∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\infty $$\end{document}-Operads via symmetric sequences

We construct a generalization of the Day convolution tensor product of presheaves that works for certain double ∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\infty $$\end{document}-categories. Using this construction, we obtain an ∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\infty $$\end{document}-categorical version of the well-known description of (one-object) operads as associative algebras in symmetric sequences; more generally, we show that (enriched) ∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\infty $$\end{document}-operads with varying spaces of objects can be described as associative algebras in a double ∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\infty $$\end{document}-category of symmetric collections.


Introduction
The theory of operads is a general framework for encoding and working with algebraic structures, first introduced in the early 70s in order to describe homotopy-coherent algebraic operations on topological spaces [5,46]. Since then, the theory has found many applications in diverse areas of mathematics-aside from algebraic topology, where operads in topological spaces, simplicial sets, and spectra have numerous uses (see for example [5,7,17,46,50], among many others), operads in vector spaces and chain complexes (also known as linear operads and dg-operads, respectively) are by now a well-studied topic in algebra (see for instance [28,42]), with applications in mathematical physics (cf. [45]) and algebraic geometry (e.g. [41]), while operads in sets have become a standard tool in combinatorics (cf. [29,47]).
Classically, an operad O in a symmetric monoidal category C consists of a sequence O(n) of objects of C, where the symmetric group n acts on O(n) (this data is called a symmetric sequence) together with a unital and associative composition operation. If C has colimits indexed by groupoids and the tensor product preserves these, then this data can be conveniently encoded using the composition product of symmetric sequences. This is a monoidal structure on symmetric sequences, given by the formula 1 the unit is the symmetric sequence where 1 is the unit in C. As first observed by Kelly [40], an operad is then precisely an associative algebra with respect to In homotopical settings, this classical notion of operads has a number of shortcomings, analogous to those afflicting topological or simplicial categories when we want to work with them only up to homotopy (i.e. consider them as models for ∞-categories). This motivates the introduction of a fully homotopy-coherent version of operads, known as ∞-operads. Just as in the case of ∞-categories, there are several useful models for ∞-operads, including those of Lurie [44] (which is currently by far the best-developed), Moerdijk-Weiss [48], Cisinki-Moerdijk [14], and Barwick [3]. These authors only consider ∞-operads in spaces, but the formalism has recently been extended to cover ∞-operads in other symmetric monoidal ∞-categories in [10].
The goal of the present paper is to provide another point of view on (enriched) ∞-operads, by extending to the higher-categorical setting the description of operads as associative algebras in symmetric sequences: Theorem Let F be the groupoid of finite sets and bijections. If V is a symmetric monoidal ∞-category compatible with colimits indexed by ∞-groupoids 2 , then there exists a monoidal structure on the ∞-category Fun(F , V) of symmetric sequences such that associative algebras are V-enriched ∞-operads. Moreover, the tensor product is described by the same formula as above.
More precisely, this gives a description of ∞-operads with a single object. It is often convenient to consider the more general notion of operads with many objects (also known as coloured operads or symmetric multicategories), and the term ∞-operad typically refers to the higher-categorical version of these generalized objects, which also have a description as associative algebras: For a set S, let . . , m} in F such that s i = s σ (i) . Then a (symmetric) S-collection (or S-coloured symmetric sequence) in V is a functor F S × S → V. The category Fun(F S × S, V) again has a composition product •, given by a more complicated version of the formula we gave above, such that an operad with S as its set of objects is precisely an associative algebra for this monoidal structure. Our work also gives an ∞-categorical version of this many-object composition product.
More generally, we can describe operads with varying spaces of objects as associative algebras in a double category. We will call a functor F S × T → V an (S, T )-collection in V. Then we can define a double category COLL(V) as follows: • Objects are sets, and vertical morphisms are maps of sets.
• Horizontal morphisms from S to T are (S, T )-collections.
• Composition of horizontal morphisms is given by a version of the composition product.
An associative algebra in COLL(V) consists of a set S together with an associative algebra in the category of horizontal endomorphisms of S with composition as monoidal structure, i.e. an associative algebra in S-collections with the composition product. Thus associative algebras are precisely operads, and moreover morphisms of algebras in COLL(V) are precisely functors of operads. We will produce an ∞-categorical version of this structure: Theorem For any symmetric monoidal ∞-category V compatible with colimits indexed by ∞-groupoids there is a double ∞-category COLL(V) such that Alg(COLL(V)) is the ∞category of V-enriched ∞-operads. 3 The double ∞-category COLL(V) admits the same description as its analogue for ordinary categories, except with ∞-groupoids as objects.
In a sequel to this paper [35] we apply this description of ∞-operads to study algebras over enriched ∞-operads. In addition, we hope that it can serve as a starting point for a better understanding of bar-cobar (or Koszul) duality for ∞-operads. Over a field of characteristic zero, Koszul duality for dg-operads was introduced by Ginzburg and Kapranov [28], and is by now well understood using model-categorical methods (see e.g. [21,22,27,42,52]). As a first step towards an ∞-categorical approach to Koszul duality, here we construct a barcobar adjunction between ∞-operads and ∞-cooperads. Following the approach proposed by Francis and Gaitsgory [20], we obtain this as the bar-cobar adjunction between associative algebras and coassociative coalgebras (constructed in great generality in [44,Sect. 5.2.2]) applied to our monoidal ∞-category of symmetric sequences. This seems likely to agree with existing constructions not only in chain complexes over a field of characteristic 0, but also in other settings such as spectra [8,9], where it is closely related to Goodwillie calculus [7], as well as in K (n)-local spectra, where bar-cobar duality has been constructed and applied in work of Heuts [37].

