variable fibrations from [HHLN21 HHLN21] in the next section.

Proof. Note that naturality of these equivalences follows directly from Span ⊥ being a functor and taking over-categories being natural in the base.

Let us start with the first equivalence. Since Span ⊥ is its own inverse, it suffices to verify that for an ingressive cartesian fibration p : Y -→ X, the dual map q = Span ⊥ (p) : Span ⊥ (Y ) -→ Span ⊥ (X) is an ingressive cartesian fibration. Recall that an ingressive map in Span ⊥ (Y ) is given by a span y 11 ∼ ← -y 01 ↣ y 00 in Y . Note that the reverse span y 11 ↢ y 01 ∼ -→ y 00 is Span(p)-cocartesian in Span(Y ) by Theorem 3.1 3.1. It follows that all ingressive maps in Span ⊥ (Y ) are q-cartesian, and that there are enough cartesian lifts of edges in X in . Therefore, q is indeed an ingressive cartesian fibration.

To see the second equivalence, we observe that given any p : Y -→ X in Cart in (X), the functor q eg : Span ⊥ (Y ) eg -→ Span ⊥ (X) eg is equivalent to the opposite of p eg : Y eg -→ X eg . This immediately implies that Span ⊥ exchanges curved orthofibrations and op-Gray fibrations, and furthermore preserves maps between them that preserve (co)cartesian ingressive arrows.

For the third equivalence, recall from Proposition 4.9 4.9 that an ambigressive square in Span ⊥ (Y ) corresponds to a diagram of the form

• • • • • • • • • ∼ α ∼ ∼ ∼ β ∼ γ ∼ in Y .
Suppose p is a cartesian fibration. The left span defines a q eg -cocartesian arrow if and only if α is p eg -cartesian. This immediately implies that β is also p eg -cartesian. Finally the top right square is cartesian, and therefore we conclude by 5.16 5.16 that γ is also p eg -cartesian, so that the right vertical span is q eg -cartesian. This implies by definition that q is an orthofibration of orthogonal triples. A dual argument (again using 5.16 5.16) shows that the dual of an orthofibration is a cartesian fibration.

Finally, for the statement about fibres, note that Span ⊥ (being an equivalence) preserves fibres, i.e. pullbacks along a map (of orthogonal adequate triples) * -→ X. Observation 5.2 5.2 then implies that the fibre of Span ⊥ (Y ) -→ Span ⊥ (X) over a point x is given by Span ⊥ (Y x , ιY x , Y x ), whose underlying category is equivalent to Y op

x by Proposition 2.15 2.15. □

Dualisation and straightening of two-variable fibrations

The purpose of this section is to use the results of the previous section to dualise and straighten various kinds of fibrations over a product of ∞-categories. The first part essentially consists of specialising Theorem 5.20 5.20 to the case where X is the particular orthogonal triple

(A, B) ⊥ := (A × B, A × ιB, ιA × B)
associated to a product of two ∞-categories. We will find that the various fibrations defined over (A, B) ⊥ in the previous section recover a subset of the fibrations which we considered in [ [START_REF] Haugseng | Lax monoidal adjunctions, two-variable fibrations and the calculus of mates[END_REF].

In particular, we will extend the explicit description of dual (co)cartesian fibrations to the situation of curved ortho-and Gray fibrations. As an application we give an explicit description of parametrised adjoints from [HHLN21 HHLN21], extending previous work of Torii [To20 To20].

Let us start by making the various types of fibrations of orthogonal adequate triples p : Y -→ (A, B) ⊥ appearing in Section 5 5 more explicit. Recall that for all these fibrations, the structure of an adequate triple on Y is uniquely determined by the underlying functor (see Observation 5.2 5.2)

p = (p 1 , p 2 ) : Y -→ A × B.

Remark.

There is an evident analogue LCocart(A, B) consisting of p : Y -→ A × B having p-cocartesian lifts over A × ιB. This ∞-category does not have a good interpretation in terms of adequate triples; one can typically not equip Y with the natural structure of an adequate triple.

6.3. Lemma. Let p : Y -→ A × B be a map in LCart(A, B). Then p defines:

(1) an op-Gray fibration in the sense of Definition 5.13 5.13 if and only if for each a ∈ A, the map p 2 : Y a -→ B is a cartesian fibration.

(2) a curved orthofibration in the sense of Definition 5.13 5.13 if and only if for each a ∈ A, the map p 2 : Y a -→ B is a cocartesian fibration.

Proof. Observe that the map Y eg -→ ((A, B) ⊥ ) eg is simply given by the restriction of p : Y -→ A × B to ιA × B. It follows from [HHLN21 HHLN21, Remark 2.2.9] that this restriction to ιA × B is a (co)cartesian fibration if and only if for each a ∈ A, the restriction p a : Y a -→ {a} × B of p is a (co)cartesian fibration. Finally, note that p a can be identified with p 2 (restricted to the fibre Y a ). □ 6.4. Definition. Let us denote by CrvOrtho(A, B) the subcategory of Cat/A × B whose objects are functors Y -→ A × B satisfying the equivalent conditions of (2) (2) and whose morphisms are maps preserving locally p-cartesian morphisms over A × ιB and locally p-cocartesian morphisms over ιA × B. We define OpGray(A, B) similarly.

We then immediately find: 6.5. Corollary. There are natural equivalences

CrvOrtho((A, B) ⊥ ) CrvOrtho(A, B)
OpGray((A, B) ⊥ ) OpGray(A, B). ∼ ∼ 6.6. Remark. As one may expect, the notion of a Gray fibration p : Y -→ A × B was introduced in [HHLN21 HHLN21] as the opposite of an op-Gray fibration, i.e. there is an equivalence (-) op : Gray(A, B) OpGray(A op , B op ).

∼

In particular, Gray(A, B) is a certain subcategory of LCocart(A, B). As in Remark 6.2 6.2, it does not have a good analogue for general orthogonal adequate triples.

y a ′ ,b (a, b) (a ′ , b) y a,b ′ y a ′ ,b ′ (a, b ′ ) (a ′ , b ′ ) α β β α
in which the vertical arrows are p-cartesian and the bottom horizontal arrow is p in -cocartesian, the top horizontal arrow is p 2 -cocartesian. This is precisely the condition of Definition 5.18 5.18, since the ambigressive squares in Y are precisely squares of the above form whose vertical maps are p-cartesian. □ Lemma 6.8 6.8 asserts that the notion of an orthofibration as defined in Definition 5.18 5.18 agrees (over X = A × B) with the notion of an orthofibration employed in [ [START_REF] Haugseng | Lax monoidal adjunctions, two-variable fibrations and the calculus of mates[END_REF]] (see in particular [HHLN21 HHLN21, Proposition 2.3.11]). Writing Ortho(A, B) ⊆ CrvOrtho(A, B) for the full subcategory spanned by the orthofibrations, we therefore obtain: 6.9. Corollary. There are natural equivalences of ∞-categories

Ortho((A, B) ⊥ ) ≃ Ortho(A, B).
All in all, we have found that over the orthogonal triple (A, B) ⊥ = (A × B, A × ιB, ιA × B), the ∞-categories of fibrations introduced in Section 5 5 coincide with those appearing under the same name in [ [START_REF] Haugseng | Lax monoidal adjunctions, two-variable fibrations and the calculus of mates[END_REF]].

We will now unravel the content of the dualisation equivalence from Theorem 5.20 5.20 over the base X = (A, B) ⊥ . To this end, let us start with the following observation: 6.10. Lemma. Let A and B be two ∞-categories. Then there is an equivalence of orthogonal adequate triples

Span ⊥ (A, B) ⊥ ≃ (A, B op ) ⊥ .
Proof. Essentially by construction, the two diagrams

LCart(A, B) LCocart(A op , B) LCart(A, B) LCocart(A op , B) Cart(A)/pr 1 Cocart(A op )/pr 1 Cart(A)/pr 1 Cocart(A op )/pr 1 SD cc D cc fgt SD cc fgt fgt D cc
fgt commute naturally in A and B. Since the vertical maps are equivalences, the agreement of the two functors in the one-variable case implies that in the two-variable case. □ 6.15. Example. As an example, consider the dual of the bifibration (s, t) : Ar(X) -→ X × X, see [Lu09a Lu09a, Corollary 2.4.7.11], which in particular defines an object in Ortho(X, X). We claim that

Tw ℓ (X) -→ X op × X ≃ SD cc (Ar(X) -→ X 2 ).
Let us mention that an analogous identification

Tw ℓ (X) -→ X op × X ≃ D cc (Ar(X) -→ X 2 )
is a direct consequence of [HMS22 HMS22, Corollary A.2.5], the argument of which is based on the universal property of twisted arrow ∞-categories from [Lu17 Lu17, Corollary 5.2.1.22]. In the (current) absence of a similar universal property for twisted arrow ∞-categories of (∞, 2)-categories, such a proof does not generalise to this more general situation. The present example should be considered a warm-up for Section 7 7 below, where we provide this extension using the span model for dual fibrations. Now for the proof: Unravelling the definitions, we find that SD cc is given by the opposite of the associated ∞-category of a certain simplicial ∞-groupoid Φ, where Φ n is given by the ∞-groupoid of diagrams ϕ : Tw r ([n]) × [1] -→ X that take edges in Tw r ([n]) eg × {0} and Tw r ([n]) in × {1} to equivalences. We now claim that the localisation of Tw r ([n]) × [1] at these subcategories is naturally equivalent to

[n] ⋆ [n] op via the map ( * ) Tw([n]) × [1] -→ [n] ⋆ [n] op , ((i ≤ j), ϵ) -→ i l ϵ = 0 j r ϵ = 1 ,
where we have used subscripts to indicate join factors. Restriction along this map then shows that ac(Φ) ≃ Tw r (X), and so ac(Φ) op ≃ Tw ℓ (X) as required. One can also check that this equivalence lives over X op × X.

To see the claim we note that both restrictions Tw([n]) × {ϵ} → X of a diagram ϕ as considered above take all squares in Tw([n]) to pushout squares in X (namely ones in which two opposite edges are equivalences). Using the pointwise formula for left Kan extensions, one readily checks that ϕ : Tw([n]) × [1] → X is then left Kan extended from the subposet J n × [1], where J n is the arch along the top of Tw([n]) consisting of all (i ≤ j) with i = 0 or j = n. It follows that the inclusion

J n × [1] → Tw([n]) × [1]
induces an equivalence upon localisation. Now note that J n consists of two copies of [n] glued along the initial vertex. The claim then follows from the fact that the localisation of [n] × [1] at [n] × {1} is given by [n + 1] and likewise for the localisation at [n] × {0}; this realises the localisation of J n as the pushout of [n + 1] and [1 + n] along [1], embedded into the former as the terminal segment, and into the latter as the initial one. Finally, observe that the localisation map

J n × [1] → [n + 1] ∪ [1] [1 + n] ∼ = [n] ⋆ [n] op
just described is indeed the restriction of ( * * ).

