David Kern

Monoidal envelopes and Grothendieck construction for dendroidal Segal objects

We propose a construction of the monoidal envelope of ∞-operads in the model of Segal dendroidal spaces, and use it to define cocartesian fibrations of such. We achieve this by viewing the dendroidal category as a "plus construction" of the category of pointed finite sets, and work in the more general language of algebraic patterns for Segal conditions. Finally, we rephrase Lurie's definition of cartesian structures as exhibiting the categorical fibrations coming from envelopes, and deduce a straightening/unstraightening equivalence for dendroidal spaces.

Introduction

Any monoidal category V ⊗ defines an operad whose colours are the objects of V and whose multimorphisms C 1 , • • • , C n → D are given by the morphisms C 1 ⊗• • •⊗C n → D. The operads which arise in this way are said to be representable; to avoid confusion -since we shall model operads as certain presheaves -we will call them representably monoidal. Indeed, the tautological multimorphism

C 1 , • • • , C n → C 1 ⊗ • • • ⊗ C n (corresponding to id C 1 ⊗•••⊗Cn
) carries the universal property that every multimorphism with source (C 1 , • • • , C n ) must factor through it, by a unique unary morphism of source C 1 ⊗ • • • ⊗ C n . This universal property can be seen as a cocartesianity condition.

Recall indeed that, if p : E → B is an opfibration (or cocartesian fibration), there is a factorisation system on E whose left class consists of p-cocartesian morphisms and whose right class consists of purely p-vertical morphisms. One can also define a notion of opfibration of multicategories, as done for example in [START_REF] Hermida | Fibrations for abstract multicategories[END_REF] for generalised multicategories, and more generally of cocartesian (multi)morphisms therein, so that a multicategory is representably monoidal if and only if its morphism to the terminal operad is an opfibration. In general, an opfibration of multicagories can be thought of as a morphism whose fibres are monoidal categories: since the selection of a colour comes from the operad generated by the nodeless edge η, which only has unary morphisms, the vertical arrows are always unary ones, while the cocartesian arrows are those exhibiting tensor products.

In the homotopical setting, there are various ways of modelling ∞-operads, each coming with its advantages and drawbacks. The model preferred by [START_REF] Lurie | Higher Algebra[END_REF], which represents an ∞-operad by its ∞-category of operators, is biased to make cocartesian fibrations and monoidal envelopes immediately accessible, but requires entangling combinatorics to recover an operadic intuition on any construction. On the other hand, the dendroidal models, based on categories of trees as the shapes for ∞-operads, is closer to the diagramatic operadic intuition, but generally requires more work to make any construction. For exemple, cocartesian fibrations and their straightening were studied in [START_REF] Heuts | Algebras over infinity-operads[END_REF] in the model of dendroidal sets, a point-set model which is not manifestly model-invariant.

A more homotopically robust model is that of Segal dendroidal spaces, or more generally Segal dendroidal objects in any complete (∞, 1)-category. This model makes it clear that all that is needed to model algebraic objects such as operads is a general category of "shapes" and their Segal decompositions. This philosophy was realised in [START_REF] Chu | Homotopy-coherent algebra via Segal conditions[END_REF] which defines a notion of algebraic pattern, a base shape category over which can be defined Segal presheaves.

Thus one would like to define a notion of cocartesian operations and fibrations for Segal objects over algebraic patterns, and in addition construct the free fibration generated by an arbitrary morphism. Here, we will not achieve this generality, but take a simpler route to defining opfibrations. In [START_REF] Hermida | Fibrations for abstract multicategories[END_REF], opfibrations of multicategories were characterised as the morphisms whose image under the functor taking monoidal envelopes is an opfibration (or cocartesian fibration) of underlying categories. We will follow this idea, and focus on constructing an envelope ∞-functor and exploiting it to define cocartesian fibrations.

The direction problem and the plus construction

To perform the above requires a notion of direction for the operations of Segal objects: in categories, there are two directions for cartesianity, from the source or from the target (giving rise to cartesian and cocartesian morphisms), while for operads or their "manyobjects" generalisation, virtual double categories, the notion of (co)cartesian morphisms which has so far been explored looks from the target (as in [CS10, Definition 5.1]), but we expect that any choice of input to separate may give a notion of cartesianity. Not all choices of direction, however, are as good as others: for operads, the choice of the output to keep apart leads to functoriality for composition, while the other choices do not (due to the absence of a duality operation as for categories). As a consequence, one may need to restrict the study to "good" orientations; this seems likely to be impossible to find for modular operads.

In this work, we have elected to eschew the problem by replacing a perhaps more canonical definition of directibility by a more practical one. The presence of a notion of direction for operations means that we can think of any composite of them as having an order of progression. This is what is captured by the objects of the simplex category ∆, chains of morphisms going from a beginning to an end. Hence we will base our notion of direction on this category, declaring an algebraic pattern to be well-directed if it can be written as the output of a certain construction involving ∆. The appropriate construction to consider turns out to be a variant of the plus construction suggested by Baez-Dolan and studied by, among others, [START_REF] Barwick | From operator categories to higher operads[END_REF] and [START_REF] Berger | Moment categories and operads[END_REF].

The plus construction of a pattern O is characterised by the fact that its Segal objects are the weak Segal O-fibrations, or "O-operads". In this language, envelopes for weak Segal fibrations of patterns have been recently constructed in [START_REF] Shaul Barkan | Envelopes for Algebraic Patterns[END_REF], building on the construction of [START_REF] Lurie | Higher Algebra[END_REF] and the properties exhibited in [START_REF] Haugseng | ∞-operads as symmetric monoidal ∞-categories[END_REF]. While in a context equivalent to ours, this construction has orthogonal goals to the one performed here, as we wish to always remain in the explicit setting of Segal objects.

Idea of the construction

In order to understand the idea of our construction of the monoidal envelope functor, we describe it here in the case of the pattern Γ op , whose Segal objects are the commutative monoids and whose plus construction parameterises operads.

Let O be an operad. Its monoidal envelope is constructed as a monoidal category Env(O) which has as set of objects the free monoid generated by the colours of O, whose elements are denoted as

C 1 ⊗ • • • ⊗ C n or simply C 1 • • • C n . If C 1 ⊗ • • • ⊗ C n is such a string of colours of O and D is one colour, a morphism C 1 ⊗ • • • ⊗ C n → D is given by a multimorphism C 1 , . . . , C n → D in O. If C 1 ⊗ • • • ⊗ C n and D 1 ⊗ • • • ⊗ D m are two such strings of colours of O, to defines a morphism C 1 ⊗ • • • ⊗ C n → D 1 ⊗ • • • ⊗ D m one needs to further select of partition of the inputs (C 1 , . . . , C n ) into m (possibly empty) parts.
The above is so far just a description of the underlying category Env(O) of the envelope of O; to define it as a monoidal category, or representably monoidal operad, one must also define multimorphisms of higher arity in the operadic structure. Let

(C i 1 ⊗ • • • ⊗ C i p i ) i∈[ [1,n] ] be n colours of Env(O) and let D 1 ⊗ • • • ⊗ D m be a further col- our. By the representability condition, a multimorphism (C 1 1 • • • C 1 p 1 , . . . , C n 1 • • • C n pn ) → D 1 ⊗ • • • ⊗ D m is given by a morphism C 1 1 ⊗ • • • ⊗ C n pn → D 1 ⊗ • • • ⊗ D m
, that is a partition of the entries and a collection of multimorphisms to each D i .

In the dendroidal model, one simply defines the object Env(O)(⋆ n ) of all n-ary morphisms without specifying their sources and targets. To describe this, it becomes useful to reverse the thinking: taking a family of multimorphisms of O, we can ask how to interpret it as a multimorphism in Env(O). If C 1 , • • • , C r is the union of the domains of the multimorphisms in the family considered, the decomposition in family provides a partition of p indexed by the targets; however, from the point of view of Env(O), this partition is completely artificial as it is only used to construct a morphism whose target may consist of several colours. Thus it must be forgotten. Meanwhile, if the family is to be interpreted as a multimorphism of specified arity n in Env(O), the set of colours r] ] must be endowed with a partition into n parts.

(C i ) i∈[ [1,
In a formula, we have that

Env(O)(⋆ n ) = r∈N λ partition of r in n m i=1 O(⋆ λ(i) ).
(1)

Recall that partitions are the same thing as active morphisms in Γ op (which is where the power of the plus construction, relating Γ op to the pattern for operads, comes into play). We can then interpret the coproduct on partitions as a colimit over morphisms in (Γ op ) act . Viewed in this way, this formula is very reminiscent to the one computing (pointwise) oplax extensions1 , with one difference: the colimit is taken only of the trees whose height is that of a corolla, i.e. compatibly with the projection to ∆. To that end, the notion of extension will need to be refined to a "fibrewise" one.

Outline of the paper

In section 3, we define the plus construction for appropriate algebraic patterns, and establish its main properties, in a manner very similar to the study of the plus construction of Γ op♮ carried out in [START_REF] Chu | Two models for the homotopy theory of ∞-operads[END_REF] or [START_REF] Chu | Enriched ∞-operads[END_REF]. Then, in section 4, we will use it to construct the "representably monoidal" envelope of a Segal object. While the construction makes sense for general well-directed patterns, we are only able to exhibit its good monoidal properties by restricting patterns such as the one Ω op♭ for operads. Finally, in section 5 we further use this envelope functor to define a straightening/unstraightening Grothendieck construction for ∞-operads; the results of this section are not new and are simply variants of results appearing in [START_REF] Lurie | Higher Algebra[END_REF], the combinatorics of whose proofs are in fact directly reused, but this language provides a new, more operadic, point of view on them.

The core results of this note were worked out as part of my PhD with Étienne Mann, and I thank him for many useful discussions.

I acknowledge funding from the grant of the Agence Nationale de la Recherche "Categorification in Algebraic Geometry" ANR-17-CE40-0014 and the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant "Derived Symplectic Geometry and Applications" Agreement No. 768679)

2 Review of the language of algebraic patterns 2.1 Algebraic patterns and Segal objects Definition 2.1.1 (Algebraic pattern). An algebraic pattern is an (∞, 1)-category endowed with a unique factorisation system and a selected class of objects called elementary. The morphisms in the left class of the factorisation system are called inert, and those in the right class active.

Notation 2.1.2. Inert maps in an algebraic pattern are usually denoted as , while active maps are denoted as .

If O is the underlying (∞, 1)-category of the algebraic pattern considered, one writes O inrt and O act for the wide and locally full sub-(∞, 1)-categories whose morphisms are respectively the inert and the active morphisms.

We also denote O el the full sub-(∞, 1)-category of O inrt on the selected elementary objects. For any object O ∈ O, we further write

O el O/ := O el × O inrt O inrt O/
, the (∞, 1)category of inert morphisms from O to an elementary object.

Example 2.1.3. Segal's category Γ , (a skeleton of) the opposite of the category FinSet * of pointed finite sets, admits an active-inert factorisation system where a morphism of pointed finite sets f : (S, s 0 ) → (T, t 0 ) is inert if for every t ∈ T \ {t 0 }, the preimage f -1 (t) consists of exactly one element, and active if the only element of S mapped to t 0 is s 0 -so that, in particular, the subcategory of active morphisms Γ opact identifies with the category of finite sets (by mapping (S, s 0 ) to S \ {s 0 }).

The induced inert-active factorisation system on Γ op gives rise to two algebraic pattern structures, from two choices of elementary objects. The algebraic pattern Γ op♭ has as elementaries the two-element sets (isomorphic to 1 ), while the pattern Γ op♮ has as elementaries the singletons (isomorphic to 0 ) and the two-element sets.

Example 2.1.4. The (non-augmented) simplicial indexing category ∆, identified with the category of ordered non-empty finite sets and order-preserving maps between them, admits a factorisation system in which a map is active if it preserves the top and bottom elements, and inert if it corresponds to the inclusion of a linear subset.

The induced inert-active factorisation system on ∆ op gives rise to two algebraic pattern structures: the algebraic pattern ∆ op♭ has as elementaries the two-element sets (isomorphic to [1]), while the pattern Γ op♮ has as elementaries the singletons (isomorphic to [0]) and the two-element sets.