Overview of results
Let us now give a more detailed overview of the results of this paper. The starting point for our construction is the "coordinate-free" definition of the composition product due to Dwyer and Hess [18,Sect. A.1]. They observe that, if F [1], denotes the groupoid of morphisms of finite sets and F [2], denotes the groupoid of composable pairs of morphisms of finite sets, then: • Symmetric sequences in Set are the same thing as symmetric monoidal functors F [1], → Set, with respect to the disjoint union in F [1], and the cartesian product of sets. • Under this identification the composition product of X and Y corresponds (by [18,Lemma A.4]) to the left Kan extension, along the functor F [2], → F [1], given by composition, of the restriction of X × Y from F [1], × F [1], to F [2], . In other words, (A→B→C)∈F [2], If C ∼ = * , then the groupoid F [2], (A→ * ) of factorizations of A → * is the groupoid of maps A → B and isomorphisms B ∼ − → B under A. An isomorphism class of such objects corresponds to a decomposition |A| = i 1 + · · · + i k where k = |B|, with a division of A into subsets of size i j . This can be rewritten to recover the previous formula (with the division of A corresponding to the product with n for a given partition of n = |A|).
After a slight reformulation this description is closely related to Barwick's indexing categorý F for ∞-operads, introduced in [3]. This is the category with objects sequences S 0 → S 1 → · · · → S n of morphisms of finite sets, with a map (S 0 → · · · → S n ) → (T 0 → · · · → T m ) given by a map φ : [n] → [m] in´and injective morphisms S i → T φ(i) such that the squares are cartesian. If (´F) [n] denotes the fibre at [n] of the obvious projection´F →´, then: • Symmetric sequences in Set are the same thing as functors X : (´F) op [1] → Set such that for every object S → T the map op [1] corresponding to d 1 : [1] → [2], of the restriction of X × Y along the functor (´F) op [2] → (´F) op [1] × (´F) op [1] corresponding to (d 2 , d 0 ). In other words, op [2] ) /(A→C) This is equivalent to the previous description since the inclusion F [2], (A→C) → (´F) op [2],/(A→C) is cofinal: given an object ξ in the target, which is a diagram where the square is cartesian, the category (F [2], (A→C) ) ξ/ is a contractible groupoid with the single object given by the factorization A → B × C C → C.
The projection´F →´is a Grothendieck fibration, and the corresponding functor :´o p → Cat is a double category, in the sense that it satisfies the Segal condition n ∼ − → 1 × 0 · · · × 0 1 .
We will obtain the composition product by applying to this double category a general construction of monoidal structures on functor categories arising from certain double ∞-categories. In fact, our construction will produce a canonical double ∞-category of which this monoidal ∞-category is a piece, with the full double ∞-category describing ∞-operads with varying spaces of objects.
The construction of this double ∞-category can be seen a variation of the Day convolution [16] structure on functor categories: If C is a small monoidal category and V is a monoidal category compatible with colimits, then the functor category Fun(C, V) has a tensor product, given for functors F and G as the left Kan extension along ⊗: C × C → C of the composite This monoidal structure has the property that an associative algebra in Fun(C, V) is the same thing as a lax monoidal functor C → V; more generally, the Day convolution has a universal property for algebras over non-symmetric operads.
Day convolution (in the symmetric monoidal setting) was implemented in the ∞categorical context by Glasman [30]. 4 In this paper we extend this to a construction of Day convolution for a class of double ∞-categories: This theorem summarizes the results of Sect. 3: We construct these double ∞-categories in Sect. 3.2 using an unfolding construction introduced in Sect. 3.1, and prove the universal property in Sect. 3.3. Note that the precise meaning of "suitable" we need is quite restrictive. We also show in Sect. 3.4 that we can extract from M S a family of monoidal ∞-categories and lax monoidal functors which suffices to describe associative algebras in M S . Moreover, we consider enriched versions of the theorem, with more general targets than S, in Sect. 3.5.
To obtain our double ∞-categories we use results on ∞-categories of spans due to Barwick [4], and Sect. 2 is devoted to a review of this work, with some slight variations, together with a brief review of non-symmetric ∞-operads and related structures.
In Sect. 4 we apply our results on Day convolution to ∞-operads. In Sect. 4.1 we describe non-enriched ∞-operads as associative algebras in a double ∞-categories of collections in S, and in Sect. 4.2 we extend this to a description of enriched ∞-operads. More precisely, we obtain an equivalence between associative algebras in a double ∞-category of collections and ∞-operads in the sense of Barwick [3], as generalized to enriched ∞-operads in [10]. In Sect. 4.3 we then apply this description of ∞-operads to obtain the bar-cobar adjunction between ∞-operads and ∞-cooperads.
As a warm-up to this description of ∞-operads, in Sect. 3.6 we also consider an additional application of our Day convolution construction, by showing that enriched ∞-categories can be described as associative algebras.