We can also use the identification of D cc and SD cc to describe the fibrewise adjoints constructed in [HHLN21 HHLN21, Section 3.1] more explicitly. For this we need to recall some notation from • For a map β : b ′ → b in B, the functor R is given on morphisms over β op in SD cc (q) by

β * x b β * R b (x b ) x b y b ′ R b (x b ) R b ′ (y b ′ ). f g
Here the left-pointing arrows are cartesian lifts of β in C and D and the right-pointing arrows are fibrewise over b ′ , with g given by the (fibrewise) adjoint to (6.17)

L b ′ β * R b (x b ) β * L b R b (x b ) β * x b y b ′ . λ β ϵ f
Proof. A direct construction of R can be carried out exactly as in [HHLN21 HHLN21, Theorem 3.1.11], so we will be brief. Applying cocartesian unstraightening to the functor L yields a curved orthofibration π : X -→ B × [1]. Since L admits fibrewise right adjoints, this map is an op-Gray fibration as well. Applying the span dualisation from Corollary 6.12 6.12 to this op-Gray fibration yields a curved orthofibration SD ct (π) : SD ct (X) -→ [1] × B op , which can be straightened over [1] yield the desired functor R.

Let us now describe the behaviour of R on arrows (and hence also on objects). Note that the functor R arises from cartesian transport in SD ct (π) : SD ct (X) -→ [1] × B op in the direction of [1]. Consequently, it suffices to understand those squares in SD ct (π) whose vertical maps are cartesian lifts of the map in [1] and whose horizontal maps cover an arrow β op in B op . Like in the proof of Theorem 5.20 5.20, unraveling the definitions shows that such a square in SD ct (X) corresponds to a diagram in the domain of π : It now remains to verify that we can indeed take g to be the adjoint to (6.17 6.17). This follows from an analysis very similar to [HHLN21 HHLN21, Proposition 3.2.7]. In fact, when f is the identity, the adjoint to (6.17 6.17) is precisely the mate of λ β ; In this case, [HHLN21 HHLN21, Proposition 3.2.7] precisely asserts that the mate makes the bottom right square commute. □

X -→ B × [1] of the form R b (x) β * R b (x) R b ′ (y) R b (x) β * R b (x) R b ′ (y) x β * x y g ∼ ∼ g ∼ f -→ (b, 0) (b ′ , 0) (b ′ , 0) (b, 0) (b ′ , 0) (b ′ , 0) (b, 1) (b ′ , 1) (b ′ , 1)
Finally, we will use Corollary 6.12 6.12 to identify various ways of straightening orthofibrations.

(C) → C × C) ≃ Map C : C × C op → Gpd,
which we will generalise to (op)lax arrow categories of (∞, 2)-categories in the next section. We shall give a fairly direct comparison between the four equivalences above in the present section, but there could be more ways of straightening an orthofibration. For example, in [START_REF] Stevenson | Model structures for correspondences and bifibrations[END_REF] Stevenson produced an equivalence Bifib(A, B) ≃ Fun(A × B op , Gpd) by comparing both sides to a model category of correspondences. In order to settle such coherence questions once and for all, we take another cue from [ [START_REF] Barwick | Dualizing cartesian and cocartesian fibrations[END_REF]] and show in the appendix that any equivalence as in the statement of 6.18 6.18 that is natural in the input categories, and restricts to the identity for A = * = B agrees with that above (in an essentially unique fashion). We also include similar statements for curved orthofibrations and bifibrations, in particular settling the comparison with Stevenson's construction.

For the direct proof (and also the naturality of the Yoneda embedding) we need: Proof. We again use the formula

Un cc (F ) ≃ colim Tw r (A × B) (s,t) ---→ (A × B) × (A × B) op F ×(A×B) -/ --------→ Cat from [GHN17 GHN17
]. Using Tw r (A × B) = Tw r (A) × Tw r (B) and (A × B) -/ = A -/ × B -/ and the fact that colimits over a product can (naturally in the indexing categories) be computed in two steps we find that this colimit agrees with that of Tw r (B) -→ Fun(Tw r (A), Cat)

colim ---→ Cat
where the first functor is curried from the original one, i.e. it takes f : x → y ∈ Tw r (B) to Tw r (A)

(s,t) ---→ A × A op (F (-,x)×A -/ ×B y/ -------------→ Cat.
But the colimit of this functor is is naturally equivalent to the unstraightening of F (-, x)×B y/ : A → Cat, so in total Un cc (F ) is naturally identified with Proof of Proposition 6.18 6.18. Recall that D cc is defined as the composite Ortho(A, B)

colim Tw r (B) -→ B × B op Un cc A (F )×B -/ ---------→ Cat ,
Str ct ---→ Fun(A op , Cocart(B)) Un cc ---→ Cocart(A op × B).
Thus Lemma 6.20 6.20 immediately identifies the functors Ortho(A, B)

D cc --→ Cocart(A × B op ) Str cc ---→ Fun(A × B op , Cat)
and Ortho(A, B)

Str ct ---→ Fun(B op , Cocart(A)) Str cc ---→ Fun(A × B op , Cat),
and the argument for the middle two functors is dual. To compare the first composite with Ortho(A, B)

D ct --→ Cart(A op × B) Str ct ---→ Fun(A × B op , Cat)
we observe that the diagram

Cocart(A × B op ) Cart(A op × B) Ortho(A, B) SD ct SD ct SD ct
commutes essentially by construction (whereas this does not seem clear from the construction for D ct ). It now follows from Corollary 6.12 6.12 that D cc : Ortho(A, B) → Cocart(A × B op ) agrees with Ortho(A, B)

D ct --→ Cart(A op × B) D cc --→ Cocart(A × B op ),
whence the result follows from Corollary 3.17 3.17. □ As another consequence of (the dual of) Corollary 6.20 6.20 we obtain that our straightening equivalence for Gray fibrations extends the usual one for cocartesian fibrations over A × B.

(Op)lax arrow and twisted arrow ∞-categories

In this section we will discuss an application of the dualisation procedure from Corollary 6.12 6.12 which extends Example 6.15 6.15: The duality between the arrow and twisted arrow category of an ∞-category X extends to a duality between the oplax arrow and twisted arrow category of an (∞, 2)-category X. More precisely, the oplax twisted arrow category of X (s, t) : Tw r (X) -→ X × X op is introduced in work of Abellán García and Stern [AGS20 AGS20] as an explicit model for the cartesian fibration classified by the enriched mapping functor of X (restricted to its underlying (∞, 1)-category X)

Map X : X op × X -→ Cat. We will show that the enriched mapping functor also classifies the orthofibration (s, t) : Ar opl (X) -→ X × X from the (∞, 1)-category underlying the oplax arrow category, defined using the (a priori unrelated) Gray tensor product of Gagna, Harpaz and Lanari from [ [START_REF] Gagna | Gray tensor products and lax functors of (∞, 2)-categories[END_REF]].

Since the oplax twisted arrow category and the Gray tensor product are both defined using the model for (∞, 2)-categories given by scaled simplicial sets [Lu09b Lu09b], we will start with a minimalistic review of these. 7.1. Notation. Recall that a scaled simplicial set is a pair (X, S) consisting of a simplicial set X and a subset S ⊆ X 2 of 2-simplices that are called thin. The category sSet sc of scaled simplicial sets carries a model structure in which the cofibrations are the monomorphisms, which is related to the model category of categories enriched over marked simplicial sets (with the categorical model structure) by a Quillen equivalence C sc : sSet sc ⇆ Cat(sSet + ) : N sc [Lu09b Lu09b, Theorem 4.2.7]. We define the ∞-category Cat 2 of (∞, 2)-categories as the ∞-category associated to any of these two Quillen equivalent model categories (or any of the other standard models, cf. [Lu09b Lu09b, Theorem 0.0.3]). 7.2. Notation. For a simplicial set X, we write X ♯ for the associated scaled simplicial set in which every 2-simplex is thin. This determines a left Quillen functor (-) ♯ : sSet -→ sSet sc , where sSet is equipped with the Joyal model structure. Its right adjoint sends a scaled simplicial set (X, S) to the sub-simplicial set X spanned by the thin 2-simplices. At the level of ∞-categories, this induces the fully faithful inclusion Cat 1 → Cat 2 of (∞, 1)-categories into (∞, 2)-categories, together with its right adjoint sending an (∞, 2)-category X to its underlying (∞, 1)-category X. The internal mapping objects induced by the Gray tensor product via

Map Cat2 A, Fun lax (B, X) ≃ Map Cat2 (A ⊠ B, X) ≃ Map Cat2 B, Fun opl (A, X)
are by definition the (∞, 2)-categories of 2-functors and (op)lax natural transformations between them, see [Ha21 Ha21, Definition 3.9].

Let us now describe the complete Segal space models for the fibrations (s, t) : Ar opl (X) -→ X × X, (s, t) : Tw r (X) -→ X × X op associated to an (∞, 2)-category X. We will start with the oplax arrow ∞-category.

7.5. Definition. Let X be a (∞, 2)-category. The oplax arrow ∞-category Ar opl (X) is the ∞-category underlying the oplax functor ∞-category Ar opl (X) = Fun opl [1], X .

Informally, Ar opl (X) is the ∞-category with objects given by arrows of X, such that morphisms from f to g are given by oplax commuting squares:

x x ′ y y ′ f g
More precisely, Ar opl (X) can be characterised in terms of the Gray tensor product by the natural equivalence Map Cat S, Ar opl (X) ≃ Map Cat2 [1] ⊠ S, X .

7.6. Example. If X is a gaunt 2-category, i.e. a 0-truncated object in Cat 2 , then the above natural equivalence becomes a natural bijection of sets. It follows that Ar opl (X) is a gaunt 1-category, i.e. a strict category without nontrivial isomorphisms, and from the description (7.4 7.4) of the Gray tensor product one sees that it coincides with the classical oplax arrow category of X.