Example 2.1.5. Of particular interest to us, the dendroidal category Ω of Moerdijk-Weiss can be described as a category of (non-planar) rooted trees, with morphisms the morphisms of free (coloured, symmetric) operads generated by these trees. We shall give an alternate construction of (a sufficient subcategory of) it in section 3.

Certain particularly interesting trees can be distinguished:

the free-living edge is the tree, denoted η, consisting of one edge but no vertex (so the operad it freely generates has one colour, and only its identity unary morphism);

the corollas are the trees, denoted ⋆ n with n ∈ N, with a single vertex and n + 1 edges (one of which is the root) attached to it. Note that the corolla ⋆ n with n leaves has as automorphism group the symmetric group S n .

The category Ω admits a factorisation system in which a morphism is active if it is boundary-preserving, and inert if it corresponds to a subtree inclusion (which, in particular, is valence-preserving on the vertices).

The induced inert-active factorisation system on Ω op gives again rise to two algebraic pattern structures: the algebraic pattern Ω op♭ has as elementaries the corollas ⋆ n , while the pattern Ω op♮ has as elementaries the corollas and the free-living edge η. Definition 2.1.6 (Morphisms of algebraic patterns). Let O and P be algebraic patterns. A morphism of algebraic patterns from O to P is an (∞, 1)-functor O → P which preserves active and inert morphisms and elementary objects.

Example 2.1.7. There is a functor ∆ → Ω which, taking the standard skeleton of ∆, sends [n] to the linear tree with n nodes and n + 1 edges. One may remark that it is fully faitful, and can also be identified with the canonical projection functor Ω /η → Ω. By translating the definition of inert and active morphisms for the given factorisation system on ∆ op , one immediately sees that this functor induces morphisms of algebraic patterns ∆ op♮ → Ω op♮ and ∆ op♭ → Ω op♭ . Definition 2.1.8 (Segal objects). Let O be an algebraic pattern. An (∞, 1)-category C is said to be O-complete if it admits limits of diagrams with shape

O el O/ for any O ∈ O. Let C be an O-complete (∞, 1)-category. A Segal O-object in C is an (∞, 1)-functor X : O → C such that X| O inrt is a lax extension of X| O el (along the inclusion). Explicitly, this means that for any O ∈ O the canonical map X(O) → lim E∈O el O/ X(E) (2) is invertible. The (∞, 1)-category Seg O (C) of Segal O-objects in C is the full sub-(∞, 1)-category of C O spanned by the Segal objects.
Example 2.1.9.

• A Segal Γ op♭ -object is a commutative algebra object (or E ∞ -algebra object).

• A Segal ∆ op♭ -object is an associative algebra object (or A ∞ -algebra object, or E 1algebra), while a Segal ∆ op♮ -object is an internal category.

• A Segal Ω op♭ -object is an internal monochromatic operad, while a Segal Ω op♮object is an internal (coloured) operad.

Remark 2.1.10. Let O be an algebraic pattern such that the inclusion

O el ֒→ O inrt is codense. Then for any O ∈ O, the corepresentable y O op (O) : O → ∞-Grpd is a Segal O-∞-groupoid -this

Morphisms of algebraic patterns I: (weak) Segal fibrations

Definition 2.2.1 (Segal morphism of algebraic patterns). A morphism of algebraic patterns F : O → P is said to be a Segal morphism if "it preserves Segal conditions", that is if for any P-complete (∞, 1)-category C, the induced (∞, 1)-functor F * | Seg P : Seg P (C) ⊂

C P → C O factors through Seg O (C) ֒→ C O .
Remark 2.2.2. By [CH21, Lemma 4.5], it is enough to check Segality of a morphism with C = ∞-Grpd, that is to check preservation of Segal ∞-groupoids. Then the condition for F to be a Segal morphism can be written in a formula as: for every O ∈ O, for every Segal P-∞-groupoid X, the morphism of ∞-groupoids

lim P el F(O)/ X → lim O el O/ X • F el (3) induced by the (∞, 1)-functor O el O/ → P el F(O)/ is an equivalence. Construction 2.2.3. Suppose V : O → (∞, 1)-Cat is an O-monoidal (∞, 1)-category.
Passing to the Grothendieck construction of the functor produces a cocartesian fibration co V → O. We shall refer to a cocartesian fibration over O whose associated (∞, 1)- 1. for every object X ∈ X, every inert arrow i :

functor O → (∞,
pX → O in O admits a p-cocartesian lift i ! : X → i ! X; 2. for every object O ∈ O, the (∞, 1)-functor X O → lim E∈O el O/
X E induced by the cocartesian morphisms over inert arrows is invertible; 3. for every X ∈ X and every choice of p-cocartesian lift of the tautological diagram i :

O el pX/ → O (of inert morphisms from pX) to an i ! : (O el pX/ ) ⊳ → X taking the cone point to X, for every Y ∈ X, the commutative square X(Y, X) lim E∈O el pX/ X(Y, i ! (E)) O(pY, pX) lim E∈O el pX/ O(pY, i(E) = E) (4) is cartesian.
Example 2.2.5.

• A weak Segal Γ op♭ -fibration is the (∞, 1)-category of operators of an ∞-operad in the sense of [Lur17, Definition 2.1.1.10], while a weak Segal Γ op♮fibration is a generalised ∞-operad in the sense of [Lur17, Definition 2.3.2.1].

• A weak Segal ∆ op♭ -fibration is the (∞, 1)-category of operators of a non-symmetric ∞-operad in the sense of [ It can be checked directly that any Segal fibration as in construction 2.2.3 is in particular a weak Segal fibration. In fact Segal O-fibrations are exactly those (∞, 1)-functors to O which are both weak Segal fibrations and cocartesian fibrations. This provides a (non-full) inclusion (∞, 2)-functor from the (∞, 2)-category of Segal fibrations into that of weak Segal fibrations.

Morphisms of algebraic patterns II: Combinatorics of enrichable structures

To finish this preliminary section, we state (a variant of) some definitions2 of the forthcoming paper [START_REF] Chu | Enriched homotopy-coherent structures[END_REF].

O ♭ := O × Γ op♮ Γ op♭ . ( 5
)
Thanks to [CH21, Corollary 5.5], we see that the fibre product appearing in eq. ( 5) has a very simple description: it consists of the category O equipped with its same factorisation system, and the choice of only those elementary objects living over 1 ∈ Γ op (i.e. excluding those over 0 ).

Definition 2.3.2 (Enrichable pattern). A combinatorial structure |-| on an algebraic pat- tern O is enrichable if, for any O ∈ O, the morphism O ♭,el O/ → Γ op♭,el |O|/ is an equivalence.
Remark 2.3.3. Viewing Γ op as (the standard skeleton of) the category of pointed finite sets, one verifies that any n ∈ Γ op admits exactly n inert morphisms to the unique elementary 1 , the pointed morphisms ρ i : n → 1 mapping i to 1 and every other element of n to the base-point. The condition of being an enrichable pattern then means that, for any O ∈ O, the (∞, 1)-category O ♭,el O/ must be equivalent to the discrete set of the (essentially unique) lifts ρ i,! of the ρ i . In particular, the O ♭ -Segal condition for a precosheaf X on O is constrained to being the finite product condition

X(O) ≃ |O| i=1 X(ρ i,! O). ( 6
)
Example 2.3.4. The functor * → Γ op picking the object 1 defines a structure of enrichable pattern on the terminal algebraic pattern.

Example 2.3.5. There is a functor ∆ op → Γ op mapping [n] to n and sending an arrow of ∆ op corresponding to φ :

[n] → [m] in ∆ to |φ| : m → n given by |φ|(i) = j if φ(j -1) < i ≤ φ(j) * otherwise. ( 7
)
It can be checked directly that this is a structure of enrichable pattern on ∆ op♮ .

Example 2.3.6. A functor

Ω op → Γ op ≃ FinSet * is defined in [CH20, Definition 4.1.16]
in the following way. A tree T with set of vertices V(T ) is mapped to the freely pointed set V(T ) + . A morphism T ′ ← T in Ω op is mapped to the pointed morphism V(T ′ ) + → V(T ) + which sends a vertex v ∈ V(T ′ ) to the unique vertex of T whose image subtree contains v, or to the basepoint if there is no such vertex. By [loc. cit., Lemma 4.1.18], this functor preserves the inert-active factorisation system, and thus defines a morphism of algebraic patterns. One checks by inspection that it is an enrichable structure. In proposition 3.2.9 we will provide another construction of this enrichable structure from the point of view of Ω as obtained from the plus construction.

We recall from [START_REF] Lurie | Higher Algebra[END_REF] the definition of semi-inert arrows in Γ op . A map of pointed finite sets f : (S, s 0 ) → (T, t 0 ) is semi-inert if for any t ∈ T \ {t 0 }, there is at most one element in f -1 (t).

Definition 2.3.7. Let (O, |•|) be a combinatorial algebraic pattern. An arrow f of O is semi-inert if |f| is a semi-inert arrow of Γ op .
Example 2.3.8. In ∆ op♮ , the semi-inert morphisms are those corresponding to the cellular morphisms of ∆ as defined in [START_REF] Haugseng | The higher Morita category of E n -algebras[END_REF] and [START_REF] Haugseng | Bimodules and natural transformations for enriched ∞categories[END_REF], that is maps of totally ordered sets f : S → S ′ such that for all s ∈ S, f(succ s) ≤ succ(f(s)).

For the product pattern (∆ op♮ ) n equipped with

(∆ op♮ ) n |-| -→ (Γ op♮ ) n ∧ -→ Γ op♮ ,
we also recover the cellular morphisms of [START_REF] Haugseng | The higher Morita category of E n -algebras[END_REF], those maps all of whose components are cellular in ∆ op .

Remark 2.3.9. As observed in [START_REF] Haugseng | The higher Morita category of E n -algebras[END_REF], an arrow f : O → O ′ of O is semi-inert if and only if, for any elementary E and any inert morphism k :

O ′ E, the composite O k•f --→ E is semi-inert.
Example 2.3.10. In Γ op , any object n admits precisely n active morphisms from the unique elementary object 1 , the functions λ i : 1 n sending 1 ∈ 1 to i ∈ n for any 1 ≤ i ≤ n. They are semi-inert. Likewise, the unique map 0 → n is also semi-inert.

It follows that, in any combinatorial pattern, any active arrow from an elementary object is semi-inert. Example 2.3.12. The pattern Γ op♮ itself is momentous: for any object n , the function λ i : 1 → n of example 2.3.10 admits as unique inert retraction ρ i . In fact λ i is also the unique active section of ρ i , so there is a one-to-one correspondence between inert morphisms to ♭-elementaries and (automaticaly semi-inert) active morphisms from ♭elementaries.

As a consequence, for any momentous enrichable pattern O and any object O ∈ O, the (semi-inert) active morphisms from ♭-elementary objects to O can be identified with a subset of O el O/ ≃ Γ op♮,el |O|/ .

3 The plus construction and its Segal objects

Categories of patterned trees

The following construction is a variant of one due to [START_REF] Barwick | From operator categories to higher operads[END_REF] in the setting of operator categories, inspired by the plus construction or "slice operads" of [START_REF] Baez | Higher-Dimensional Algebra III. n-Categories and the Algebra of Opetopes[END_REF], and which was also used in [START_REF] Chu | Two models for the homotopy theory of ∞-operads[END_REF] and studied in [START_REF] Berger | Moment categories and operads[END_REF] in the setting of hypermoment categories -as well as [START_REF] Batanin | Operadic categories as a natural environment for Koszul duality[END_REF], in a different form, for operadic categories. O is equivalent to ∆ /O act -where ∆ is seen as a subcategory of (∞, 1)-Cat -or also to ∆ / N • O act , where N • : (∞, 1)-Cat ֒→ ∞-Grpd ∆ op is a nerve functor (for example incarnating (∞, 1)categories as complete Segal ∞-groupoids, or as quasicategories, to the reader's preference). Thus an

object of ∆ pre O consists of a pair ([n], O • ) where [n] ∈ ∆ and O • : [n] → O act is a linear diagram in O act , that is a sequence O 0 O 1 • • • O n of active morphisms in O, while a morphism ([n], O • ) → ([n ′ ], O ′ • ) consists of a pair (φ, f • ) where φ : [n] → [n ′ ] is a map in ∆ and f • : O • ⇒ O ′ φ(•) = O ′ • • φ is a natural transformation of [n]-shaped diagrams in O act .
We define ∆ O to be the wide and locally full (i.e. containing all objects and all higher morphisms between a selection of

1-morphisms) sub-(∞, 1)-category of ∆ pre O on those morphisms (φ, f • ) such that • f • is component-wise semi-inert (in addition to active), that is each f i : O i O ′ φ(i) , for i ∈ [n], is semi-inert, and
• f • is a cartesian (or equifibred) natural transformation, that is for any morphism i < j in [n] the naturality square Example 3.1.5. For Γ op♭ , we obtain a category of forests, i.e. disjoint unions of trees, with level structures, whose misodendric subcategory is identified in [START_REF] Berger | Moment categories and operads[END_REF] with a full subcategory of Ω. Indeed, recalling that the active subcategory of Γ op is equivalent to the category of finite sets, the object O n is to be thought of as the set of roots of the forest, and each O i is the set of leaves at level ni. The morphisms in (Γ op ) act give partitions of the leaves at levels ℓ corresponding to the node (recognised by its unique output leaf at level ℓ + 1) to which they lead.