Related work
There are at least two other approaches to constructing the composition product on symmetric sequences ∞-categorically:

Composition product from free presentably symmetric monoidal categories
An alternative approach to defining the composition product of S-coloured symmetric sequences in Set starts with the observation that Fun( ∞ n=0 S n h n , Set) is the free presentably symmetric monoidal category generated by S. If C is a presentably symmetric monoidal category we therefore have a natural equivalence where the right-hand side denotes the category of colimit-preserving symmetric monoidal functors. Taking C to be we get a natural equivalence Here the right-hand side has an obvious monoidal structure given by composition of functors, and this corresponds under the equivalence to the composition product of S-coloured symmetric sequences. This construction is described in [2,Sect. 2.3]. The one-object variant is much better known; it is attributed to Carboni in the "Author's Note" for [40], and it is also found in Trimble's preprint [51]. There is also an enriched version of this construction, for (coloured) symmetric sequences in a presentably symmetric monoidal category. More recently, this approach has been further developed in [19,23] where it is shown to arise from a 2-categorical construction that produces a 2-category of operads with varying sets of objects (but with bimodules of operads as morphisms rather than functors).
In the ∞-categorical setting, it is not hard to see that Fun( ∞ n=0 X n h n , S) is again the free presentably symmetric monoidal ∞-category generated by a space X . One can thus take the same route to obtain a composition product on X -coloured symmetric sequences in the ∞-category of spaces. In the one-object case this approach (including its enriched variant) is worked out in Brantner's thesis [6,Sect. 4.1.2]. However, this approach has not yet been compared to any of the established models for ∞-operads.

Polynomial monads
In [26] we show that ∞-operads with a fixed space of objects X are equivalent to analytic monads on the slice ∞-category S /X . These analytic monads can be viewed as associative algebras under composition in an ∞-category of analytic endofunctors of S /X . The latter can be identified with X -coloured symmetric sequences in S, so this gives an alternative description of ∞-operads as associative algebras for the composition product. Compared to our approach here, this has a number of advantages: • it makes it clear that an ∞-operad can be recovered from its free algebra monad, • it clarifies the relation between ∞-operads and trees (because free analytic monads can be described in terms of trees).
It also seems likely that versions of polynomial monads in other ∞-topoi can be used to define operad-like structures that occur in equivariant and motivic homotopy theory. On the other hand, polynomial monads do not seem to extend usefully to give a description of enriched ∞-operads.