7.7. Remark. Of course, the lax arrow category Ar lax (X) is defined by a similar universal property:

Map Cat S, Ar lax (X) ≃ Map Cat2 S ⊠ [1], X .
The description of the Gray tensor product of simplices (7.4 7.4) shows that for any simplex [m], there is a natural equivalence of gaunt 2

-categories [1] ⊠ ([m] op ) ≃ [m] ⊠ [1] op op . By adjunction, this determines a natural equivalence (7.8) Ar opl (X) op ≃ Ar lax (X 1-op )
where X 1-op has only the directions of the 1-morphisms inverted. On objects, this equivalence sends an arrow in X to the opposite arrow in X 1-op . 7.9. Proposition. Suppose X is an (∞, 2)-category. Then the functor (s, t) : Ar opl (X) → X × X is an orthofibration where an edge σ, given by an oplax square

x x ′ y y ′ , f g ρ
over ιX × X is cocartesian if and only if ρ is invertible, and similarly an edge over X × ιX is cartesian precisely when the same square commutes. More formally, a morphism σ :

[1] ⊠ [1] → X over ιX × X (resp. X × ιX) is cocartesian (resp. cartesian) if and only if it factors through the 2-functor [1] ⊠ [1] → [1] × [1].
Proof. Let us first show that the cocartesian morphisms σ over ιX × X indeed correspond precisely to those σ :

[1] ⊠ [1] -→ X that factor over [1] × [1].
To see this, consider the following two equivalent unique lifting problems:

Λ 0 [2] Ar opl (X) Λ 0 [2] ⊠ [1] Λ0[2]×{0,1} [2] × {0, 1} X [2] X × X [2] ⊠ [1], (s,t)
where Λ 0 [2] is the ∞-category 1 ← 0 → 2. Using the explicit description of the Gray tensor product of simplices of (7. Now suppose that the morphism 0 ≤ 1 in Λ 0 [2] projects to ιX × X and that the corresponding map

[1] ⊠ [1] -→ X factors over [1] × [1]
. This means that in the above diagram, the left oplax square commutes and that 00 -→ 10 is an equivalence. The space of such diagrams is given by the space of 2-functors from the (∞, 2)-category 00 20 11 01 21 to X, where the left triangle commutes. For example this follows by decomposing the 2-category before as a pushout in Cat 2 using Proposition 7.3 7.3, and then observing that the 2-category

• • • ∼ ∼ is equivalent to [1]. Similarly, the space of diagrams [2] ⊠ [1] → X such that the morphism 0 ≤ 1 in [2] projects to ιX × X and the corresponding map [1] ⊠ [1] -→ X factors over [1] × [1]
is also equivalent to the space of 2-functors from the (∞, 2)-category above to X. Therefore we conclude that an essentially unique lift exists in the diagram above. Note that for any arrow in ιX × X and a lift of its domain, there exists an (s, t)-cocartesian lift of this form, and that lifts of this form are closed under equivalence; this implies that up to equivalence, all (s, t)-cocartesian lifts over ιX × X are of the form asserted in the proposition.

A dual argument proves the existence and form of (s, t)-cartesian arrows over X × ιX. It follows that (s, t) is a curved orthofibration in the sense of Definition 6.4 6.4.

The map ϕ is then given by the identity on objects and on (nerves of) mapping categories it (a) (a) either sends all chains to the point or (b) (b) takes the height of a chain. It suffices to verify that these maps between mapping categories are exactly the localisations at the marked arrows. This is evident in case (a) (a), since the poset of chains is either empty or a cube (hence contractible). In case (b) (b), the functor sending a chain to its height has a fully faithful right adjoint, sending each height k to the maximal chain i

≤ i + 1 ≤ • • • ≤ k -1 ≤ k ≤ k ≤ k -1 ≤ • • • ≤ j + 1 ≤ j.
□ 7.17. Corollary. Let X be an (∞, 2)-category. Then the simplicial ∞-groupoid

[n] Map Cat2 [n] ⋆ opl [n] op , X
is a complete Segal ∞-groupoid, whose associated ∞-category is naturally equivalent to Tw r (X).

Proof. Corollary 7.13 7.13 and Lemma 7.16 7.16 identify the simplicial ∞-groupoid Map Cat2 [-]⋆ opl [-] op , X with N(Tw r (X)). □

We will now compute the image of Ar opl (X) under the dualisation functor SD ct : Ortho(X, X) → Cart(X, X op ), and show it is equivalent to the functor (s, t) : Tw r (X) → X × X op .

Recall that SD ct is given as the composite of (-) op and Span ⊥ . Since Remark 7.7 7.7 identifies the opposite of the oplax arrow category of X with the lax arrow category of X 1-op , we find that the dual of (s, t) : Ar opl (X) -→ X × X is given by SD ct (s, t) : D := Span ⊥ (Ar lax (X 1-op ))

X × X op .

Unwinding definitions, we find that D is the associated ∞-category of the complete Segal ∞groupoid given in level n by the sub-∞-groupoid of

Map Cat2 Tw r ([n]) ⊠ [1], X 1-op spanned by the functors f : Tw r ([n]) ⊠ [1] -→ X 1-op which send (i) Tw r ([n]) in ⊠ {1} to ιX op , (ii) Tw r ([n]) eg ⊠ {0} to ιX op , and (
iii) every 2-morphism σ ⊆ Tw r ([n]) in ⊠ [1] to an invertible 2-morphism in X.
This is of course naturally equivalent to the space of functors f : Tw

r ([n]) ⊠ [1] 1-op -→ X
satisfying analogous conditions. To obtain a functor D → Tw r (X) we will write down a natural transformation of cosimplicial objects in Cat 2

ϕ n : Tw r ([n]) ⊠ [1] 1-op [n] ⋆ opl [n] op .
We will produce this map at the level of scaled simplicial sets, using that for a scaled simplicial set (Y ′ , T ) modeling Y, the opposite Y 1-op is modelled by ((Y ′ ) op , T ). Let us therefore define

ϕ n : (Tw r ([n]) × [1]) op -→ ∆ n ⋆ ∆ n,op ; ((i ≤ j), ϵ) -→ ī ϵ = 0 j ϵ = 1
exactly as in Example 6.15 6.15. By inspection this defines a map of scaled simplicial sets. For example ϕ n (σ, σ ′ ) = ∆ {i,j, k} for i ≤ j ≤ k when σ ′ equals s 1 : ∆ 2 → ∆ {0,1} . By definition this is thin in ∆ ⋆ sc ∆ n,op . Similarly, one can readily show that the image of the other thin simplicies is thin. We conclude that the ϕ n determine a natural transformation of cosimplicial objects in sSet sc . 7.18. Example. Let us describe the induced map ϕ n : Tw(

[n]) ⊠ [1] op -→ [n] ⋆ opl [n] op in Cat 2
a bit more precisely in the case n = 1. In this case, the domain is the pushout in Cat 2 of two lax squares along a common 1-morphism, which is a gaunt 2-category by Proposition 7.3 7.3. In particular, the domain and codomain of ϕ 1 are gaunt 2-categories, and the map ϕ 1 is then given by the strict 2-functor For each X, the maps ϕ n induce a natural transformation of simplicial ∞-groupoids

N D -→ S κr (X),
which after taking associated ∞-categories gives a functor Φ : D → Tw r (X) which evidently commutes with the canonical functors to X × X op . 7.21. Theorem. The functor

Φ : D -→ Tw r (X)
is an equivalence of ∞-categories. Consequently, the orthofibration (s, t) : Ar opl (X) → X × X classifies the mapping ∞-category functor Map X (-, -) : X op × X -→ Cat.

Proof. Given the first part of the theorem, the second part follows from Corollary 6.14 6.14 and Theorem 7.12 7.12. Because both sides of the equivalence are the associated ∞-category of Segal ∞-groupoids, and the functor is induced by a natural transformation of Segal ∞-groupoids, it suffices to prove that Φ 0 and Φ 1 are equivalences of ∞-groupoids. Note that the map ϕ 0 is simply the identity on [1], so that Φ 0 is certainly an equivalence.

To see that Φ 1 is an equivalence, note that its domain N(D) 1 is the space of 2-functors from the gaunt 2-category (Tw r ( sending the marked 1-and 2-cells to equivalences in X. It therefore suffices to verify that the functor ϕ 1 (7.19 7.19) is the universal functor of (∞, 2)-categories that collapses these marked morphisms and 2-morphisms.

[1]) × [1]) op
To see this, we observe that the above gaunt 2-category can be obtained by "cell attachments", along the lines of the following picture: Here we start from the middle vertical arrow, attach the bottom horizontal arrows, then the 2-cells and finally the factorisations over 00 and 11. Each step is given by a pushout of gaunt 2-categories as described in Proposition 7.3 7.3 (which is hence also a pushout in Cat 2 ).

7.24. Remark. For the sake of completeness, let us also mention the behaviour of Ar opl and Ar lax under the two kinds of taking opposites. Recall that we have already seen in (7.8 7.8) that Ar opl (X 1-op ) ≃ Ar lax (X) op . We are left to note Ar opl (X 2-op ) ≃ Ar opl (X 1-op ) op .

To see this, observe that both are orthofibrations over X × X and that the functors X op × X → Cat which both classify are equivalent by Theorem 7.21 7.21 and the corresponding statements for the twisted arrow ∞-categories.

Naturality of the Yoneda embedding

In this short final section we deduce the following result as an application of Lemma 6.20 6.20: Before giving a proof of Theorem 8.1 8.1, let us briefly comment on related results that already appear in the literature. First, the characterisation of P(A) as the free cocompletion of A [Lu09a Lu09a, Section 5.1.5] implies that the assignment A → P(A) extends to a functor equipped with a natural transformation A -→ P(A) given pointwise by the Yoneda embedding (essentially by construction). However, it is not a priori clear that this second functor agrees with the one described in Theorem 8.1 8.1; let us write P free and P Kan to distinguish these two functorialities, the first being the functoriality via the free cocompletion and the second the functoriality of the statement.

Without considering the naturality of the Yoneda embedding, one can produce a natural equivalence P free ≃ P Kan as follows: Both of these functors are easily checked to factor as Cat

(-) ♮ ---→ Cat ♮ P -→ Cat ccpt ⊂ Cat,
where the second term denotes the ∞-category of small idempotent complete ∞-categories (and the first functor idempotent completion), and the third term the ∞-category of cocomplete ∞categories admitting a set of completely compact objects that jointly detect equivalences, and functors among them preserving both colimits and completely compact objects. By an argument analogous to [Lu09a Lu09a, Proposition 5.5.7.8], P free is an equivalence between the middle two terms.