O i O ′ φ(i) O j O ′ φ(j) f i O i<j O ′ φ(i<j) f j (8) is cartesian in O act . Remark 3.1.2. Given fixed φ : [m] → [n] in ∆ and ([n], P • ) in ∆ O over [n], morphisms (φ, f • ) : ([m], O • ) → ([n], P • ) in ∆ O
Example 3.1.6. For ∆ op♭ , we similarly obtain a category of planar trees (or rather, forests thereof) with level structures.

The pattern structure on the plus construction

Lemma 3.2.1. The ∞-functor d O : ∆ O ⊂ ∆ pre O → ∆ is a cartesian fibration.
Proof. We first observe as in [START_REF] Chu | Two models for the homotopy theory of ∞-operads[END_REF] that, due to the pullback condition in eq. ( 8), a morphism (φ, f

• ) : ([m], O • ) → ([n], P • ) in ∆ O is d O -cartesian if and only if f • is a natural equivalence. Indeed, to say that (φ, f • ) : ([m], O • ) → ([n], P • ) is d O -cartesian is to say that ∆ O,/([m],O• ) ∆ O,/([n],P•) ∆ /[m] ∆ /[n] (φ,f•)•- φ•- (9)
is a cartesian square. Let then ([k], L • ) be any object of ∆ O , and take the fibre of eq. ( 9) at ([k], L • ), which we decompose as follows:

∆ O ([k], L • ), ([m], O • ) ∆ O ([k], L • ), ([n], P • ) ∆ pre O ([k], L • ), ([m], O • ) ∆ pre O ([k], L • ), ([n], P • ) ∆([k], [m]) ∆([k], [n]). iff f• invertible (10) Since ∆
pre O → ∆ is constructed -and described -as a Grothendieck construction, the lower square is cartesian if and only if f • invertible. It is then enough to observe that the condition for a morphism in ∆ pre O to be in ∆ O is detected by postcomposition with a componentwise invertible morphism of ∆ O , which is clear.

This characterisation now makes it easy to determine cartesian lifts. Let φ : [m] → [n] be a morphism in ∆, and let We shall say that an algebraic pattern is well-directed if it can be written as the plus construction of some combinatorial algebraic pattern.

([n], O • ) be a lift of [n] in ∆ O . We define a cartesian lift of ([n], O • ) along φ to be ([m], O φ(•) ) with (φ, id O••φ ). Corollary 3.2.2. Say that an arrow (φ, f • ) in ∆ O is inert if φ is inert in ∆ and
Example 3.2.4. The main result of [START_REF] Chu | Two models for the homotopy theory of ∞-operads[END_REF] shows that ∆ Γ op♮ op♮ is Morita-equivalent to Ω op♮ in the sense that their (∞, 1)-categories of Segal ∞-groupoids are equivalent. We will henceforth conflate operad objects with Segal ∆ Γ op♮ op♮ -objects.

Scholium 3.2.5. The Segal condition for a Segal ∆ O

op♮ -object X can be explicitly written as the following set of conditions: level decomposition For any ([n], O • ), the canonical arrow

X [n], O • → X [1], (O 0 O 1 ) × X([0],(O 1 )) • • • × X([0],(O n-1 )) X [1], (O n-1 O n ) (11) is an equivalence.
root decomposition For any height-0 forest of the form ([0], O 0 ), the canonical map

X [0], (O 0 ) → lim E O 0 X [0], (E) (12)
is an equivalence (where the limit is taken over O act el/O ) shrub decomposition For any [1], (O 0 O 1 ) , the canonical arrow

X [1], (O 0 O 1 ) → lim E O 1 X [1], (O 0,E E) (13) is an equivalence, where O 0,E is the fibre product O 0 × act O 1 E -with × act indicating that it is a fibre product in O act .
Proof. The only part that bears commenting upon is the shrub decomposition. Recall that the inert maps in ∆ O are simply those whose projection to ∆ is inert, so the relevant inert morphisms in ∆ O are those from the elementaries over [1] ∈ ∆. If we consider such a morphism, represented by

P O 0 E O 1 , (14)
the equifibred condition implies that P is the active pullback O 0 × act O 1 E.

Example 3.2.6. When O is Γ op♮ , or more generally a momentous enrichable patter, the limits appearing in the root and shrub decompositions take on a simpler form: the root decomposition now becomes the condition that

X [0], (O 0 ) → lim E O 0 X [0], (E O 0 i ) (15)
be an equivalence (where

E O i
O is the unique lift of O along the map λ i of example 2.3.12, and we put a trivial factor if this lift is inexistent), and the shrub decomposition is the condition that

X [1], (O 0 O 1 ) → |O 1 | i=1 X [1], (O 0,i E O 1 i ) (16)
be an equivalence, where O 0,i is the fibre product

O 0 × act O 1 E O 1 i ,

16]).

There is an equivalence of (∞, 1)-categories between ∆ O op♭ -Segal objects and weak Segal O-fibrations.

Proof (Sketch of construction).

The proof is a direct generalisation of that of [HM22, Theorem 3.1.1], which is quite technically involved, so we do not reproduce it here. However, since it will be useful later on, we recall the construction of the adjunction.

We first explain what is, for our purposes, the most important part: how to get from Segal objects to weak fibrations.

We start by constructing an ∞-functor ω * : ∞-Grpd ∆ O op → ∞-Grpd ∆ /O op . Recall that presheaves on ∆ O can be seen equivalently as presheaves on ∞-Grpd ∆ O op satisfying the representability condition of being colimit-preserving. We now define an ∞-functor

ω : ∆ /O → ∞-Grpd ∆ O op as the composition i * • y ∆ /O where i * is restriction (or inverse image) of presheaves along the canonical ∞-functor i : ∆ O ֒→ ∆ /O act (O act →O) ! -------→ ∆ /O . We thus get an inverse image ∞-functor ω * : ∞-Grpd ∞-Grpd ∆ O op op
→ ∞-Grpd ∆ /O op , which we can restrict to the (full) sub-(∞, 1)-category of those presheaves which are representable and satisfy the Segal condition. To finish, recall also (from [Lur09, Corollary 5.1.6.12]) that ∞-Grpd ∆ /O op ≃ ∞-Grpd ∆ op / N • O , so we eventually have

ω * : Seg ∆ O op♮ (∞-Grpd) → ∞-Grpd ∆ op / N • O . ( 17
)
In the case where O = Γ op♮ , our functor ω corresponds to the one described in [HHM16, §5.1]. By [HM22, §3.3], this functor does land first in the subcategory Seg ∆ op♮ (∞-Grpd) / N • O which we interpret as (∞, 1)-Cat /O , and in fact in the further subcategory of those ∞functors to O which are weak Segal fibrations.

To go the other way, we consider ̟ :

∆ O y -→ ∞-Grpd ∆ O op → WkSegFib /O and the "conjoint" profunctor it corepresents: then (ω * ) -1 is the ∞-functor mapping (E → O) to the presheaf WkSegFib /O (̟(-), E) : ∆ O → ∞-Grpd.
Note that -again for O = Γ op♮ -an alternate proof, using the monoidal envelope ∞-functor, appears in[HK21, Corollary 4.2.8]. However, this proof uses crucially an explicit description of the Grothendieck construction of Env(1) seen as a Segal O-∞category, which do not have in general.

Monoidal envelopes and Grothendieck opfibrations 4.1 Construction of the monoidal envelope functor

As explained in the introduction, we wish to interpret the usual formula computing monoidal envelopes of operads as a fibrewise oplax extension along a certain functor appropriately mixing in partitions. To define the necessary fibrewise extensions, we need to briefly work in the generality of formal ∞-category theory introduced in [START_REF] Riehl | Elements of ∞-category theory[END_REF], that is in the framework of ∞-cosmoi, modelling the (∞, 2)-category of ∞categories. Recall from [RV22, Proposition 9.1.8, Theorem 9.3.3] that (pointwise) oplax extensions along an ∞-functor can be expressed as colimits weighted by the conjoint of this ∞-functor. We will define fibrewise extensions similarly, replacing this conjoint by a relative version.

The reader who wishes to limit their intake of abstract nonsense may use the formula of eq. ( 23) as a definition, and take its functoriality properties on faith. Construction 4.1.1 (Relative comma ∞-category). In an ∞-cosmos K, consider a cocorrespondence in the sliced ∞-cosmos K /B :

E F G B f g p q (18)
We let f ↓ /B g denote the comma object in K /B , and call it the relative comma ∞category over B.

Remark 4.1.2. By [RV22, Proposition 1.2.22, (iv)-(vi)], the relative comma ∞-category can be constructed as

f ↓ /B g ≃ (F × B G) × (E× B E) (B × B 2 E 2 ) (19)
(where the map B → B 2 is the diagonal). That is, informally, an object of f ↓ /B g consists of a triple (F, G, α) where F and G are objects of F and G respectively and α : f(F) → g(G) is an arrow in E, such that all data live above the same object of B (i.e. there is an object B ∈ B and isomorphisms pF q -→ B be two ∞categories defined over a base B, and let K : E → F be an ∞-functor defined over B. Let D : E → G be an (∞, 1)-functor, so that we have the solid diagram

E G F B p K D q (20)
A fibrewise lax extension of D along K (relative to B) is a limit

Lex K/B D := K /B * , D : F → G (21)
of D weighted by the ∞-profunctor K /B * : E -→ F associated with the relative comma ∞-category id F ↓ /B K.

A fibrewise oplax extension of D along K (relative to B) is a colimit

Opex K/B D := K * /B ⋆ D : F → G (22) of D weighted by the ∞-profunctor K * /B : F -→ E corresponding to the relative comma ∞-category K ↓ /B id F .
Remark 4.1.5. We have the explicit formulae, deduced from [RV22, Lemma 9.5.5], computing fibrewise extensions: the fibrewise oplax extension of D along K, evaluated at

F, is Opex K/B D(F) = colim K(E)→F E∈E q(F) D(E), (23)
and the fibrewise lax extension of D along K, evaluated at F, is

Lex K/B D(F) = lim F→K(E) E∈E q(F) D(E). (24)
Definition 4.1.6 (Envelope). Let X :

∆ O op → ∞-Grpd be a precosheaf on ∆ O op . Its envelope is the precosheaf Env(X) = i * Opex i/∆ op X (25)
where i * denotes the (∞, 1)-functor of (fibrewise) restriction along i :

∆ O op ֒→ (∆ pre O ) op ,
right-adjoint to fibrewise oplax extension.

Remark 4.1.7. From the formula for fibrewise extensions ( eq. ( 23)), we see that the value taken by the envelope of

X ∈ Seg ∆ O op♮ (C) at an O-tree T • of length [n] is computed by Env(X)(T • ) = colim B• T• ∈((∆ pre O ) [n] ) /T• lim E Bn X(B •,E ) (26)
where (B •,E ) E Bn denotes the forest decomposition in fibres (as in eq. ( 13)), and where we denote the indexing arrows for the colimit as to emphasise that we view them as families of n active arrows of O.