Background on spans and Non-symmetric ∞-operads
In this section we first review non-symmetric ∞-operads and related structures in Sect. 2.1, and then recall some definitions and results regarding spans from [4], with some minor variations to get the generality we need in the next section.

Review of non-symmetric ∞-operads
For the reader's convenience, we will briefly review some definitions and results related to non-symmetric ∞-operads that we will make frequent use of below. For more details, as well as motivation, we refer the reader to [24,32,44].
. If C is an ∞-category with products, then an associative monoid in C is a functor A :´o p → C such that for every n the map A n → n i=1 A 1 , induced by the maps , is an equivalence.

Definition 2.1.3
If C is an ∞-category with finite limits, then a category object in C is a functor X :´o p → C such that for every n the map X n → X 1 × X 0 · · · × X 0 X 1 induced by the maps ρ (i−1)i and ρ ii , is an equivalence.

Remark 2.1.4
A category object in the ∞-category S of spaces is a Segal space in the sense of Rezk [49]. The structure of a Segal space describes precisely the "algebraic" structure of an ∞-category, i.e. a homotopy-coherent composition with identities, but to capture the correct equivalences between ∞-categories one must invert the fully faithful and essentially surjective maps between Segal spaces, or equivalently restrict to the full subcategory of complete Segal spaces. Definition 2.1.5 A monoidal ∞-category is a cocartesian fibration C ⊗ →´o p such that the corresponding functor´o p → Cat ∞ is an associative monoid. Similarly, a double ∞-category is a cocartesian fibration M →´o p such that the corresponding functor´o p → Cat ∞ is a category object. Notation 2.1. 6 We will use the following terminology to describe double ∞-categories M → op : • an object of M 0 is an object of the double ∞-category, • a morphism of M 0 is a vertical morphism of the double ∞-category • an object of M 1 is a horizontal morphism, • a morphism in M 1 is a square, • composition of vertical morphisms is composition in the ∞-category M 0 , • vertical composition of squares is composition in M 1 • composition of horizontal morphisms, as well as horizontal composition of squares, is given by the functor , and call this the ∞-category of horizontal morphisms from X to Y . Given its simplicial origin, it is usually less confusing to write composition of horizontal morphisms in the non-standard order, and we denote it We will also write 1 X ∈ M(X , X ) for the horizontal identity.  [1] , induced by the cocartesian morphisms over the maps ρ (i−1)i and ρ ii , is an equivalence, We refer to the cocartesian morphisms over inert morphisms in´o p as inert morphisms in O. A morphism of generalized non-symmetric ∞-operads is a functor over´o p that preserves inert morphisms; we also refer to a morphism of generalized non-symmetric ∞-operads O → P as an O-algebra in P and write Alg O (P) for the ∞-category of these. More generally, if O and P are generalized non-symmetric ∞-operads over Q we write Alg O/Q (P) for the analogous ∞category of commutative triangles of morphisms of generalized non-symmetric ∞-operads O P Q.