Since P free and P Kan are homotopic on individual morphism spaces [Lu09a Lu09a, Proposition 5.2.6.3], the same follows for P Kan . But by a minor modification of Toën's theorem [To05 To05], Cat ♮ has discrete automorphism space consisting of two objects (the identity and op), whence there is a unique natural equivalence between P free and P Kan .

To conclude that the Yoneda embedding extends to a natural transformation as in Theorem 8.1 8.1, it thus remains to verify that this equivalence is given pointwise by the identity of P(A); the trouble is that this latter identification is not clear.

Proof of Theorem 8.1 8.1. Recall that for any ∞-category A, the Yoneda embedding A → P(A) is defined as the adjoint of the functor Map A : A op × A → Gpd, which in turn is given by the This can be verified using the techniques of [GHN17 GHN17, Appendix A]; we refrain from working out further details, as we do not need the statement. A.3. Remark. As mentioned the analogue of Theorem A.1 A.1 for the functor A → Fun(A, Cat) is one step in the proof of [BGN18 BGN18, Theorem 1.4] and our proof below follows their strategy in the large. There is, however, one crucial difference: Barwick, Glasman and Nardin use Yoneda's lemma to deduce that the automorphisms of the functor Cat op → Gpd, A → Map Cat (A, Cat) are given by Aut(Cat). This suffices to establish the analogue of Theorem A.1 A.1 after taking ∞-groupoid cores, but it is unclear to us how to obtain the actual statement from this information.

By contrast, our more elaborate proof of Theorem A.1 A.1 below also makes use of intermediate steps of Toën's results (and the strategy applies equally well in the situation of [START_REF] Barwick | Dualizing cartesian and cocartesian fibrations[END_REF]).

In the proof we will make use of two tangential results. For the first, recall that f : which is cartesian if and only if f * is faithful.

Proof of Lemma A.4 A.4. The first assertion is immediate from Fun(hC, hB) ≃ Fun(C, hB) and the analogous assertion for A in place of B. The second assertion then follows, since the lower horizontal functor is clearly faithful and faithful functors are closed under pullback: This is most easily seen from the characterisation that all induced maps on morphism complexes have empty or contractible fibres, which is evidently stable under pullback. The third statement similarly follows from the analogue for ordinary categories by applying cores to the diagram of the lemma. □

In the following, recall that a full subcategory I → C is said to be dense if the restricted Yoneda embedding C -→ P(I) is fully faithful, or equivalently, if for each c ∈ C the canonical map colim i∈I/c i → c in C is an equivalence.

where the limits run over [f : (n, m) → (n ′ , m ′ )] ∈ Tw(∆ × ∆). The second term is a set of path components in the third; indeed, this is so before taking limits, so that the fibre over a point in the target is the limit of a diagram only taking values ∅ and * , and thus also either empty or contractible itself. Similarly, we find In particular, the claim that Aut(G, G) is discrete with two components will follow from the analogous statement for H.

Step 3: automorphism group of the generators. Recall that H(n, m) is spanned by the functors h (i,j), [k] for (i, j) ∈ [n] × [m] and k ≥ 0, defined by the universal property (A.7 A.7) or by the explicit formula (A.9 A.9). The latter formula reduces to

h (i,j),[k] : [n] × [m] -→ Cat, (a, b) -→ [k] a ≥ i, b ≥ j ∅ otherwise.
Combined with (A.7 A.7), one sees that the mapping spaces between such functors are sets and that there are equivalences

∼

In particular, the ∞-categories H(n, m) are 0-truncated, i.e. equivalent to ordinary categories with discrete core; one can use this to conclude that the above equivalence is natural in (n, m) (which is now a property, rather than a structure), for the obvious functoriality on the left leaving ∆ fixed. Since all H(n, m) are 0-truncated, H is a 0-truncated object in the ∞-category of bicosimplicial ∞-categories and Aut(H) is discrete. Furthermore any automorphism of H induces one on H(0, 0) = ∆, and this restriction determines the entire transformation: The composite The other two cases. To conclude, we briefly describe the modifications to be made to the above argument to prove that F Gpd and F Gray have trivial automorphism groups.

For F Gpd , we repeat the entire argument, but in Step 2 we use the natural subcategory H Gpd (n, m) ⊆ Fun( Again, these full subcategories can also be characterised as those spanned by the essential images of ∆ under all structure maps G Gray (0, 0) → G Gray (n, m). Formula (A.9 A.9) shows that the object h (i,j), [k] in H Gray (n, m) is given by the 2-functor The explicit description of the Gray tensor product (7.4 7.4) (see [HHLN21 HHLN21, Proposition 5.1.9]) identifies the mapping ∞-category Map [n]⊠[m] ((i, j), (a, b)) with the poset of maximal chains from (i, j) to (a, b) in the grid [n] × [m], ordered using that for each square, "right-after-down" is smaller than "down-after-right". Using this and the equivalence (A.7 A.7) to compute mapping spaces, one readily sees that each H Gray (n, m) is a 0-truncated ∞-category. This already implies that Aut(H Gray ) is discrete, so it remains to verify that it has only one component.

h (i,
To see this, consider the fully faithful functor (which we only need for gaunt 2-categories) Let us now consider the map of sets Aut(H Gray ) -→ Aut(H Gray (0, 0)) restricting an automorphism φ of H Gray to the component φ 0,0 . For an automorphism ψ of H Gray (0, 0), the fibre Aut(H Gray ) ψ can be identified with the set of isomorphisms Since such isomorphisms are determined by their behaviour on objects, compatibility with the vertex inclusions shows that there is a unique natural φ (the identity) and no natural φ ′ . We conclude that Aut(F Gray ) ⊆ Aut(H Gray ) ≃ * . □ Mathematisches Institut, WWU Münster, Germany Email address: f.hebestreit@uni-muenster.de

Φ
Mathematisches Institut, RFWU Bonn, Germany Email address: linskens@math.uni-bonn.de IMT, Université de Toulouse III, France Email address: joost.nuiten@math.univ-toulouse.fr

  op . Then S B (C) ≃ N Tw r (C): This follows from the observation that Tw r arises from a right Quillen functor between the Joyal model structures whose left adjoint sends [n] to [n] ⋆ [n] op (see [Lu17 Lu17, Section 5.2.1] or [HNP17 HNP17, Proposition 4.13]). In particular S B takes values in complete Segal spaces.

Proof.

  Let us point out that any colimiting cocone G : I £ → E arises as the image of a colimiting cocone G ′ : I £ → P(D) under |-| f : indeed, one can take G ′ to be the colimit of the diagram S f • G |I : I -→ P(D). Consequently, a functor out of E preserves colimits if and only if its composition with |-| f does. Using this, (1) ⇔ (2) is an immediate consequence of [Lu09a Lu09a, Lemma 5.1.5.5].

  Proposition 5.2.7.12]. In turn, |-| F •f is the left Kan extension of F • f along the Yoneda embedding h : D -→ P(D) [Lu09a Lu09a, Lemma 5.1.5.5]. The claim then follows from transitivity of Kan extensions. Finally, for (3) ⇒ (2) note that (ii) implies that |-| F •f descends along |-| f to some functor G : E → C, which is then automatically its left Kan extension along |-| f . Part (i) and transitivity of Kan extensions then imply that G ≃ F , or in other words that |

  satisfies conditions (1) (1) and (2) (2) as well.Let us start by computing Φ(F ). To this end, let us write Ar(B) s = Ar(B) × Fun({0},B) ιB. Then t : Ar(B) s -→ B is the left fibration classified by the functor b → ι(B/b) and s : Ar(B) s -→ ιB admits a fully faithful left adjoint cst : ιB → Ar(B) s taking degenerate arrows. Let us now consider the following two pullbacks of ∞-categories

  ) is the left fibration classifying ι(B /-) : Tw(B op ) → B → Gpd and F ′ is constant along the fibres of q, the left Kan extension q ! F is naturally equivalent to the diagram F (-) × ι(B /-) of which we want to compute the colimit [Lu09a Lu09a, Proposition 4.3.3.10]. By transitivity of Kan extensions, it therefore suffices to compute colim C F ′ . Note that the vertical arrows in the above diagram are all cartesian fibrations, so that the fully faithful inclusion D → C admits a (localising) right adjoint [Lu09a Lu09a, Corollary 5.2.7.11] p = (id, s) : C = Tw r (B op ) × B Ar(B) s Tw r (B op ) × B ιB = D.

  2.8]. It consists of two classes of arrows C l (left) and C r (right), denoted ↠ and ↣, such that (i) both classes of maps are closed under retracts, (ii) each solid commutative diagram dashed filler whenever f ∈ C l and g ∈ C r , and (iii) every map in C factors as • • • f g with f ∈ C l and g ∈ C r . Let us point out that the notation for the left and right classes is inspired by the notation for the epi-mono factorisation system, and is opposite to the convention typically employed for model categories. Both C l and C r then define subcategories of C that contain all equivalences and each class uniquely determines the other [Lu09a Lu09a, Proposition 5.2.8.6 & 5.2.8.11]. 4.1. Remark. The factorisation in the last item is essentially unique, and this uniqueness is in fact equivalent to the first two items [Lu09a Lu09a, Proposition 5.2.8.17]. More precisely, writing Fact(C) ⊆ Fun([2], C) for the full subcategory of functors sending 0 ≤ 1 to C l and and 1 ≤ 2 to C r , one has that restriction to the arrow 0 ≤ 2 defines an equivalence compose : Fact(C) Ar(C). ∼ An (essentially unique) section provides a functorial factorisation of morphisms in C.