Remark 4.1.8. In the case O = Γ op♭ , we may understand the envelope in the following way. Recall that an active morphism in Γ op can be seen as giving a partition of its source indexed by its target (possibly with empty parts). Then the colimit creates several copies of X(⋆ n ), each equipped with a new partition specifying how to distribute its inputs.

To work with envelopes in an intuitive way, it will then be convenient to introduce the following terminology. Proof. By definition, Env(1)(T • ) is given by the colimit

Env(1)(T • ) = colim B• T• 1 (27)
(where the last 1 is the terminal ∞-groupoid). Now Env(1)(T • ) is an ∞-groupoid so it is equivalent to its localisation, and furthermore the localisation functor L : (∞, 1)-Cat → ∞-Grpd, being left-adjoint to the inclusion I : ∞-Grpd ֒→ (∞, 1)-Cat, preserves colimits, so we may compute the colimit on the right-hand side in (∞, 1)-Cat and then localise.

As a colimit of a constant diagram of (∞, 1)-categories, the colimit in (∞, 1)-Cat is equivalent to the tensor of the constant value 1 by the indexing (∞, 1)-category

(∆ pre O ) [n]/T• .
From the adjunction L ⊣ I and the tensor-cotensor adjunction, we have that for any (∞, 1)-category C and any ∞-groupoids G and H,

∞-Grpd (LC) ⊗ G, H ≃ ∞-Grpd LC, H G ≃ (∞, 1)-Cat C, I(H G ) . (28)
Since functor (∞, 1)-categories between ∞-groupoids are automatically ∞-groupoids, I(H G ) is also the cotensor of (∞, 1)-categories (IH) IG , and thus

(∞, 1)-Cat C, I(H G ) ≃ (∞, 1)-Cat C, (IH) IG ≃ (∞, 1)-Cat C ⊗ IG, IH ≃ ∞-Grpd L(C ⊗ IG), H , (29)
which shows that L(C ⊗ IG) ≃ (LC) ⊗ G. Hence, in our case, we obtain -now writing C[ -1 ] = LC for the total localisation of an (∞, 1)-category C -the tensor of ∞-groupoids

(∆ pre O ) [n]/T• [ -1 ] ⊗ 1, which is simply (∆ pre O ) [n]/T• [ -1 ]. Definition 4.1.11 (Linearisable pattern). A collection {C i } i∈I of objects in an (∞, 1)- category C is mock multiterminal if the projection C /{C i } i → C is an equivalence. A combinatorial algebraic pattern O |-| -→ Γ op♮ is linearisable if the collection of ♭- elementary objects is mock multiterminal in O act .
We note that while, obviously, if a collection {C i } i∈I is multiterminal, it is in particular mock multiterminal, this is is not the only case: it can also happen if the only morphisms between objects of {C i } i are ismorphisms. Notation 4.1.12. When O is linearisable, we have a simple classification of the elementaries of ∆ O op♮ . Recall that those were defined to be the trees (i.e. the forests whose final value is elementary in O ♭ ) of length [0] and [1].

If E • is of length [0]
, it is simply given by an elementary object of O ♭ , so we write those as η E with E ∈ O ♭,el , and call them the edges.

If E • is of length [1], it must be given by a diagram in O act of the form O E with E ∈ O ♭,el . The linearisability condition ensures that for any given O, there is only essentially one such morphism, and so we denote the corresponding object as ⋆ O , and dub such elementaries the corollas (as they are the elementaries considered in construction 3.2.8). [n], which is specified by (0 < 1) → (0 < n). Finally, an active morphism in ∆ O op♭ must have its morphism to O n be an isomorphism, so E 1 is also forced to be O n (while E 0 was already seen to be equivalent to O 0 ).

In particular, when O is linearisable, evaluating example 4.1.10 on the elementaries, we find that:

• E∈O ♭,el Env(1)(η E )

Opfibrations and representable monoidality

Env(X)(T • ) = colim B•→T• X(B n ) = colim B•→T• lim B•→E X(E • ) (30)
and lim

T•→E• Env(X)(E • ) = lim T•→E• colim B 0,1 →E• X(B 0,1 ). ( 31
)
Their equality is the condition of distributivity of limits over colimits as made explicit in [CH21, Definition 7.12], and using [CH21, Corollary 7.17] which shows that ∞-Grpd is admissible (or even any (∞, 1)-topos when the ∞-category of active maps from a given object of O to an elementary is finite). Alternatively, we can use the explicit form of the Segal conditions given in scholium 3.2.5, which can be checked directly.

For the cocartesian properties of the envelope, we now need to specialise to the case where O is linearisable. Indeed, following [Her04, Theorem 2.4], we will characterise cocartesian fibrations of Segal P-objects (for P = ∆ O op♮ a well-directed algebraic pattern) in terms of cocartesian fibrations of the underlying P P 0 / -objects of their envelopes. This presupposes having already a good understanding of cocartesian fibrations of P P 0 / -objects, which is only the case when P P 0 / is ∆ op♮ , in particular when P = Ω op♮ (and P 0 = {η}).

Remark 4.2.2. In a combinatorial pattern O, the ♭-elementaries lie over 1 ∈ Γ op . An active and semi-inert morphism to 1 can have as source only 1 or 0 . So if O is linearisable, for any object O the unique active morphism to an elementary will only be semi-inert in |O| is 1 or 0 , in which case it will be so because there are no non-semiinert morphisms 0

→ 1 or 1 → 1 . Lemma 4.2.3. Suppose O is linearisable and O 1 is an ∞-groupoid. The slice (∆ O ) /η is equivalent to ∆ × O 1 , itself equivalent to colim O 1 ∆. Proof. Consider a morphism T → η from some tree T = ([n], O • ), consisting of a map φ : [n] → [0] and a transformation O • → φ * E
where E is an elementary defining the edge η E = ([0], (E)). Now φ * E is a constant diagram, so the equifibred condition for the transformation implies that all components of O • must be equal to O n . By the previous remark, O n only admits a semi-inert active morphism to an E if it lies over 1 (and it exists uniquely by the linearisability condition).

By [Bar22, Proposition 2.37], there is a canonical pattern structure on a coslice of a pattern. The equivalence of categories underlies one of algebraic patterns. If F is the name of a Segal ∆ O op♮ -object, we will generally denote its underlying category object F. Remark 4.2.6. The functor p clearly admits unique liftings of inert morphisms, and it is extendable, so by [CH21, Proposition 7.13] the restriction ∞-functor p * has a leftadjoint (given by oplax extension), which is fully-faithful, and has as its essential images the presheaves with empty value on any non-linear tree. Definition 4.2.7 (Cocartesian fibration). A morphism X → B of Segal ∆ Γ op♭ op♮ -∞groupoids is a cocartesian fibration of Segal objects if Env(X) → Env(B) is a cocartesian fibration (of (∞, 1)-categories), where Env(X) denotes the underlying (∞, 1)-category of Env(X).

A Segal object X is representably monoidal if the unique morphism X → 1 is a cocartesian fibration of Segal objects. Proposition 4.2.8. For any Segal object X, Env(X) is representably monoidal.

Proof. Since families of cocartesian fibrations can be checked componentwise, we shall assume that O 1 is comprised of one single ♭-elementary object.

Let φ : O O ′ be a morphism in Env(1), seen as an O-partition of O by O ′ , and let C ∈ Env(Env(X)) lying over O. Under the identification

Env Env(X) (η) ≃ colim P∈O act Env(X)([0], P) ≃ colim P∈O act colim Q P lim E Q X(η), (32)
where the condition of "lying over O" means that the index P in the first colimit is fixed to be O, one passes to φ ! C over O ′ by composing the maps Q O in the second colimit with O O ′ . For clarity, let us write it in the momentous enrichable case, where this simplifies to

Env Env(X) (η) ≃ colim P∈O act |P| i=1 Env(X)(η) ≃ colim P∈O act |P| i=1 colim Q i ∈O act |Q i | j=1 X(η), (33)
and so we write C as

(C 1,1 • • • C 1,|Q 1 | ) • • • (C |O|,1 • • • C |O|,|Q |O 0 | | ) (34)
for some choice of

(Q i ) |O| i=1 . We set φ ! C = (C 1,1 • • • C n 1 ,in 1 ) • • • (C n m-1 +1,1 • • • C nm,in m ) (35) which lies over m = |O ′ |, where (n 1 , . . . , n m ) is the partition of n in m furnished by |O O ′ |. A morphism C → φ ! C in Env(Env(X)) is given by a moprhism C 1,1 • • • C nm,in m → C 1,1 • • • C nm,in m in Env(X)
along with a partition of m k=1 i n k into n parts. We define the lift C → φ ! C of φ to be given by the identity arrow of C 1,1 • • • C nm,in m along with the partition exhibited in eq. ( 34). This lift is cocartesian.

Thus the construction X → Env(X) defines an (∞, 1)-functor Env :

Seg ∆ O op♮ (∞-Grpd) → O 1 -Mon(∞, 1)-Cat.
Remark 4.2.9 (Monoidal structure on monoidal ∞-categories). Using the idea from the proof of proposition 4.2.8, we can construct a product on the colours of a representably monoidal ∞-operad X. First note that (regardless of monoidality) the colours of Env(X) in the image of the unit map X → Env(X) are exactly those whose image under the morphism Env(X) → Env(1) is 1.

Consider an n-uple of colours of X, given by n morphisms C 1 , • • • , C n : yη → X. Those define a morphism (C 1 , • • • , C n ) : yη → Env(X), whose image lies over the colour n of Env(1). Now since X is representably monoidal, the morphism n → 1 in Env(1) has a cocartesian lift from (C 1 , • • • , C n ), whose target is then a colour C 1 ⊗ • • • ⊗ C n of X. Clearly, the same construction can be applied to obtain a product of morphisms as well, with appropriate functoriality. Theorem 4.2.10. The (∞, 1)-functor Env is left-adjoint to the inclusion Mon(∞, 1)-Cat ֒→ ∞-Oprd.

Proof. By the construction of the envelope, we have a unit morphism η X : X → Env(X) for any ∞-operad X. We need to construct a counit ε V : Env(V) → V for any representably monoidal ∞-operad V, which is a morphism of monoidal ∞-categories.

This morphism is provided by the construction of remark 4.2.9: since V is representably monoidal, it admits a monoidal product, and its envelope simply corresponds to adding a second level a parenthesising to the products.

We will construct ε V componentwise, as a natural transformation of ∞-functors ∆ Γ op ♭ op → ∞-Grpd. Let T • be a tree. By the formula eq. ( 26), giving a map Env(V)(T • ) → V(T • ) is equivalent to giving, for each morphism B • T • , a map X(B • ) → X(T • ). But recall that B • T • can be interpreted as a T • -partition of B • . Following the previous remark, we can use the product to simply reorganise the parenthesising levels according to the partition, which produces the desired morphism.

Finally, it is directly checked that ε and η satisfy the triangular equalities, so that they do exhibit an adjunction.

Cartesian monoidal structures and application to the Grothendieck construction

Cartesian monoidal structures

Defining the monoidal (∞, 1)-category associated with an (∞, 1)-category admitting finite coproducts is easy enough: by [Lur17, Proposition 2.4.3.9], the ∞-functor realising this construction is right-adjoint to the one taking the underlying (∞, 1)-category of a unital ∞-operad. For cartesian monoidal ∞-categories, however, it is not so easy.

We follow here the idea used independently in [START_REF] Barwick | Spectral Mackey functors and equivariant algebraic K-theory, II[END_REF] and [START_REF] Dyckerhoff | Higher Segal Spaces[END_REF] to define cartesian monoidal ∞-categories: defining them not as ∞-operads, but as "∞-antioperads", the variant of ∞-operads whose operations have one input but many outputs. We now give an alternate construction of the cartesian structure as a ∆ O op♮ -object, in the spirit of [START_REF] Lurie | Higher Algebra[END_REF].

Construction 5.1.7. Let U ≤-1 be the subobject classifier ((-1)-truncated universe) in the (∞, 1)-topos ∞-Grpd ∆ O op . Define a presheaf Env(C × ) over Env(1) by the specification that hom(K, Env(C × )) ≃ hom(K × Env(1) Env(U ≤-1 ), C) for any presheaf K. Note that the canonical (mono)morphism true: 1 U ≤-1 induces a transformation Env(C × ) → C.