( 4 )

 4 Tw (2) ([n]) 4 is the subcategory spanned by the edges abcd → abcd ′ . 4.16. Proposition. Let X ∈ AdTrip ⊥ and let [n] ∈ ∆. Then the natural transformation (4.14 4.14) identifies the domain with those path components in Map Cat (Tw

  (2) ([n]) from Construction 4.15 4.15. Note that Tw r (Tw r ([n])) in consists of maps of tuples of the form abcd ≤ a ′ bcd ′ , while Tw r (Tw r ([n])) eg consists of maps abcd ≤ ab ′ c ′ d. On the other hand, Tw r (Tw r ([n]) in ) consists of maps of the form abcc ≤ a ′ b ′ cc and Tw r (Tw r ([n]) eg ) consists of maps aacd ≤ aac ′ d ′ .Furthermore, Span ⊥ (X) in is the subcategory consisting of spans of the form • • •∼, where the right map is in X in . Similarly, Span ⊥ (X) eg is the subcategory whose morphisms are spans of the form • • • ∼

  (2) ([1]) -→ Ar([1]) exhibits Ar([1]) as the localisation of Tw (2) ([1]) at the maps in Tw (2) ([n]) 2 and Tw (2) ([n]) 4 . The map z is given by the map 0000 ←-0001 -→ 0011 ←-0111 -→ 1111 00 -→ 01 -→ 11 collapsing the two left-pointing arrows. This is manifestly a localisation. Furthermore, under the map z, the noninvertible arrow in Tw([1]) 1 , i.e. 0111 → 1111, is sent to the ingressive 01 → 11 and the noninvertible arrow in Tw([1]) 3 , i.e. 0001 → 0011, is sent to the egressive 00 → 01. We conclude that the map LTw (2) ([1]) -→ Ar([1]

  id b ), where b/p denotes the pullback b/B × B A along p, admits a right adjoint, given by taking (a, f : b → p(a)) to a p-cartesian lift of f ending at a. Thus |p -1 (b)| ≃ |b/p|, whence the claim follows from Joyal's cofinality criterion [Lu09a Lu09a, Theorem 4.1.3.1].To finally see that (3) ⇒ (1), consider for some ∞-category C the functorp * : Fun(B, C) -→ Fun w (A, C),where the superscript on the right denotes those functors inverting all maps that p inverts. We have to show that p * is an equivalence. We first claim that it has a right adjoint p * . By [Lu09a Lu09a, 4.3.3.7] this will follow from admitting a limit for every b ∈ B and F : A → C that inverts the fibrewise maps. But as we just discussed above, the inclusion p -1 (b) ⊆ b/p admits a right adjoint and is thus coinitial, so we may instead consider p -1 (b) ⊆ A F -→ C, which factors through the localisation p -1 (b) → |p -1 (b)| since F inverts all fibrewise maps by assumption. Thus the assumption |p -1 (b)| ≃ * implies that F (a) is a limit of the above diagram for any a ∈ p -1 (b). It also follows from the formula for the adjoint as a right Kan extension that the unit and counit of the adjunction (p * , p * ) are equivalences, as desired. □ 5.6. Example. A cartesian fibration p : X → S is called pointed if both X and S have a terminal object and p preserves it. In [La19 La19] Lanari shows that the ∞-category of pointed cartesian fibrations is equivalent to a certain subcategory of factorisation systems (he calls them cartesian).

Proof.

  The forgetful functor factors as Cart -→ AdTrip ⊥ -→ Cat. The first functor has left adjoint L eg by Proposition 5.4 5.4 and the second has left adjoint Ar by Proposition 4.8 4.8. The composite precisely sends A to s : Ar(A) -→ A, since this has contractible fibres and the associated orthogonal triple structure on Ar(A) is precisely that from Example 4.7 4.7. □ We now come to the proof of Proposition 5.3 5.3. Proof of Proposition 5.3 5.3. Let us start by recalling the model for X[(X eg ) -1 ] given by the relative Rezk nerve [MG19 MG19]: for each [n], let us write

By

  Proposition 4.4 4.4, such p determines an ingressive cartesian fibration if and only if all (ingressive) arrows in A × ιB have enough cartesian lifts. By [HHLN21 HHLN21, Corollary 2.2.2] or [Lu09a Lu09a, Proposition 2.4.1.3(3)], this is equivalent to of cartesian fibrations over A preserving cartesian arrows. In particular, each α : a -→ a ′ in A gives rise to a cartesian transport functor α * : Y a ′ -→ Y a between the fibres of p 1 . Following [HHLN21 HHLN21], let us write LCart(A, B) for the ∞-category of p : Y -→ A × B having p-cartesian lifts over A × ιB and maps preserving such p-cartesian lifts. In these terms, we have: 6.1. Observation. There are natural equivalences of ∞-categories Cart in ((A, B) ⊥ ) ≃ LCart(A, B).

•

  [HHLN21 HHLN21, Section 3]. Fix a parametrised left adjoint . a map between two cartesian fibrations (though not necessarily preserving cartesian edges), such that the restrictions L b : D b -→ C b to the fibres admit right adjoints R b : C b -→ D b . In [HHLN21 HHLN21] we constructed from this data a diagram D cc (p) D cc (q) B op , R , such that R restricts to the functors R b under the identifications D cc (p) b ≃ D p and D cc (q) b ≃ C b arising from the naturality of D cc , and showed that this gives an equivalence between the (∞, 2)category of parametrised left adjoints and that of parametrised right adjoints (after taking opposites appropriately). One can use the equivalence SD cc ≃ D cc to give an explicit description of the functor R; see also [To20 To20, Section 3.1] for a point-set variant of this construction of fibrewise adjoints. To formulate the statement, observe that for each β : b ′ → b, the left adjoints L b and L ′ b and the cartesian transport functors of p and q are related by a natural transformation λ β : L b ′ β * → β * L b . We then have: 6.16. Proposition. For a parametrised left adjoint L the associated parametrised right adjoint SD cc (p) SD cc (q) B op , R can be described as follows: For an object y b in the fibre SD cc (q) b ≃ C b over b ∈ B, one has R(y b ) ≃ R b (y b ).

  right and bottom left squares are (cartesian) ambigressive in X. In particular, the left pointing arrows are all π-cartesian lifts of β. Let us explain the rest of the diagram in more detail. For the left and right vertical spans to describe cartesian arrows in SD ct (X), one needs their upwards pointing leg to be an equivalence (as indicated) and their downwards pointing leg to define a cartesian arrow for the map π b : X b -→ {b} × [1]. Because each π b is both a cocartesian and a cartesian fibration classifying the adjoint pair (L b , R b ), the objects in the middle row are then given by R b (x), β * R b (x) and R b ′ (y). Finally, the right vertical square is entirely contained in the fibre X b ′ . Since R b ′ (y) -→ y was π b ′ -cartesian, the map g is therefore the unique one making the square commute.

6. 20 .

 20 Lemma. The three functorsCocart(A × B)Str cc A×B -----→ Fun(A × B, Cat) Cocart(A × B) → Fun(A × B, Cat)are pairwise equivalent.

  which is itself unstraightening over B; here we regard Un cc A as the functor Fun(A × B, Cat) -→ Fun(B, Cocart(A)) -→ Fun(B, Cat)/ cst A by forgetting the map to A. In total this process identifies Un cc A×B : Fun(A × B, Cat) -→ Cocart(A × B) as the composite Fun(A × B, Cat) Un cc A ---→ Fun(B, Cocart(A)) Un cc B ---→ Cocart(A × B) as desired.□

6. 21 .

 21 Remark. (1) Straightening of bifibrations is also discussed in detail in [HLAS16 HLAS16, Section 5], [HMS22 HMS22, Appendix A] and [CDH+20a CDH+20a, Section 7.1], in each case by choosing one of the last two equivalences from 6.18 6.18 as the definition. For example, the stable ∞-category underlying the Poincaré ∞-category Pair(C, Ϙ) from [CDH+20a CDH+20a, Section 7.3] is simply the orthocartesian unstraightening of Ω ∞ B Ϙ : C op × C op → Gpd in the language of the present paper. (2) The equivalence Gray(A, B) ≃ Fun(A ⊠ B, Cat) constructed in [HHLN21 HHLN21, Section 5.2] by definition restricts to the composite Cocart(A × B) Str cc ---→ Fun(A, Cocart(B)) Str cc ---→ Fun(A × B, Cat); see [HHLN21 HHLN21, Remark 5.2.10].

  See [[START_REF] Gagna | Gray tensor products and lax functors of (∞, 2)-categories[END_REF] for the details of this construction. Furthermore, by [HHLN21 HHLN21, Proposition 5.1.9] its value on the pair ([m], [n]) is naturally equivalent to the expected (gaunt) (∞, 2

  4 7.4), the right vertical map is the inclusion of the sub-(∞, 2)-category of [2] ⊠ [1

8. 1 .

 1 Theorem. The Yoneda embedding A → P(A) canonically extends to a natural transformation of functors Cat → Cat from the inclusion to the compositeCat Fun(-op ,Gpd) ---------→ (Cat R ) op ≃ Cat L ⊆ Cat.8.2. Remark. As our model of the equivalence (Cat R ) op ≃ Cat L we take the equivalence of [HHLN21 HHLN21, Theorem 3.1.11]. Intuitively it acts as the identity on objects, and sends a left adjoint L : C → D to a choice of right adjoint R : D → C. 8.3. Remark. This question was recently posed to the second author (among others) by D. Clausen during a visit to the University of Copenhagen, as it can be used to simplify a number of arguments in [CØJ21 CØJ21, Section 2] and appears missing from the literature so far. It is a lucky accident that our methods answer it. Let us also mention that, building on Theorem 8.1 8.1, Ben Moshe and Schlank have recently upgraded the monoidal (and modal) versions of the Yoneda embedding to natural transformations as well, see [BMS21 BMS21, Theorem D].

  A → B is called a faithful functor if each Map A (a, b) → Map B (f (a), f (b)) is an inclusion of path components, or equivalently if hf is faithful and the commutative square Lemma. If f : A -→ B is faithful then the diagram Fun(C, A) Fun(C, B) Fun(hC, hA) Fun(hC, hB) is cartesian and, in particular, f * : Fun(C, A) -→ Fun(C, B) is again faithful for any C ∈ Cat. If, furthermore, the restriction ι(A) → ι(B) is an inclusion of path components, then so is ι(Fun(C, A)) -→ ι(Fun(C, B)). A.5. Remark. Note that the functor hFun(C, A) -→ Fun(hC, hA) is not usually an equivalence, so the square of the lemma does not agree with the square Fun(C, A) Fun(C, B) hFun(C, A) hFun(C, B)

Map c 2

 2 Cat (H, H) ≃ lim Map Cat (H(n, m), H(n ′ , m ′ ))⊆ lim Map Cat (H(n, m), G(n ′ , m ′ )) ≃ Map c 2 CAT (H, G).Now notice that any automorphism φ of G preserves the sub-diagram H. Indeed, the induced automorphism on G(0, 0) = Cat preserves the full subcategory ∆ ⊂ Cat by [Lu09b Lu09b, Corollary 4.4.11 & Proposition 4.4.13] and naturality with respect to left Kan extension then implies that φ preserves the full subcategory H(n, m) ⊂ G(n, m) as well. The inclusions of path components above therefore refine to inclusions of path components Aut(G, G) ⊆ Aut(H, H) ⊆ Map c 2 CAT (H, G).