When K is yη, and the transformation η → Env(1) selects an object O of O act , we obtain functors from the poset of subobjects of O to C. We define Env(C × ) to be the subfunctor of Env(C × ) on the functors as above which exhibit their image as product of the elementaries in their source. It is a strong cartesian structure if it induces an equivalence of underlying (∞, 1)categories.

  Definition 2.3.1 (Combinatorial structure). A combinatorial pattern is an algebraic pattern O equipped with a morphism |-| : O → Γ op♮ . When (O, |-|) is a combinatorial pattern, we let O ♭ → Γ op♭ denote the base-change

  Definition 2.3.11 (Momentous pattern). A combinatorial algebraic pattern (O, |-|) is momentous if any active morphism from a ♭-elementary admits an essentially unique inert retraction. Momentous patterns which are enrichable are closely related to (an ∞-categorical version of) the hypermoment categories of [Ber22].

Construction 3.1. 1 .

 1 Let O be a combinatorial algebraic pattern. Consider the restrictiony (∞,1)-Cat (O act )| ∆ : [n] → O act [n] to ∆ ⊂ (∞,1)-Cat of the (∞, 2)-functor represented by O act and let ∆ pre O → ∆ be its Grothendieck construction. Note that ∆ pre

  lifting φ are essentially determined, if they exist, by their underlying arrow f m : O m → P φ(m) at the terminal object m ∈ [m]. Indeed, for each i ∈ [m] the object O i and arrow f i are required by the pullback condition in eq. (8) to be the base-change of O m and f m along i ≤ m. Definition 3.1.3. Let O be a combinatorial algebraic pattern. The (∞, 1)-category ∆ O is called the (∞, 1)-category of O-forests.The full sub-(∞, 1)-category ∆(1)O of ∆ O on the ([n], O • ) such that O n ∈ O ♭,el is called the (∞, 1)-category of misodendric O-forests, or O-trees.Example 3.1.4. For the terminal algebraic pattern * , the (∞, 1)-categories of * -forests and of * -trees both recover the simplex category ∆.

  Forests and trees have the same Segal objects: the (∞, 1)-functor Seg ∆ O op♮ (∞-Grpd) → Seg ∆ (1) O op♮ (∞-Grpd) of restriction along the inclusion ∆ (1) O ֒→ ∆ O is an equivalence of (∞, 1)-categories. Proof. Let ([n], O • ) be an O-forest, and denote the decomposition of |O n | into its fibres. We then write any forest as a union of trees, and use the forest decomposition condition of scholium 3.2.5. Construction 3.2.8. We define a corolla of an O-forest F ∈ ∆ O to be an equivalence class of inert morphisms F → E where E is elementary lying over [1] ∈ ∆. We can now define a functor |-| op : ∆ O → Γ by counting the numbers of corollas in an O-forest, as in [CH20, Definition 2.2.10]. Proposition 3.2.9. The functor |-| : ∆ O op → Γ op gives a structure of enrichable pattern on ∆ O op♮ . Proof. By direct verification; this follows essentially by definition of corollas. Proposition 3.2.10 ([Bar18, Theorem 10.

≃-

  → B, qG ≃ -→ B, and α ≃ = ⇒ id B ).Lemma 4.1.3. The canonical projection f ↓ /B g → F × G is a discrete two-sided fibration in K. Proof. As in [RV22, Proposition 7.4.6]. Definition 4.1.4 (Fibrewise (op)lax extensions). Let E p -→ B and F

  Definition 4.1.9 (O-Partitions). Let O be an algebraic pattern, and let O and P be objects of O. An O-partition of O by P, or simply a P-partition of O when there is no ambiguity, is an active morphism O P. Example 4.1.10. If 1 denotes the terminal ∆ O op♮ -Segal ∞-groupoid, then its envelope Env(1) evaluates, on any forest T • of length n, the the groupoidal localisation of (∆ pre O ) [n]/T• .

  Proposition 4.1.13. For any combinatorial pattern O, the misodendric plus construction ∆ (1) O op♮ is linearisable. Proof. Let ([n], O • ) be an object of ∆ O . By the condition for misodendric objects, O n must be in O ♭,el . Recall from remark 3.1.2 that a morphism to ([n], O • ) is determined by the data of a morphism to [n] in ∆ and a (semi-inert and active) morphism to O n . Now the elementaries of ∆ O op♭ are of the form ([1], (E 0 E 1 )) with E 1 ∈ O ♭,el . There is only one active morphism [1]

  is the localisation of O act ; • for any corolla C O = [1], (O E O ) (with E O the unique elementary receiving an active morphism from O), an object of Env(1)(C O ) consists of an active morphism P 0 P 1 of O -a P 1 -partition of P 0 -along with an O-partition P 0 O (such that, necessarily, E P 1 is also compatibly equivalent to E O ).

  Remark 4.2.4. Since hom ∞-functors preserve limits, we have that Seg colim O 1 ∆ op♮ (C) ≃ lim O 1 Seg ∆ op ♮ (C). In other words, a Segal (∆ O ) op η/ -object can be seen as a family of category objects indexed by the ∞-groupoid O 1 , which we shorten to O 1 -category. Definition 4.2.5 (Underlying O 1 -category). Suppose O is linearisable and O 1 is an ∞-groupoid, and denote p : (∆ O ) op η/ → ∆ O op . The underlying O 1 -category, or simply underlying category, object of a Segal ∆ O op♮ -object X is its restriction along p.

  Construction 5.1.1 (Anti-plus construction). We let ∆ pre O denote the Grothendieck construction of y (∞,1)-Cat (O act )| ∆ • (-) op , and define a locally full sub-(∞, 1)-category ∆ O by imposing the same conditions as in the plus construction. Scholium 5.1.2. There is an equivalence of (∞, 1)-categories between Segal ∆ O op♮ -objects and (∞, 1)-functors p : P → O op such that p op is a weak Segal fibration. Definition 5.1.3 (Reduced pattern). A combinatorial pattern O is reduced if O el 0 is the terminal ∞-groupoid. We denote its (essentially) unique object ∅. Definition 5.1.4 (Unitality). Let O be a reduced pattern. A ∆ O op♮ -object X is anti-unital if it is local with respect to the (op-)morphism ⋆ ∅ ← η. Remark 5.1.5. The (∞, 1)-category of anti-unital Segal ∆ O op♮ -objects in an (∞, 1)-category C is evidently a localisation of Seg ∆ O op♮ (C). It can also be seen a category of Segal objects for a different pattern. The pattern for anti-unital ∆ O op♮ -objects is the same as ∆ O op♮ , with ⋆ ∅ removed from the elementaries. Lemma 5.1.6. The restriction morphism Seg ∆ O op ♮,unit. (C) → Seg ∆ O op♮ η/ (C) (taking underlying ∞-categories) preserves Segal equivalences, and thus admits a right-adjoint (by lax extension).

(

  Proposition 5.1.8. The structure morphism Env(C × ) → Env(1) is the image of a cocartesian fibration of ∆ O op♮ -objects. Proof. As in [Lur17, Proposition 2.4.1.5.]. Weak cartesian structure). Let X be a Segal ∆ O op♮ -∞-groupoid, and let D be a Segal ∆ O op♮ η/ -∞-groupoid. A lax cartesian structure from O to D is a morphism Env(X) → D that takes decompositions to products.

  (∞, 2)-category of (∞, 1)-Cat /O spanned by the weak Segal O-fibrations and Segal morphisms thereof.

GH15, Definition 2.2.6, Definition 3.1.3] while a weak Segal ∆ op♮ -fibration is virtual double ∞-category, or generalised ∞-operad in [GH15, Definition 2.4.1, Definition 3.1.13]. Definition 2.2.6 (Morphisms of weak Segal fibrations). By [CH21], the source X of a weak Segal O-fibration p : X → O inherits an algebraic pattern structure in which active morphisms are those lying over an active morphism in O, inert morphisms are the pcocartesian morphisms lying over inert arrows of O, and elementaries are the objects lying over elementary objects. The (∞, 2)-category of weak Segal O-fibrations WkSegFib /O is the locally full sub-

  active if φ is active in ∆ and f • is an equivalence. Then ((∆ O op ) inrt , (∆ O op ) act ) defines a factorisation system

	on ∆ O	op .
	Proof. By [Lur17, Proposition 2.1.2.5], since d O : ∆ O ⊂ ∆ the factorisation system on its base can be lifted as required. pre O → ∆ is a cartesian fibration,
	Definition 3.2.3. Let O be a combinatorial algebraic pattern. Its plus construction ∆ O op♮ is the (∞, 1)-category ∆ O op equipped with the inert-active factorisation of corollary 3.2.2
	and as elementary objects those ([n], O (1) O	op .

• ) with [n] ∈ ∆ op♮,el (i.e. either [0] or [1]) and O n ∈ O ♭,el .

Its misodendric plus construction is the algebraic pattern induced on ∆

  as since O is momentous and enrichable, E O 1 must be one of the lifts E O 1 i .

	Proposition 3.2.7 ([CHH18, Lemma 2.11]).

  Let T • be an object of ∆ O of length [n]. The Segal condition requires comparing

	Proposition 4.2.1. For any Segal ∆ O	op♮ -object X, its envelope Env(X) is a Segal ∆ O	op♮ -
	object.
	Proof.

Oft known as left Kan extensions

which were presented in the seminar talk available at https://www.msri.org/seminars/25057

Acknowledgements

We thus obtain the following (un)straightening correspondence.

Corollary 5.2.6. For every Segal ∆ O op♮ -∞-groupoid X, there is an equivalence of (∞, 1) categories between cocartesian fibrations over X and morphisms X → (∞, 1)-Cat × .

Remark 5.2.7. The straightening/unstraightening equivalence for Segal ∆ O op♮ -∞-groupoids takes as input the one for ∆ O op♮ η/ -∞-groupoids. This is reminiscent of the straightening correspondence for Segal (∆ n ) op♮ -∞-groupoids of [START_REF] Nuiten | On straightening for Segal spaces[END_REF] which is inductive on n.


HfC-alloy

Co

Ni Fe Mn Cr Hf

Introduction

Microstructures of the HEA-based TaC-alloy and HfC-alloy in their as-cast states [1, 2]

• patrice.berthod@univ-lorraine.fr

Three points flexural creep tests at 1100°C in argon flow

• Principle of the test and apparatus used:

• Examples of samples after test:

• Deformation curves obtained for the three studied alloys:

• Comparison of the TaC-alloy with two nickel alloys reinforced by (Ta,Hf)C carbides:

Summary / Conclusion

  Cast high entropy CoNiFeMnCr alloys can become alternative solutions to cast cobalt-based and nickel-based superalloys able to resist creep at 1000°C and beyond. The partial substitution of Co and Ni by Fe and Mn allow lower cost and lower dependence on the Co and Ni critical elements. However, their grain boundaries need to be strengthened. In this work equimolar CoNiFeMnCr alloys were elaborated by high frequency induction melting under inert atmosphere, after having added carbon and either Ta or Hf, in quantities rated to favor the development of an intergranular network of script-like eutectic carbides, either TaC or HfC. These carbides were successfully obtained with the required location and morphology. 3 points bending creep tests were carried out at 1100°C for an induced maximal tensile stress = 20 MPa.

  TaC-alloy: double-phased polycrystalline alloy with a dendritic structure; matrix equivalent to the equimolar CoNiFeMnCr Cantor alloy but containing 2 wt.%Ta; interdendritic network made of script-like TaC carbides resulting from final eutectic solidification (L → matrix + TaC) ; presence of rare coarse TaC possibly precipitated as pre-eutectic crystals • HfC-alloy: double-phased polycrystalline alloy with a dendritic structure; matrix equivalent to the equimolar CoNiFeMnCr Cantor alloy (no Hf in matrix); interdendritic network made of script-like TaC carbides resulting from final eutectic solidification (L → matrix + HfC) ; presence of rare coarse HfC possibly precipitated as pre-Strengthening against creep at elevated temperature of HEA alloys of the CoNiFeMnCr type using MC-carbides Patrice Berthod Faculté des Sciences et Technologies, Campus Victor Grignard, F-54500 VANDOEUVRE-LES-NANCY (France) Institut Jean Lamour, CNRS (UMR 7198), 2 allée André Guinier, Campus Artem, F-54000 NANCY (France)

The two alloys studied

  to which this study was devoted, the HfC alloy and the TaC alloy, result from the successful combination of an efficient interdendritic MC carbide network very stable at elevated temperature, and a matrix made of a High Entropy Alloy (CoNiFeMnCr equimolar)case of the HfC alloy -or made of a HEA matrix with minor presence of Ta atoms in solid solution -the TaC alloy -which demonstrated here a particularly interesting creep resistance. Both can be considered as being metal matrix composites made of a HEA matrix (especially the HfC alloy) and strengthening short fibers (HfC or TaC carbides). Obtaining such microstructures was reasonably expected since they were earlier successfully obtained in alloys based on Co-Cr (HfC and TaC), Ni-Cr (HfC) or (Ni,Fe)-Cr and Fe-Cr (TaC) systems. From such Co-based, Ni-based and even (Ni,Fe)-based alloys in which the critical Co and Ni elements are present in rather high quantities, one derived thus new less {Ni and Co}dependent alloys with more Fe and Mn substituting parts of Co and Ni with similar level of high creep-resistance for a temperature at which gamma/gamma prime single-crystalline Ni-based superalloys are significant weakened by the loss of part of their strengthening intermetallic particles.

Acknowledgements:

Lionel Aranda who performed the creep tests.

The corresponding supplemental proceeding TMS is available; see pp. 1093-1102.


Xiaotao Wu

Claude H Moog

L A Marquez Martinez

Yueming Hu

Nonlinear Control of a Buoyancy-Driven Airship

The control of a new kind of airship is presented. By restricting its flight to a vertical plane, the mathematical model is reduced. The simplified model is proved to be minimum phase, and a nonlinear controller based on inputoutput linearization is designed. Since the performance of the controller is significantly impacted by the choice of parameters, simulations of three different pole placement strategies are presented. The nonlinear controller shows better performances than a linear LQR controller when the initial condition is significantly away from the desired equilibrium.

I. INTRODUCTION

Airships attract people's interest again in the recent ten years for their new applications [START_REF] Tozer | High-altitude platforms for wireless communications[END_REF], [START_REF]Lighter Than Air Systems for Future Naval Missions Briefing: Flag Officers and Senior Executive Service[END_REF], and there is a lot of research on its stability, control, planning and tracking of trajectories [START_REF] Gomes | Airship dynamic modeling for autonomous operation[END_REF], [START_REF] Ouyang | Research on modeling and control of an unmanned airship[END_REF], [START_REF] Purandare | A buoyancy-propelled airship[END_REF]. Almost all of these works are based on airships which are driven by propulsors along the hull, and the attitude is controlled by elevators and rudders or vectorpropulsors. These mathematical models are all derived from the Gomes model [START_REF] Gomes | Airship dynamic modeling for autonomous operation[END_REF].

In this paper, a new kind of airship will be considered, which moves forward by a cyclic change of the net buoyancy of the craft and the position of the ballast. This concept of airship is derived from underwater gliders who replace traditional thrust propulsion by a cyclic change of the net buoyancy and of the position of the centroid [START_REF] Leonard | Model-based feedback control of autonomous underwater gliders[END_REF], which has proved to be efficient in water [START_REF] Leonard | Stability of a bottom-heavy underwater vehicle[END_REF]. The structure of this new kind of airship, the inertial frame {G, i, j, k}, and the non inertial frame {O, e 1 , e 2 , e 3 } are shown in Figure 1. A blower is used to fill an hermetic inner bladder with ambient air, and the valve is used to release the inner air. The ballast can move backward and forward.

The mechanism to operate this kind of airship is as follows. When releasing air from ballonets, the mass of the airship reduces, the lift becomes positive. Accompanying the ballast moves to the tail, the airship gets a positive pitch angle θ. Then it moves upward and forward, see Figure 2. Oppositely, when pumping air into ballonets, the airship's mass increases, the lift decreases and becomes negative. With the ballast moving to the head, the pitch angle θ becomes negative. So it moves downward and forward, see Figure 3. If the ballast is moving to the side, then the airship will roll. Due to the coupling of roll and rotation moments, the airship flies to the right or the left [START_REF] Purandare | A buoyancy-propelled airship[END_REF], [START_REF] Leonard | Model-based feedback control of autonomous underwater gliders[END_REF].

This new kind of airship flies more efficiently than airships with thrusters under some conditions. At present, the only monograph available on buoyancy-driven airship is the doctoral dissertation [START_REF] Purandare | A buoyancy-propelled airship[END_REF]. This pioneering studies the feasibility and physical model of this kind of airship, and gives a PID control for the system. However, the complete mathematical model of 6DOF is not given, and the control method also needs to be improved. These motivates this work. The complete equations of 6DOF motion can be found in [START_REF] Wu | Modelling and linear control of a buoyancy-driven airship[END_REF], and a linear LQR controller was designed. The model considered herein is based on the mathematical model in [START_REF] Wu | Modelling and linear control of a buoyancy-driven airship[END_REF].

In the rest of this paper, a simplified model valid for a motion in the vertical plane is introduced without considered disturbances. A suitable choice of the output is proven to be locally minimum phase. The system is then input-output linearized. Some simulations are presented to illustrate the The rest of the paper is organized as follows. In section II, the mathematical model of the buoyancy-driven airship presented in [START_REF] Wu | Modelling and linear control of a buoyancy-driven airship[END_REF] is recalled. The model is further simplified based on some principles in section III. In section IV, it is proved that the simplified model is minimum phase with the selected output and a nonlinear controller is designed to linearize it. The impact of the pole placement for closed loop stabilisation is illustrated by simulation results.

II. PRELIMINARIES

The following complete mathematical model of the 6DOF buoyancy-driven airship has been derived in [START_REF] Wu | Modelling and linear control of a buoyancy-driven airship[END_REF].

          Ṙ ḃ Ω V ṙp Ḃp ṁb           =           R Ω RV J -1 s M m -1 s F 1 m B p -V -Ω × r p mg(R T k) + B p × Ω + u u 4           (1)
where, M is a moment matrix, and F is a force matrix. The values of M and F are as follows,

M = (J s Ω + rp B p ) × Ω + m s V × V +R T M -rp R T F p F = (m s V + B p ) × Ω + R T F -R T F p
The state variables and the rest of the parameters are explained in Table I.

To control the motion of the airship, we start from the most simple case, when the airship only flis in the vertical plane i -k, see Figure 1. In this case the equations are

v1 = 1 m 1 (-m 3 v 3 Ω 2 + ( m -m 0 ) g sin θ +Z a sin α -X a cos α -u 1 ) (2)
v3 = 1 m 3 (m 1 v 1 Ω 2 + ( m + m 0 ) g sin θ -Z a cos α -X a sin α -u 3 ) (3)
Ω2 = 1 J 2 ((m 3 -m 1 ) v 1 v 3 + M a + r p1 u 3 -r p3 u 1 ) (4) θ = Ω 2 (5) ṙp1 = 1 m B p1 -v 1 -r p3 Ω 2 (6) ṙp3 = 1 m B p3 -v 3 -r p1 Ω 2 (7)
Ḃp1 = -B p3 Ω 2 -mg sin θ + u 1 (8) Ḃp3 = B p1 Ω 2 + mg cos θ + u 3 (9) ṁb = u 4 (10) ẋ = v 1 cos θ + v 3 sin θ (11) ż = -v 1 sin θ + v 3 cos θ (12) TABLE I NOMENCLATURE Notation Description α attack angle, α = arctan(v 3 /v 1 ) θ pitch angle Ω Ω = (Ω 1 , Ω 2 , Ω 3 ) T ,
u u = (u 1 , u 2 , u 3 ) T , vector of control inputs ũ ũ = (ũ 1 , ũ2 , ũ3 ) T , transformed vector of control inputs u 4
the control input of the mass of the ballast 2)-( 12), the position of the ballast is represented by r p1 and r p3 , which are unbounded in the system. That means the ballast can freely move in the vertical plane. But the ballast's movement along e 3 axis has little contribution to the control of the attitude and velocity of the airship, and the movement of the ballast along this direction also leads to unnecessary complexity and unstability to the system, because the airship should have "bottom-heaviness". The ballast's movement on the negative direction of e 3 axis contributes to the unstable "topheaviness" [START_REF] Leonard | Stability of a bottom-heavy underwater vehicle[END_REF]. So it is assumed that the ballast is fixed at a constant height. That means a fixed value of r p3 and to cancel u 3 and B p3 . A fixed height of the ballast also makes the actuators more simple and easy to realize. Under these cosiderations we may proceed to simplify the math model.

V V = (v 1 , v 2 , v 3 ) T , velocity along e 1 ,
First, we simplify ( 8) and ( 9) as follows,

Ḃp1 = ũ1 Ḃp3 = ũ3
where, ũ1 and ũ3 denote the total external forces on the ballast, which is different from u 1 and u 3 that only denote the external forces from the body of airship. With fixed r p3 , the right hand side of equation ( 7) is equal to zero. So

B p3 = m(v 3 -r p1 Ω 2 ) and ũ3 = Ḃp3 = m( v3 - ṙp1 Ω 2 -r p1 Ω2 ).
Combining with equations ( 4), ( 2) and ( 6), the new functions of Ω2 and v3 can be derived.

To further simplify the model, it is assumed that during the routine fly, we just adjust the position of the ballast to cope with general disturbances. This assumption is reasonable, because compared with changing the mass of the ballonets, it is easy and energetically economic to change the position of the ballast, and in this paper, we just focus on the stabilization of airship during straight line cruise. So it is unnecessary to change the mass of the ballonets which is denoted by m b .

x and z are the position of the airship in the vertical plane with respect to the inertial frame. Since our control objective will be tracking the attitude angle ξ, and not path following. equations ( 11) and ( 12) can be ignored.

Integrating the above suggestions to simplify the model ( 2)-( 12), it reduces to six states x = (θ, Ω 2 , v 1 , v 3 , r p1 , B p1 ) T and one control input ũ1 :

θ = Ω 2 (13) Ω2 = T 1 H 1 + T 2 H 2 (14) v1 = H 3 /m 1 (15) v3 = T 2 H 1 + T 3 H 2 (16) ṙp1 = B p1 / m -v 1 -r p3 Ω 2 (17) Ḃp1 = ũ1 (18)
where

T 1 = m 3 + m J 2 (m 3 + m) + mm 3 r 2 p1 T 2 = mr p1 J 2 (m 3 + m) + mm 3 r 2 p1 T 3 = J 2 + mr 2 p1 J 2 (m 3 + m) + mm 3 r 2 p1 H 1 = (m 3 -m 1 ) v 1 v 3 -mg (r p1 cos θ + r p3 sin θ) -(r p1 B p1 + r p3 m (v 3 -r p1 Ω 2 )) Ω 2 + M a -r p3 ũ1 + mr p1 Ω 2 (v 1 + r p3 Ω 2 ) -r p1 Ω 2 B p1 H 2 = m 1 v 1 Ω 2 + B p1 Ω 2 + m 0 g cos θ -Z a cos α -X a sin α -mΩ 2 (v 1 -r p3 Ω 2 ) + B p1 Ω 2 H 3 = -m 3 v 3 Ω 2 -m(v 3 -r p1 Ω 2 )Ω 2 -m 0 g sin θ +Z a sin α -X a cos α -ũ1 IV. NONLINEAR CONTROLLER
In [START_REF] Wu | Modelling and linear control of a buoyancy-driven airship[END_REF], a linear controller was derived using a LQR optimal method. This linear controller displays good control performances, but valid only in a small neighborhood around the equilibrium. In this section, a nonlinear controller based on I/O linearization is designed for the system whose internal stability is proved.

A. Stability of zero dynamics

Define x = (θ, Ω 2 , v 1 , v 3 , r p1 , B p1 ) T . Then equations (13)-( 18) are rewritten in the following form,

ẋ = f (x) + g(x)ũ 1 .
Consider the output of the system y = x, that is the six states. Both variables θ and r p1 have relative degree 2 [START_REF] Isidori | Nonlinear control systems: an introduction[END_REF]. However, it is easy to check that considering the output r p1 yields a non minimum phase systems and thus, input-output linearization can not be applied; whereas the output function θ defines a minimum phase system.