  [n] op × [m] op × ∆ H(n, m); (i, j, [k]) h (i,j),[k] .

  [n] op × [m] op × ∆ [n] op × [m] op × ∆ ∆ φn,mis determined by naturality for the codegeneracy map (n, m) → (0, 0) and the composite[n] op × [m] op × ∆ [n] op × [m] op × ∆ [n] op × [m] op φn,mby naturality with respect to the boundary maps (0, 0) → (n, m). Thus Aut(H) ≃ Aut(∆) = Z/2 as desired.

  [n] × [m], Gpd) of diagrams of spaces given by left Kan extensions along (i, j) : [0] -→ [n] × [m] of the constant diagram on the point. One then identifies the diagram H Gpd , with functoriality by left Kan extension, with the obvious diagram sending (n, m) → [n] op × [m] op . This has trivial automorphisms. For F Gray , one uses the dense subcategories H Gray (n, m) = ∆ [n]⊠[m] ⊆ Fun([n] ⊠ [m], Cat).

  j),[k] : [n] ⊠ [m] -→ Cat, (a, b) -→ Map [n]⊠[m] (i, j), (a, b) × [k].

: Cat 2

 2 Cart(∆) defined as follows: for each (∞, 2)-category C and [k] ∈ ∆, let Φ(C) k be the ∞-category obtained by applying the monoidal functor Map Cat ([k], -) : Cat -→ Gpd to each mapping object. By naturality in [k], this defines a simplicial diagram of ∞-categories, whose underlying simplicial diagram of spaces of objects is constant; we define Φ(C) -→ ∆ to be the cartesian unstraightening of this simplicial diagram of ∞-categories. Unraveling the definitions then shows that the codegeneracy map H Gray (n, m) -→ H Gray (0, 0) ∼ = ∆ can be identified with the cartesian fibration between 0-truncated ∞-categories Φ([n] ⊠ [m]) -→ ∆ (naturally in [n] and [m]).

H

  Grayψ * H Gray of bicosimplicial diagrams in the over-∞-category Cat/H Gray (0, 0). By the above discussion, H actually determines a cosimplicial diagram in the subcategory Cart(H Gray (0, 0)) ⊆ Cat/H Gray (0, 0). By Lemma A.4 A.4, it then suffices to compute the sets of isomorphisms H Gray -→ ψ * H Gray within Cart(H Gray (0, 0)).Identifying H(0, 0) ∼ = ∆, it now suffices to compute the set of isomorphisms of bicosimplicial diagrams in Cart(∆)Φ [-] ⊠ [-] ψ * Φ [-] ⊠ [-]for all ψ ∈ Aut(∆) ∼ = {id, op}. Since Φ : Cat 2 -→ Cart(∆) is fully faithful (at least on gaunt 2-categories), this comes down to computing the sets of natural isomorphismsφ : [n] ⊠ [m] -→ [n] ⊠ [m] and φ ′ : [n] ⊠ [m] -→ [n] ⊠ [m] 2-op .

  Vers une axiomatisation de la théorie des catégories supérieures. K-theory, vol. 34 no. 3, 233-263 (2005) [To20] T. Torii, On quasi-categories of comodules and Landweber exactness. Bousfield classes and Ohkawa's theorem. Springer Proceedings in Mathematics & Statistics, vol. 309, 325-380, Springer (2020) Institutt for matematiske fag, NTNU Trondheim, Norway Email address: rune.haugseng@ntnu.no

  g ingressive and f egressive or (d) vice versa. The squares (c) and (d) are furthermore cartesian. Now, F preserving ambigressive (cartesian) squares of type (a) is equivalent to F ′ taking values in Fun AdTrip (X, Y ). Preserving the ambigressive (automatically cartesian) squares of type (c) and (d) then corresponds precisely to F ′ : Z -→ Fun AdTrip (X, Y ) preserving ingressive and egressive morphisms. Finally, F preserves the cartesian ambigressive squares (b) if and only if F ′ preserves ambigressive pullbacks.

□

One can generate more interesting examples of adequate triples using the following criterion: 2.6. Proposition. Let p : Y -→ X be a functor and let (X, X in , X eg ) be an adequate triple such that Y has all p-cartesian lifts over X in . Then Y is part of an adequate triple Y, Y †

  Map AdTrip Tw r ([-] op ), X ≃ Map AdTrip Tw r ([-]) rev , X ≃ Map AdTrip Tw r ([-]), X rev

	. For each [n], consider Tw r ([n]) and Tw r ([n] op ) with the structure of an adequate triple
	as in Example 2.9 2.9. There is an equivalence of adequate triples Tw r ([n] op ) ≃ Tw r ([n]) rev , natural
	in [n], sending an object (i ≥ j) in Tw r ([n] op ) to the object (j ≤ i) in Tw r ([n]). This induces a
	natural equivalence of simplicial objects
	so that the simplicial ∞-groupoid defining Span(X) is the opposite of that defining Span(X rev ).
	The result follows since generally ac(T ) op ≃ ac(T op ) as this is true on simplices.	□
	2.15. Proposition. Let A ⊆ B be a wide subcategory. Then the triples (B, A, ιB) and (B, ιB, A)
	are adequate and
	Span(B, A, ιB) ≃ A and Span(B, ιB, A) ≃ A op .

  and only if it is right Kan extended from its restriction to J n , which has to lie in J n . The claim now follows from [Lu09a Lu09a, Proposition 4.3.2.15], since right Kan extension from a full subcategory is fully faithful.

  Proof. Apply Theorem 3.8 3.8 to the identity functor on AdTrip and use that the cartesian fibration classified by the identity is equivalent to ( * // lax AdTrip) op -→ AdTrip op , with the adequate triple Remark. Applying the conclusions of Remark 3.16 3.16, we obtain that if we declare a map (f, µ) in * // lax AdTrip to be ingressive if µ is ingressive and egressive if µ is egressive and f is an equivalence, then we obtain a different adequate triple structure on ( * // lax AdTrip) Mackey functors. By inspecting the definition of A eff [Na16 Na16, Definition 4.10], one observes that A eff = Span(Y ), where Y is the cartesian unstraightening of the functor underlying

	structure from 3.5 3.5 corresponding precisely to the one from Construction 3.18 3.18 under taking opposite
	categories.	□
	3.21.

3.19.

Remark. If we write * // lax Cat for the domain of the cocartesian fibration classified by (-) op : Cat -→ Cat, then * // lax AdTrip = * // lax Cat × Cat AdTrip. We will show in Section 7 7 (see specifically Remark 7.22 7.22) that * // lax Cat is equivalent to the lax under-category, defined a priori via a construction in (∞, 2)-category theory. 3.20. Corollary. This defines the structure of an adequate triple on * // lax AdTrip op and the map Span(p) : Span ( * // lax AdTrip) op -→ Span(AdTrip op , ιAdTrip op , AdTrip op ) ≃ AdTrip is a cocartesian fibration, classified by the functor Span : AdTrip -→ Cat. op , which gives rise to a cartesian fibration Span(( * // lax AdTrip) op ) -→ AdTrip op , classifying the functor Span : AdTrip -→ Cat. 3.22. Remark. In [Na16 Na16], Nardin proves that the T -∞-category of T -commutative monoids in a T -∞-category C with finite T -products is equivalent to the T -∞-category Fun × T (A eff (T ), C), where A eff is a cocartesian fibration whose fibre over t ∈ T is equivalent to Span((F T ) /t ), where F T is the finite coproduct completion of T . This indicates that T -commutative monoids are computed by a generalisation of

  explicitly, this means that i * and j * jointly detect equivalences and j * i * sends every object to the terminal object, but there are several other characterisations, see [CDH+20b CDH+20b, Appendix A.2]. In particular, by [CDH+20b CDH+20b, Lemma A.2.5] i * admits a further right adjoint i ! such that i

	U	j *	X	i *	Z;
		j *		i *

categories admits the structure of an orthogonal adequate triple, denoted (A, B) ⊥ , with ingressives A × ιB and egressives ιA × B. 4.6. Example. Let X be a stable ∞-category, together with reflective stable subcategories exhibiting X as a recollement

[START_REF]Higher Algebra[END_REF] Definition A.8.1]

  6.7. Remark. In [HHLN21 HHLN21, Section 2] we recorded various other ways to recognise (op-)Gray fibrations and curved orthofibrations. For example, (p 1 , p 2 ) : Y -→ A × B is a curved orthofibration if p 1 is a cartesian fibration and p 2 is a cocartesian fibration [HHLN21 HHLN21, Proposition 2.3.3], i.e.

CrvOrtho(A, B) = LCart(A, B) ∩ RCocart(A, B). 6.8. Lemma. A map p = (p 1 , p 2 ) : Y -→ A × B is an orthofibration in the sense of Definition 5.18 5.18 if and only if it is a curved orthofibration and for each map α : a -→ a ′ , the cartesian transport functor α * : Y a ′ -→ Y a preserves p 2 -cocartesian arrows. Proof. Let us start by noting that an arrow in some fibre Y a ⊆ Y is p 2 -cocartesian if and only if it defines a cocartesian arrow for the base change of p to ιA × B; see [HHLN21 HHLN21, Corollary 2.2.7]. This base change coincides with p eg : Y eg -→ ((A × B) ⊥ ) eg , so an arrow in Y a is p 2 -cocartesian if and only if it defines a p eg -cocartesian arrow. Using this, it follows that α * preserves p 2 -cocartesian arrows if and only if the following holds: for each square in Y y a,b

  6.18. Proposition. The four natural equivalences Ortho → F given by × B op , Cat) : Un oc .

	Ortho(A, B)	D cc		Cocart(A × B op )	Str cc	Fun(A × B op , Cat)
	Ortho(A, B)	D ct		Cart(A op × B)	Str ct	Fun(A × B op , Cat)
	Ortho(A, B)	Str cc	Fun(A, Cart(B))	Str ct	Fun(A × B op , Cat)
	and
	Ortho(A, B)	Str ct	Fun(B op , Cocart(A))	Str cc	Fun(A × B op , Cat)
	are pairwise equivalent.
	6.19. Definition. We shall refer to any of the functors above as the orthocartesian (un)straightening
	equivalence, in formulae
	Str oc : Ortho(A, B) Fun(A For example we learn from Example 6.15 6.15, that
	Str oc ((s, t) : Ar

  To see that this agrees with the description of ϕ 1 in terms of scaled simplicial sets, it suffices to note that this strict 2-functor is uniquely determined by its values on objects and 1-morphisms. 7.20. Remark. More generally, since [n] ⋆ opl [n] op is a gaunt 2-category, ϕ n is adjoint to a (strict) functor Tw r ([n]) op -→ Ar opl [n] ⋆ opl [n] op into its strict oplax arrow category (see Example 7.6 7.6). This functor sends each (i ≤ j) in Tw r ([n]) to the arrow j → i in [n] ⋆ opl [n] op of minimal height.

	00	∼	01		11	0	1
	7.19)			∼
	00		01	∼	11	0	1
	obtained by collapsing the 1-and 2-morphisms marked by ∼.

Proof. Note that (A, B) ⊥ decomposes as a product of the triples (A, A, ιA) and (B, ιB, B), both of which are evidently orthogonal and adequate (and both these properties are preserved under taking products). Since the span construction preserves products, being a right adjoint, the claim follows from Proposition 2.15 2.15. □ Consequently, Theorem 5.20 5.20 provides equivalences between certain types of fibrations over A × B and A × B op , respectively. Recall from Theorem 5.20 5.20 that these equivalences take opposite categories at the level of fibres. To conform with the conventions of [HHLN21 HHLN21], we will therefore compose the equivalences of Theorem 5.20 5.20 with taking opposites, resulting in: 6.11. Definition. For ∞-categories A and B we define functors SD cc : LCart(A, B)

LCart(A, B op ) LCocart(A op , B)

Span ⊥ (-) op (-) op Span ⊥ using the identifications from Observation 6.1 6.1 and Lemma 6.10 6.10.

Using Theorem 5.20 5.20 and the identifications Corollary 6.5 6.5 from Corollary 6.9 6.9 we then immediately find: 6.12. Corollary. The functor SD cc : LCart(A, B) → LCocart(A op , B) is an equivalence with inverse SD ct , and restricts to equivalences where the left map z → y is p-cartesian. Dually, for a map q : X → A × B in LCocart(A, B) we find SD ct (Y ) → A op × B has morphisms from y to y ′ given by diagrams

where the right map y ′ → z is q-cocartesian.

Taking B = * , Corollary 6.12 6.12 reproduces the dualisation equivalence between cartesian fibrations over A and cocartesian fibrations over A op from [ [START_REF] Barwick | Dualizing cartesian and cocartesian fibrations[END_REF]], which their main result shows is equivalent (naturally in A and B) to Cart(A)

We gave a more direct proof of this fact as Theorem 3.17 3.17 above (it also follows from the unicity results from Appendix A A). In [HHLN21 HHLN21, Section 2.5] we extended this procedure to an equivalence

by straightening in one variable. We deduce: 6.14. Corollary. The functors

LCocart(A op , B) : SD ct agree with the equivalences D cc and D ct constructed in [HHLN21 HHLN21, Section 2.5].

To manipulate simple diagrams in (∞, 2)-categories, such as lax commuting squares and triangles, let us recall from [START_REF] Barwick | On the Unicity of the Homotopy Theory of Higher Categories[END_REF]] that a gaunt 2-category is a strict 2-category whose only invertible 1and 2-cells are the identities. For example, a gaunt 1-category A gives rise to a gaunt 2-category

Every gaunt 2-category determines an object in Cat(sSet + ) by taking the nerves of its mapping categories (with degenerate marking) and this determines a fully faithful functor Gaunt 2 → Cat 2 from the (ordinary) category of gaunt 2-categories into the ∞-category of (∞, 2)-categories, with essential image given by the 0-truncated objects [BSP21 (1) Pushouts of the form

(2) Pushouts of the form [2]

γ -→ X, freely adding a factorisation γ = αβ.

(3) Pushouts of the form

-→ X for i = 0, 1, where γ : x → y is an arrow in X such that x has no nontrivial incoming arrows and y has no nontrivial outgoing arrows.

Proof. Let us first give explicit descriptions of these three types of strict pushouts in Cat + ∆ , for an arbitrary marked simplicial category X. These will show that when X arises from a gaunt 2-category, then so does the pushout X ′ .

Case (1) (1): We only treat the pushout along {0} -→ [1]. The pushout X ′ contains X as a full subcategory, together with a new object z, so that for any object u ∈ X

Composition is defined by acting on the right factor of A × Map X (u, x). Case (2) (2): The pushout X ′ then freely adds a factorisation γ = αβ. More precisely X ′ has one additional object z and mapping categories

is an isomorphism for all u ∈ X. The composition is then the evident one, using the (formal) relation that αβ = γ. Case (3) (3): The pushout X ′ has the same objects and mapping categories as X, except for Map X ′ (x, y), which is given by a (homotopy) pushout [1] ←-{i} -→ Map X (x, y). Note that when Map X (x, y) is the nerve of a gaunt 1-category, this pushout of marked simplicial sets is weakly equivalent to the nerve of a gaunt 1-category (essentially by Case (1) (1)).

It remains to show that the above (strict) pushouts also model the pushout in the ∞-category Cat 2 . From the above descriptions, one sees that a Dwyer-Kan equivalence of marked simplicial categories X -→ Y induces a Dwyer-Kan equivalence between the resulting pushouts: It clearly induces weak equivalences on mapping objects, and since we add at most one extra object it remains essentially surjective on homotopy categories. Since Cat(sSet + ) is left proper [Lu09a Lu09a, Proposition A.3.2.4], this means that the above pushouts are all homotopy pushouts, and the result follows.

□

The ∞-category Cat 2 admits a closed (non-symmetric) monoidal structure given by the (oplax) Gray tensor product ⊠. The Gray tensor product is induced by a Quillen bifunctor ⊠ : sSet sc × sSet sc -→ sSet sc . The scaled simplicial set (X, S)⊠(Y, T ) equals (X×Y, S ⊠T ), where a 2-simplex (σ, σ ′ ) is in S ⊠ T if the following conditions hold:

Finally, let α : a ′ -→ a, β : b -→ b ′ and let f : a -→ b be an object in the fibre Ar opl (X) (a,b) . Then the following (commuting) cube in X

a square in Ar opl (X) (actually contained in Ar(X)) in which two sides are cocartesian and the other two sides are cartesian arrows. This means that the corresponding interpolating edge is an equivalence. □

Let us now introduce the model for the enhanced twisted arrow ∞-category given in [AGS20 AGS20], which is constructed via the model of scaled simplicial sets. 7.10. Notation. In the following we will denote the image of i under the inclusion ∆ n ⊂ ∆ n ⋆ ∆ n,op by i and the image of j under the image of [n] op ⊂ ∆ n ⋆ ∆ n,op by j. 7.11. Definition. Consider the cosimplicial object

sending each n to the join ∆ n ⋆ ∆ n,op where a 2-simplex σ ∈ ∆ n ⋆ ∆ n,op is thin if one of the following conditions holds:

The functor κ r induces a functor Tw r = S κr : sSet sc -→ sSet from scaled simplicial sets to simplicial sets. We enhance this to a functor with values in marked simplicial sets by declaring an edge in Tw r (X, S) to be marked if it arises from a totally thin diagram (∆ 1 ⋆ ∆ 1,op ) ♯ -→ (X, S) and write Tw r + (-) : sSet sc sSet +

for the resulting functor. Note that the two inclusions

7.12. Theorem (Abellán García, Stern). Let p : (Y, T ) -→ (X, S) be a fibration between fibrant scaled simplicial sets. Then the map

is a fibration between fibrant objects in the cartesian model structure on (sSet + ) /Y ×Y op (where all arrows in X × X op and Y × Y op are marked).

For each fibrant scaled simplicial set (X, S), the cartesian fibration Tw r (X, S) -→ X × X op is classified by the functor Map X : X op × X -→ Cat.

Proof. For the first part, it suffices to verify that Tw r + (Y, T ) -→ Tw r + (X, S) × X×X op Y × Y op has the right lifting property with respect to all marked anodyne maps [Lu17 Lu17, Proposition B.2.7]. Let us write L for the left adjoint of Tw r + and note that for each marked simplicial set (A, E), there is a natural map

The desired right lifting property now follows from the fact that for each (generating) marked anodyne map (A, E) -→ (B, F ), the map

is a scaled anodyne map [AGS20 AGS20, Lemma 2.9 and 2.10]. The second part of the theorem is [AGS20 AGS20, Theorem 3.3]. □ 7.13. Corollary. The functor Tw r : sSet sc -→ sSet is a right Quillen functor, where sSet is endowed with the Joyal model structure.

Proof. The left adjoint to Tw r preserves cofibrations, which are simply monomorphisms. Furthermore, Theorem 7.12 7.12 implies that the map

) is a composite of fibrations in the cartesian model structure. Since fibrations in the cartesian model structure are in particular categorical fibrations [Lu09a Lu09a, Proposition 3.1.5.3], we conclude that Tw r sends fibrations between fibrant scaled simplicial sets to categorical fibrations between quasicategories. □ 7.14. Definition. The right Quillen functor from Corollary 7.13 7.13 induces a right adjoint functor of ∞-categories which we will denote Tw r : Cat 2 -→ Cat 1 and refer to as taking the oplax twisted arrow ∞-category.

To compare with the oplax arrow category, it will be convenient to give a slightly different presentation of Tw r (X) in terms of Segal spaces (which a posteriori does not make use of scaled simplicial sets). To this end, let us make the following observation: Proof. We will provide a natural weak equivalence of cosimplicial diagrams in Cat(sSet + )

where the target is viewed as a marked simplicially enriched category by taking nerves of the mapping categories.

Unraveling the definitions, C sc ∆ n ⋆ sc ∆ n,op is given as follows. Its objects are the objects of the poset 0 For each pushout where we add a marked cell, collapsing the resulting cell in Cat 2 simply has the result of not adding a cell at all. Consequently, the (∞, 2)-category obtained from (Tw r ([1]) × [1]) op by collapsing the marked cells is the gaunt 2-category obtained by performing only those cell attachments where no marked cell was added. The result of this is then precisely

Again by Theorem 7.21 7.21, this classifies the enhanced mapping functor of X 1-op , which is equivalent to the functor

Applying this in the case X = Cat, we obtain that the functor

is a cartesian fibration which classifies the identity on Cat. Because the fibre of t : Ar lax (Cat) → Cat over { * } is one definition for the lax over-category, this makes precise the statement that ( * // lax Cat) op → Cat op is the universal cartesian fibration.