The unforced system has an equilibrium x e which can obtained from solving f (x e ) = 0. For the chosen parameter, x e = (θ e , Ω 2e , v 1e , v 3e , r p1e , B p1e ) T = (0.44, 0, 9.97, -0.8, -1, 299) T . The linear approximation around this point is

ẋ = Ax + B ũ1 (19)
where,

A =         0 1 0 0 0 0 -0.
0 -2 -1 0 0 0.03 0 0 0 0 0 0         B =         0 -0.0002 -0.002 0.00001 0 1        
and whose transfer function is easily computed as T F 1 = -0.02s 4 -0.02s 3 -0.1s 2 -0.08s -0.005 100s 6 + 67s 5 + 1.9s 4 -2.3s 3 -5s 2 -s for y = θ, and T F 2 = 3.6s 4 + 2.4s 3 + 0.3s 2 + 0.2s + 0.01 100s 6 + 67s 5 + 1.9s 4 -2.3s 3 -5s 2 -s for y = r p1 .

The four zeros of T F 1 have negative real part, whereas some of T F 2 have positive real part and thus are unstable.

Since the linear approximation of the system (13)-( 18) with the output y = θ has stable transmission zeros, the nonlinear system has locally stable zero dynamics and thus the system is minimum phase [START_REF] Conte | Algebraic Methods for nonlinear control systems-theory and applications[END_REF].

B. I/O linearization of θ

Since the relative degree of the output y = θ is 2, we propose the following desired error equation

ë + λ 1 ė + λ 0 e = w 1
where e = y -θ e , θ e is the desired value for θ, and w 1 denotes a new control input. Equivalently,

θ + λ 1 θ + λ 0 (θ -θ e ) = w 1 (20)
λ 1 and λ 0 assign the poles of the error dynamics. Substituting ( 13) and ( 14) into (20), the equation can be solved and the control ũ1 derived, as follows,

ũ1 = T 4 (H 4 -w 1 + λ 1 Ω 2 + λ 0 (θ -θ e )) (21)
where, 21) is the nonlinear feedback controller of the system. Let w 1 = 0, and substituting (21) into ( 13)-( 18), the closed loop system will be derived.

T 4 = J 2 m 3 + J 2 m + mr 2 p1 m 3 r p3 (m 3 + m) H 4 = T 1 (H 1 + r p3 ũ1 ) + T 2 H 2 Equation (
The choice of the values of λ 1 and λ 0 not only has a direct impact on the motion of θ and Ω 2 , but also a significant one on the states v 1 , v 3 , r p1 and B p1 . The impact on the θ and Ω 2 are easy to analyze, but the transient response of v 1 , v 3 , r p1 and B p1 remains unclear due to the nonlinear equations. By trying different values of λ 1 and λ 0 , different performances of the controller are presented.

C. Simulation results

The airship is desired to fly with angle ξ e and velocity V e , see Figure 1 and4. In this paper, we only consider the control during the upward and downward phase, and do not consider the transfer between these two phases. The conditions in the downward phase are the same as the upward phase with only reversed values of some states and parameters. The parameters used in this project are taken from [START_REF] Wu | Modelling and linear control of a buoyancy-driven airship[END_REF]. The value of the states on the equilibrium x e are presented in the section IV.A of this paper.

When deciding the values of λ 1 and λ 0 , there is a tradeoff between the performance of θ and Ω 2 with respect to the one of states v 1 , v 3 , r p1 and B p1 , and thus they must be carefully chosen. To this end, we will present three different sets of λ 1 and λ 0 . All of these simulations are under initial errors ∆θ = +5 • and ∆v 1 = +2 m/s.

To compare the results of different parameters, the poles under different sets of parameters should have the same negative real part, so θ stabilizes in a similar period, see Figure 5.

First strategy: two complex poles

The poles are chosen to be of the form s = -a(1 ± i). For the reported simulation, a = 1/4, so λ 1 = 1/2 and λ 0 = 1/8, see Figure 5.a. This is called Controller No. 1.

Second strategy: a double real pole

Both poles are placed at -a, with a = -1/4, see Figure 5.b. So the controller No. 2 is with λ 1 = 0.5 and λ 0 = 0.0625.

Third strategy: a dominant real pole

The last considered strategy is to obtain a first-order like response by placing a dominant pole. Controller No. 3 has two different negative real poles. One pole close to the imaginary axis, another is far away from the imaginary axis. Let s 1 = - 1 4 , s 2 = -100 4 , see Figure 5.c, so λ 1 = 25.25 and λ 0 = 6.25.

The dynamics of θ under these controllers are presented in Figure 6. Figure 7 and 8 are the dynamics of v 1 and v 3 with the controller No. 3. To different controllers, the mainly difference of dynamics of v 1 and v 3 is the final values. Figure 9 to 11 are the dynamics of r p1 which presents the movement of the ballast.

D. Analysis of the Simulation Result

Generally, from the simulation results, the system is stable under certain initial errors with these controllers, but the differences between three controllers are significant. In detail, from Figure 6, the three controllers have similar stationarity and rapidity, which is due to the same negative real parts, see Figure 5. Both θ and Ω 2 stabilize at the equilibrium point quickly, which is the goal of these feedback controllers.

With respect to v 1 and v 3 , they have small periodic oscillations under three conditions, but the final amplitudes are different. With controller No. 1 and No.2, the final amplitudes are the same, which is 0.14 m/s for v 1 , and v 3 is small enough to be ignored. The controller No. 3 has much better performance. To v 1 , the final amplitude is 0.01 m/s, and v 3 attenuates to v 3 e quickly.

From Figure 9 to 11, we can more clearly find that controller No. 3 is better than the two others. The amplitude of r p1 under controller No. 3 continually declines with the time, but it is constant under controllers No. 1 and No. 2.

These controllers are also sensitive to the initial errors as with the LQR controller in [START_REF] Wu | Modelling and linear control of a buoyancy-driven airship[END_REF], since large initial errors will lead to unstability. However, the bound is much more broad.

V. CONCLUSIONS AND FUTURE WORK

A. Conclusions

With these controllers, v 1 , v 3 , r p3 and B p1 have periodic oscillations, with a period approximately of 3 s, which is acceptable for the system. These oscillations are associated to the complex zeros of the linear approximation close to the imaginary axis. Here, two of the zeros of T F 1 are -2.85 × 10 -4 ± 2.16i. These nonlinear controllers are better than linear LQR controllers for autonomous buoyancy-driven objects, since the nonlinear controllers have larger stable domain of initial conditions than the linear LQR one, which can be illustrated by simulations. When the initial error of v 1 is larger than +4.5 m/s, the linear controller no more ensures stability and the airship crashes down [START_REF] Wu | Modelling and linear control of a buoyancy-driven airship[END_REF]. But these nonlinear controllers also ensure stability in a certain domain, and large initial errors lead to larger oscillations.

B. Future Work

In this paper, we presented a nonlinear controller based on input-output linearization, using an output with relative degree 2. An open research line is the search of outputs with maximal relative degree to obtain the smallest internal dynamics. Also the behavior of the internal dynamics suggests the existence of stable limit cycles, which produces a continuous change of speed and orientation. Further research is required to eliminate these limit cycles, or maximally reduce their amplitude at least.

Fig. 1 .Fig. 2 .

 12 Fig. 1. Structure of Buoyancy-Driven Airship V

Fig. 3 .

 3 Fig. 3. Descent of Buoyancy-Driven Airship

  angular velocity in the body frame b the position of the airship in the inertial frame Bp Bp = (B p1 , B p2 , B p3 ) T , the momentums of the ballast F the total external forces on the airship Fp total external forces on the ballast Js the moment of inertia of the airship J 2 sum of the moment of inertia and added inertia about e 2 k a unit vector along the direction of the gravity m b the mass of the ballonet ms the stationary mass of the airship m the mass of ballast m 1 , m 3 sum of body and added mass along e 1 , e 3 axis m 0 the net buoyancy of the airship M the total external moments on the airship Ma aerodynamic pitch moment rp rp = (r p1 , r p2 , r p3 ) T , the position of the ballast R rotation matrix

  Fig. 4. Flying Sketch

4 Fig. 5 .

 45 Fig. 5. Three Different Assignments of Poles

Fig. 6 .Fig. 7 .

 67 Fig. 6. Comparison of Dynamics of θ

Fig. 8 .Fig. 9 .

 89 Fig. 8. Dynamics of v 3

Fig. 10 .Fig. 11 .

 1011 Fig. 10. Dynamics of r p1 with Controller No.2


Guillaume Le Guludec

Christine Guillemot

JOINT NEURAL REPRESENTATION FOR MULTIPLE LIGHT FIELDS

Keywords: Light fields, Neural Implicit Representations, Singular value decomposition

Neural implicit representations have appeared as a promising technique for representing a variety of signals, among which light fields, offering several advantages over traditional gridbased representations, such as independence to the signal resolution. While some work has been done to find good initial representations for a given type of signal, usually via metalearning approaches, exploiting the features shared between different scenes remains an understudied problem. We provide a step towards this end by presenting a method for sharing the representation between thousands of light fields, splitting the representation between a part that is shared between all light fields and a part which varies individually from one light field to another. We show that this joint representation possesses good interpolation properties, and allows for a more light-weight storage of a whole database of light fields, exhibiting a ten-fold reduction in the size of the representation when compared to using a separate representation for each light field.

INTRODUCTION

Compared to traditional two-dimensional images, light fields provide a richer modelling of natural visual scenes by discriminating the received intensities not only spatially, but also angularly, meaning that different radiances are recorded for different directions of incident rays. Light fields, when viewed as collections of sub-aperture images, are therefore very costly to store, as the number of pixels grows quadratically in both the spatial and the angular resolution. While signals are traditionally represented as discrete arrays of data points, neural implicit representations have recently gained a lot of attention as efficient alternative means for representing and storing signals.

Since their success in the context of view synthesis [START_REF] Mildenhall | Nerf: Representing scenes as neural radiance fields for view synthesis[END_REF] and shape representation [START_REF] Park | Deepsdf: Learning continuous signed distance functions for shape representation[END_REF][START_REF] Vincent Sitzmann | Scene Representation Networks: Continuous 3D-Structure-Aware Neural Scene Representations[END_REF], neural implicit representations have been applied to a great variety of tasks for many types of signals, including to light field representation [4, 5, This work was supported by the french ANR research agency in the context of the artificial intelligence project DeepCIM. [START_REF] Attal | Learning neural light fields with ray-space embedding[END_REF]]. Those methods typically rely on some encoding of the input coordinates [START_REF] Tancik | Fourier features let networks learn high frequency functions in low dimensional domains[END_REF], followed by a multi-layer perceptron that predicts the value of the signal at the given coordinate. Feng et al. were able to achieve excellent representation quality of dense light fields using an architecture which they dubbed SIGNET [START_REF] Yushan | Signet: Efficient neural representation for light fields[END_REF]. Their method, however, can only be used to represent a single light field at a time. Building upon their architecture, we show that it is possible to jointly represent an entire database of dense light fields, and demonstrate our approach on a collection of two thousand light fields of natural scenes captured using a Lytro Illum.

LIGHT FIELD REPRESENTATION

Light fields parameterization

A light field (LF) can be viewed as a function that associates a radiance in each of the color channels of interest to any geometrical ray in space. We consider the widespread twoplane parameterization of light fields. In this parameterization, each ray in space is uniquely characterized by the coordinates of its intersection with two fixed planes, the spatial and angular planes, corresponding usually (albeit not necessarily) to the plane of the camera sensor and the plane of the camera aperture respectively. In such a parameterization, illustrated in Fig. 1a, any ray is described by a tuple (x, y, u, v). We consider RGB light fields; a LF is thus simply a function L : R 4 → R 3 from the space of coordinates to the space of color values. In the remainder of the article, for the sake of brevity, we use the symbol x to denote the 4D vector of spatio-angular coordinates. We denote the space of spatioangular coordinates for which data is available by X.

Representing a single light field

We seek to approximate a LF using a synthesis network f (.; θ) parameterized by some θ ∈ R d , called a representation of the LF. The quality of the representation depends on a chosen metric; in our experiments we choose to use the pixel-wise mean squared error as an objective metric to measure the distortion of the signal. Our goal of finding a good representation therefore comes down to solving the following regression problem: min x for a given LF, using the LF representation θ = (θ 1 , . . . , θ K ). The representation θ k of layer k is produced by the representation provider P k . σ k is a tensor of size (J, r) that denotes the concatenation of the σ k j for all LFs; b k is a tensor of size (J, m) concatenating the biases for all LFs; select denotes the operation of selecting a given row; * denotes broadcast multiplication of a vector by a matrix, while × denotes matrix multiplication. This problem is efficiently solved using a stochastic gradient descent solver, such as ADAM [START_REF] Diederik | Adam: A method for stochastic optimization[END_REF], combined with standard backpropagation for computing the gradient of the objective with respect to the representation θ.

θ x∈X f (x; θ) -L(x) 2 (1)
(u, v) • (x, y) (a

Architecture of the representation network

Feng et al. [START_REF] Yushan | Signet: Efficient neural representation for light fields[END_REF] were able to obtain a good neural implicit representation of LFs by combining a fully connected multilayer perception with sinusoidal activation with a polynomial positional encoding. We briefly present here the architecture, which is based on the SIREN architecture [START_REF] Vincent Sitzmann | Implicit neural representations with periodic activation functions[END_REF]. The layers of their network are defined by:

a 0 = Encode(x) (2)
y k = sin(a k W k + b k ) (3)
z k = y k + a k (4)
a k+1 = LayerNorm(z k ) (5)
where Eq. 4 holds for all layers k except for the first and last ones, for which z k = y k and Eqn. 5 holds for all layers k except for the last one, which satisfies a k+1 = z k instead.

In our experiments, we reuse their architecture without any change, except for the last layer, in which the activation is replaced by a sigmoid. We have therefore θ = (W k , b k ) k≤K .

Positional encoding

In [START_REF] Yushan | Signet: Efficient neural representation for light fields[END_REF], the authors use the Gegenbauer polynomials G α up to a certain order as the positional encoding (Eq. 2) and show that α = 0.5 is empirically optimal, which corresponds to using the Legendre polynomials. We found that replacing this polynomial positional encoding scheme with a random Fourier encoding [START_REF] Tancik | Fourier features let networks learn high frequency functions in low dimensional domains[END_REF] did not hurt the ability of the model to fit the LFs and reduces some artifacts, provided the (integer) input coordinates x are normal-ized to an appropriate scale. Formally, we use an encoder with trainable parameters W and b defined by:

Encode(x) = sin((x -x 0 )W + b) (6)
where x 0 is the mean coordinates. W is a 4 × n matrix in which the elements in the two first rows (corresponding to the spatial coordinates x and y) are randomly initialized by sampling the uniform distribution U(-s, s), while the elements in the last two rows (corresponding to the angular coordinates u and v) are initialized from a uniform distribution U(-t, t). We found that using s = 0.2 and t = 0.06 yielded good results. The elements of b are obtained by sampling from U(-π, π).

REPRESENTING MULTIPLE LIGHT FIELDS

Problem formulation

Let us now consider the more general case in which we have a collection of LFs (L j ) indexed by some j ∈ J. We call representation provider a function P (j; φ) parameterized by some vector φ, that maps the index j of a LF to a representation θ of that LF. The goal of the joint representation is to find a representation provider by learning the parameters φ, such that P (j; φ) provides on average a good representation of L j .

In other words, we want to minimize the following loss:

L(φ) = j∈J x∈X f (x; P (j; φ)) -L j (x) 2 (7)
The rationale for finding a joint representation relies on the notion that all representations θ lie on the same manifold. Since light fields of natural scenes share some common features (e.g parallax and laws of optics, common shapes, etc.), it is natural to expect some structure on this manifold. One approach is thus to learn this structure by training from a dataset of various LFs.

How to parameterize the representation manifold?

Hypernetworks [START_REF] Ha | Hypernetworks[END_REF] could be used as a means to parameterize this manifold. The hypernetwork might be used to map a discrete representation of a LF to a neural representation θ, e.g. using a CNN. It could also be an auto-decoder [START_REF] Vincent Sitzmann | Scene Representation Networks: Continuous 3D-Structure-Aware Neural Scene Representations[END_REF][START_REF] Park | Deepsdf: Learning continuous signed distance functions for shape representation[END_REF] mapping some latent code z of a given LF, to a neural representation θ.

However, while this approach might be yielding good results in the case of simple signals such as low-resolution 2D images, it becomes intractable in practice when working with light fields even of average resolution. Indeed, representing a LF usually requires a larger network when compared to a 2D image due to a higher complexity. The output of the hyper-networks therefore becomes very large, potentially several millions, which makes the approach intractable.

As an alternative, FiLM conditioning [START_REF] Vincent Dumoulin | Feature-wise transformations[END_REF][START_REF] Perez | Film: Visual reasoning with a general conditioning layer[END_REF] is an approach that modulates the pre-activations in an affine manner using a scaling and offset coefficient. In the same vein as FiLM, we first designed an approach that shared the same weight matrices W k between all LFs, while learning individual biases for each LF. While we found that this yields good results when the number of LFs in the dataset is small, as the number of LFs increases, the faithfulness of the reconstruction degrades.

In order to mitigate this problem, we increased the capacity of the representation provider by allowing the weight matrix in the representation to vary between LFs. In the following sections, we describe the structure of the representation provider.

PROPOSED REPRESENTATION

Structure of the joint representation

A joint representation φ is obtained by solving Eqn. 7 using stochastic gradient descent and gradient back-propagation. As f is a neural network, each forward and backward pass is costly. Therefore we want the representation provider to be as light-weight as possible.

The first idea is to split the parameterization φ of the representation provider into two parts: a shared part φ sh and individual part (φ ind j ) j∈J . For a LF of index j, the two parts are combined by a simple procedure described below to provide the representation θ.

In our approach, the shared part of the joint representation is composed, for each layer k in the synthesis network with input dimension m and output dimension n, of a pair of matrices:

φ sh = (U k , V k ) k ( 8
)
where U k has size m×r and V k has size r×n, m and n being respectively the input and output dimensions of the layer; r is a hyper-parameter controlling the trade-off between the num-ber of parameters and the quality of the joint representation. The individual parts of the representation are given by:

φ ind j j = (σ k j , b k j ) k for all j ∈ J (9)
where σ k and b k j are vectors of size r and n respectively. The full joint representation for the collection of LFs L j is the pair φ sh , φ ind j j

Representation provider P

The representation provider P (.; φ) that maps the index j of a LF to its representation θ is defined by:

P (j; φ) = (W k j , b k j ) k ( 10
)
where the weight matrix W k j is given by

W k j = U k diag(σ k j )V k for all j ∈ J and layer k ( 11
)
where diag(x) is the diagonal matrix whose diagonal elements are those of x. The matrix product U k diag(σ k j ) is computed by broadcast multiplication of the rows of U k by the elements of σ k j . The complete representation pipeline is illustrated in Fig. 1b. We motivate our choice of architecture in the section below.

Inspiration from the singular value decomposition

Any matrix W of size m × n can be written as its well-known singular values decomposition (SVD):

W = UΣV (12)
where U and V are orthonormal matrices of size m × r and r × n respectively and Σ is a diagonal matrix of size r × r containing the non-zero singular values of W, where r ≤ min(m, n) is the rank of W, equal to the number of non-zero elements in Σ.

Assuming that we have weight matrix W k j for a layer k and a LF j ∈ J, we could separately apply the SVD to each matrix as:

W k j = U k j Σ k j V k j (13)
However, this is not going to lead to any joint representation.

Our idea was to assume that the weight matrices can be jointly decomposed across LFs as

W k j = U k Σ k j V k (14)
with U k and V k now being shared between the different LFs.

Since an arbitrary collection of matrices is unlikely to be amenable to joint decomposition, we relax the constraint that r be less than or equal to min(m, n).

It is worth noting that such a parameterization enforces the representation manifold to be a vector space. The shared part of the representation effectively provides a basis for this space, while the individual parts correspond to coordinates in the representation manifold. We performed experiments using the architecture for the synthesis network described in section 2.3 and the representation provider described in section 4.2. We experimented with values for hyper-parameter r ranging in {4096, 8192, 16384}. Our dataset is composed of the first 2,000 LFs of the dataset introduced by [START_REF] Pratul | Learning to synthesize a 4d rgbd light field from a single image[END_REF], consisting of 375 × 540 × 8 × 8 RGB dense light fields of flowers captured using a Lytro Illum.

To be consistent with the singular value decomposition idea, for each layer k, the matrices U k and V k are initialized as random orthogonal matrices, while the elements of σ k j are drawn from the uniform distribution U(0, √ 6). During training however, neither U k nor V k are constrained to be orthogonal, and the elements of σ k j can become negative. We used K = 10 layers for our synthesis network, with 512 hidden channels per layer.

We trained each model for 16 million iterations, with batch size of 4096. This is equivalent to 2.5 epochs of training; that is, each pixel of each LF is fed on average 2.5 times to the network. At each iteration, we randomly sample a LF index j along with a random batch of coordinates x. The forward pass is described in Algorithm 1. Following [START_REF] Yushan | Signet: Efficient neural representation for light fields[END_REF], we use a learning rate with cosine decay from 10 -5 to 10 -8 . Training the largest model (r = 16384) took 20 days on a Nvidia RTX 2080 Ti.

Parameters reduction

To demonstrate the ability of our approach to effectively reduce the total number of parameters required to fit an entire dataset, we separately trained a model (as defined by Eqs. 2 to 6) on several LFs of the whole dataset. We considered a random subset J of 100 LFs and computed the average PSNR as an estimate of the average PSNR on the whole dataset. We trained a new set of weights for each LF j ∈ J for 3 epochs, so as to provide a fair comparison with the joint training. We adjusted the capacity of the individual models so that the reconstruction quality after 3 epochs approximately matches that of the joint training. To do so, we chose a depth of 5 layers for the synthesis network.

Results

Table 2 shows the average PSNR on the whole dataset for different values of r, as well as the size of the models per LF (i.e. the total number of parameters divided by the number of LFs), and additionally provides a comparison with the separately trained models. We see that, for the same reconstruction quality, the joint representation results in a ten-fold reduction in the number of parameters. Fig. 3 provides a visual illustration of the reconstruction quality of our joint representation. In addition to the central view, we computed interpolated epipolar plane images (EPI) by querying the model on non-integer coordinates. More precisely, we interpolated by a factor 8, converting the 8 × 8 LF into a 64 × 64 LF.

The smoothness and quality of the interpolated EPIs show the ability of the joint representation to capture parallax and perform angular interpolation on the learned LFs.

CONCLUSIONS

We have presented a method to jointly represent a large collection of light fields using neural implicit representations.

We have shown that our technique yields good reconstruction quality both visually and in terms of PSNR. Additionally, we showed that the joint representation greatly reduces the number of parameters required to represent an entire collection of light fields when compared a set of scene-by-scene representations.

Fig. 1 :

 1 Fig.1: (a) Two-plane parameterization of a LF. (x, y) and (u, v) parameterize the intersection of the ray with the spatial and angular planes respectively. (b) Representation pipeline: the synthesis network f computes the RGB value y from ray coordinates x for a given LF, using the LF representation θ = (θ 1 , . . . , θ K ). The representation θ k of layer k is produced by the representation provider P k . σ k is a tensor of size (J, r) that denotes the concatenation of the σ k j for all LFs; b k is a tensor of size (J, m) concatenating the biases for all LFs; select denotes the operation of selecting a given row; * denotes broadcast multiplication of a vector by a matrix, while × denotes matrix multiplication.

Algorithm 1

 1 Joint training forward passRequire: LF index j, coordinates x a = Encode(x)for k ∈ [0..K -1] do b ← b k j W ← U k diag(σ k j )V k a ← Layer k (a, W, b

Fig. 2 :

 2 Fig. 2: Reconstruction quality and model size for the different models.

Fig. 3 :

 3 Fig. 3: Central views along with EPIs for 3 LFs in the Flowers dataset [14]. Top row are the LFs produced by the joint model with rank r = 16384; bottom row corresponds to a model specifically fitted for one LF.