7.23. Remark. Let us also briefly discuss the cocartesian fibration encoding the enhanced mapping functor of an (∞, 2)-category X. Just as for ∞-categories it is given by a version of the left twisted arrow ∞-category. However, simply taking the opposite of Tw r (X), as one does for ∞categories, would also apply (-) op to the fibres. Therefore we instead consider the simplicial object κ l = (κ r ) rev,2-op in Cat 2 obtained from that defining Tw r by reversing the simplicial direction and taking opposites for 2-morphisms at each level, and define Tw ℓ (X) = S κ l (X). By direct inspection one then finds Tw ℓ (X) ≃ Tw r (X 2-op ) op . In particular, it follows from Theorem 7.12 7.12 that (s, t) : Tw ℓ (X) -→ X op × X is a cocartesian fibration, which classifies the composition

We claim this composite is the enhanced mapping functor of X. One way to see this is to note that the functor (-) 2-op : Cat 2 → Cat 2 is modelled on categories enriched in marked simplicial sets (which we used to define the enhanced mapping functor) by changing the enrichment via (-) op : sSet + → sSet + . Let us remark that the model of scaled simplicial sets does not seem to admit a simple implementation of reversing 2-morphisms, so in particular we do not know of a simplicial object in scaled simplicial sets giving rise to κ rev,2-op , and consequently also no explicit model for Tw ℓ (X) as a scaled simplicial set even if X is given as such. Nevertheless, applying the equivalence of Lemma 7.16 7.16 one can write down a cosimplicial diagram of gaunt 2-categories equivalent to κ l . Note also that Tw r (X 1-op ) ≃ Tw r (X) as cartesian fibrations over X × X op , since (κ r ) 1-op ≃ κ r .

cocartesian unstraightening of Tw ℓ (A) → A op × A. The definition of the twisted arrow category makes this left fibration functorial in A. More precisely, we can consider the pullback square

and note that Tw ℓ defines a functor from Cat to the full subcategory of E spanned by the left fibrations. This subcategory can also be regarded as the cartesian unstraightening of the functor (A, B) → LFib(A op × B). There are equivalences LFib(A op × B)

which are natural in the input ∞-categories (where the contravariant functoriality on P(-) is given by precomposition). Writing F : (Cat op ) 2 → Cat for the functor sending (A, B) → Fun(B, P(A)), we therefore obtain a functor Tw ℓ : Cat -→ Un ct (F).

On objects, this takes an ∞-category A to its Yoneda embedding and on morphisms it witnesses the lax commutativity of the diagrams Let Un oc (G) -→ Cat × Cat be the orthocartesian unstraightening of G. We claim that there is a canonical equivalence Un ct (F) ≃ Un oc (G). To this end invoke Corollary 6.20 6.20 to write the cartesian unstraightening functor as Fun(B op , Fun(A op , Cat))

and the orthocartesian one as Fun(B op , Fun(A, Cat))

But now by definition, G and F have the same image in Fun(Cat op , Bicart(Cat)), viewed as a subcategory of Fun(Cat op , Cocart(Cat)) and Fun(Cat op , Cart(Cat)) respectively. This gives the claim that Un ct (F) ≃ Un oc (G).

The resulting functor Tw ℓ : Cat → Un oc (G) again takes A to its Yoneda embedding, but this time morphisms witness lax commuting squares

But by [Lu09a Lu09a, Proposition 5.2.6.3] (or rather the second step of its proof), the natural transformation µ is an equivalence, i.e. the diagram actually commutes.

Armed with this information consider the diagram Un oc (G) Un oc (Fun(-, -))

whose top right corner is the oplax arrow ∞-category Ar opl (Cat) of large ∞-categories by Theorem 7.21 7.21. Via the inclusion Map Cat (A, B) ⊆ Fun(A, B) it contains the actual arrow ∞-category Ar(Cat) as a wide subcategory and by the previous observation the composite

actually takes values in this subcategory. The resulting functor Cat → Ar(Cat) is then the natural transformation we set out to construct. □

Appendix A. Uniqueness of straightening and dualisation

To resolve coherence questions surrounding the dualisation and straightening equivalences once and for all, we will prove a rigidity result in this appendix, implying that any two ways of straightening or dualising a two-variable fibration are naturally equivalent; namely, we compute the automorphism groups of the functors

following ideas of [BGN18 BGN18], who treated the case of a single variable. Here ⊠ denotes the Gray tensor product (see Section 7 7) and Fun(A ⊠ B, Cat) denotes the ∞-category of functors from the (∞, 2)-category A ⊠ B to the (∞, 2)-category of ∞-categories. By [HHLN21 HHLN21, 5.2.9] the functor F Gray is equivalent to the functor Gray(-, -) which sends a pair (A, B) to the ∞-category of Gray fibrations over A × B (see Remark 6.6 6.6). Before we describe the result of our calculations, note that the functor F carries a natural action of the group Aut(Cat), acting pointwise on Fun(A × B, Cat) by postcomposition. When evaluated at ([0], [0]) ∈ Cat ×2 , this simply gives the canonical action of Aut(Cat) on Cat. By a theorem of Toën, Aut(Cat) is discrete with two path components, corresponding to the identity and (-) op : Cat → Cat [To05 To05]. Our goal will be to prove: A.1. Theorem. Acting by postcomposition and evaluation at

In particular, Aut(F) is discrete with π 0 Aut(F) = Z/2 having its nontrivial element induced by postcomposition with (-) op : Cat → Cat. Furthermore, Aut(F Gpd ) ≃ * and Aut(F Gray ) ≃ * . A.6. Lemma. Let X be an (∞, 2)-category and Fun(X, Cat) the ∞-category of 2-functors X -→ Cat. For each x ∈ X and A ∈ Cat, there exists an object h x,A ∈ Fun(X, Cat) with the universal property

The full subcategory ∆ X ⊆ Fun(X, Cat) spanned by all h x,[k] with x ∈ X and k ≥ 0 then forms a dense subcategory.

A.8. Observation. For any functor f : X -→ Y, let f ! : Fun(X, Cat) -→ Fun(Y, Cat) be the left adjoint to the restriction functor. By the universal property (A.7 A.7), this sends h x,A to h f (x),A and hence restricts to

Proof. Let us start by observing that by the enriched Yoneda lemma [Hi20 Hi20, Proposition 6.2.7], these universal functors h x,A indeed exist and are given by (A.9)

Moreover, Hinich's results show that the ∞-category Fun(X, Cat) can be described as that of X-modules in Cat (in the sense of algebras for a many-object module ∞-operad whose underlying many-object algebra is the Cat-enriched ∞-category X). From this description it follows that Fun(X, Cat) admits all colimits, and so the inclusion j : ∆ X → Fun(X, Cat) induces an adjoint pair j ! : P(∆ op X ) ⇆ Fun(X, Cat) : j * and we have to verify that the counit map (A.10)

is an equivalence for all F . We will prove this in increasing levels of generality. First, suppose that F = h y,A . In this case, we claim that there is a cofinal functor h y : ∆/A -→ ∆ X /h y,A sending each [k] → A to h y,[k] -→ h y,A . The result then follows from the fact that A -→ h y,A preserves colimits and that A ≃ colim [k]∈∆/A [k]. To see that h y is indeed cofinal, it suffices to verify that the natural map

Map Fun(X,Cat) h x,[n] , h y,A is an equivalence. From the universal property (A.7 A.7) and the explicit formula (A.9 A.9) for h y,A , one sees that this map is equivalent to colim

Taking the constant factor Map Cat [n], Map X (y, x) out of the colimit, one sees that this is an equivalence since ∆ → Cat is a dense subcategory [Re01 Re01].

Next, note that the class of F for which (A.10 A.10) is an equivalence is closed under all colimits that are preserved by j * . For example, one easily sees from the universal property (A.7 A.7) that j * preserves all coproducts.

To conclude, we now write each functor F as the geometric realisation of a bar construction. More precisely, let us fix a set S and an essentially surjective functor f : S -→ X. The description of Fun(X, Cat) as modules then gives a free/forgetful adjunction (A.11) f ! : S Cat Fun X, Cat : f * where f * evaluates at each object in S. Note that f * preserves all limits and colimits and detects equivalences, so that (f ! , f * ) is a monadic adjunction. The left adjoint f ! sends each (A s ) s∈S ∈ Cat to s∈S h s,As . Any F : X -→ Cat can now be written as the geometric realisation F ≃ |B • | where B • = Bar(f ! , f * f ! , f * F ) is the usual bar resolution of F for the monadic adjunction (A.11 A.11). The functor j * preserves these geometric realisations, i.e. for any x ∈ X and k ≥ 0, the map Map

x) is an equivalence. Indeed, up to equivalence we may take x = f (s), in which case this follows from the augmented simplicial object f * B • → f * F having extra degeneracies [Lu17 Lu17, Example 4.7.2.7]. Since each term in the bar construction is a coproduct of functors of the form h f (s),A , for which we have already verified that the counit map (A.10 A.10) is an equivalence, it follows that (A.10 A.10) is an equivalence for F as well. □ Proof of Theorem A.1 A.1. We will spell out the case of F and discuss the modifications required for F Gpd and F Gray at the end of the proof.

Step 1: reducing to simplices. Recall that precomposition with the localisation ac : sGpd → Cat gives a fully faithful embedding Fun(Cat op × Cat op , X) -→ Fun(sGpd op × sGpd op , X)

for any ∞-category X. Since ac preserves colimits, this inclusion furthermore preserves the property of commuting with small limits in each variable separately, and on the right the full subcategory spanned by such functors is equivalent to Fun(∆ op × ∆ op , X). Since F : Cat op × Cat op → Cat does indeed preserve small limits each variable, we find that Aut(F) agrees with the automorphisms of the (large) bisimplicial ∞-category F |∆ 2 given by (n, m) → Fun([n] × [m], Cat), with functoriality arising from restriction.

Step 2: restricting to generators. Note that F |∆ 2 takes values in the subcategory Pr R of Cat consisting of presentable ∞-categories and right adjoint functors. Although the inclusion Pr R ⊆ Cat is not fully faithful, it does induce a fully faithful map on ∞-groupoid cores, and by Lemma A.4 A.4 this feature persists to simplicial objects. We may therefore compute the automorphisms of F |∆ 2 as a bisimplicial object in Pr R instead.

From the equivalence Pr R ≃ (Pr L ) op , given by taking adjoints, we find Fun(∆ op × ∆ op , Pr R ) ≃ Fun(∆ × ∆, Pr L ) op .

By another application of