Notation 2.1.10
If O is a generalized non-symmetric ∞-operad and x is an object of O n , we will often write x → x i j for the cocartesian morphism over ρ i j for 0 ≤ i ≤ j ≤ n.    The ∞-categorical analogue of Day convolution was first constructed by Glasman [30] for symmetric monoidal ∞-categories. It was generalized by Lurie [44,Sect. 2.2.6] to Omonoidal ∞-categories where O is a (symmetric) ∞-operad and further extended by Hinich to flat ∞-operads [39]. The following is a special case of another generalization, proved in [11]: then there is a fully faithful O-monoidal functor given over X ∈ O [1] by the Yoneda embedding Proof This is a special case of [ is an equivalence. We write Seg O (C) for the full subcategory of Fun(O, C) spanned by the Segal O-objects.
for the cartesian fibration corresponding to the functor X → Alg O X /O (U) and refer to its objects as O-algebroids in U.
Example 2.1.21´o p -algebroids in a monoidal ∞-category V are algebras in V for the fam-ily´o p X (X ∈ S) of generalized non-symmetric ∞-operads. These were called categorical algebras in [24], where they were used to model ∞-categories enriched in V.
natural in , and so an equivalence with image those Segal U op,⊗ -spaces such that for every x ∈ O [1] , p ∈ (x 00 ), and q ∈ (x 11 ) the presheaf induced by the cocartesian morphisms over f , preserves K -colimits in each variable.
is an equivalence, then π is a cocartesian fibration.
Proof We first prove that O has locally cocartesian morphisms over any active map α : [n] → [m] in´. Given x ∈ O m and y ∈ O n , we have where α i j is the active part of α • ρ i j . By assumption we have locally cocartesian morphisms x α(i)α( j) → α i j,! x α(i)α( j) (if i = j this is just the identity), so we can rewrite this as where the first map is cocartesian over ι and the second is locally cocartesian over α: for y ∈ O n we have an equivalence x is cocartesian. This shows that O →´o p is a locally cocartesian fibration. Before we prove part (ii), we make a further observation in the general case: Suppose Then it follows from the decomposition above of α ! in terms of locally cocartesian morphisms over the unique active maps It remains to prove (ii), for which we have to check that the assumption implies that locally cocartesian morphisms over active maps compose, i.e. for active morphisms the natural map (βα) ! X → α ! β ! X is an equivalence for X ∈ O k . Using the decomposition of locally cocartesian morphisms above we can immediately reduce to the case where m = 1. Now if α is surjective, we must have n = 0 or 1; if n = 0 then β = id [0] , while if n = 1 then α = id [1] -in either case the claim is trivially true. We can therefore assume that α is not surjective, in which case we can find a factorization of α as where α (1) = α (0), α (2); using this factorization we get for X ∈ O k a commutative square Here our assumption guarantees the horizontal maps are equivalences, and we want to show the left vertical map is an equivalence. It thus suffices to show the right vertical map is an equivalence, for which it's enough to prove (βα ) ! X → α ! β ! X is an equivalence since d 1,! is a functor. Our assumption on α (1) means this decomposes as a pair of maps This means we can reduce to our assumption by inducting on n. Combined with our previous observations we have then shown that locally cocartesian morphisms compose in general, since it holds for all combinations of active and inert maps. Thus π is a cocartesian fibration, as required.

∞-Categories of spans
For compatibility with [4] we will work with ∞-categories as quasicategories, i.e. simplicial sets satisfying the horn-filling condition for inner horns, in this and the next subsections.
This induces a functor * : Set → Set given by composition with ; this functor is the edgewise subdivision of simplicial sets. If C is an ∞-category, we will write Tw r C := * C and refer to this as the twisted arrow ∞-category of C.
is a right fibration.

Remark 2.2.3
If C is an ordinary category, then it is easy to see that Tw r C can be identified with the twisted arrow category of C. This has morphisms c → d in C as objects, and diagrams Unwinding the definition of Tw r C for C an ∞-category, we see that its objects and morphisms admit the same description in terms of C. ; let us call the resulting simplicial set Tw C -this is the definition of the twisted arrow ∞-category used in [4] (there called O(C)). We clearly have Tw r C ∼ = (Tw C) op , which explains why op's appear in different places here compared to [4].

Definition 2.2.6
The functor * has a right adjoint * : Set → Set , given by right Kan extension. Explicitly, * X is determined by Hom( n , * X ) ∼ = Hom(Tw r ( n ), X ). If C is an ∞-category, we write Span(C) for the simplicial set * C.
We say a simplex n → Span(C) is cartesian if the corresponding functor F : Tw r ( n ) → C is the right Kan extension of its restriction to Tw r ( n ) 0 , or equivalently if for all integers 0 ≤ i ≤ k ≤ l ≤ j ≤ n, the square is cartesian. We write Span(C) for the simplicial subset of Span(C) containing only the cartesian simplices. Definition 2.2.10 Following Barwick [4], we say a triple is a list (C, C F , C B ) where C is an ∞-category and C B and C F are both subcategories of C containing all the equivalences. We will call the morphisms in C B the backwards morphisms and the morphisms in C F the forwards morphisms in the triple. We say a triple is adequate if for every morphism f :

Example 2.2.11
If C is any ∞-category, we have the triple (C, C, C) where all morphisms are both forwards and backwards morphisms. We call this the maximal triple on C; it is adequate if and only if C has pullbacks.

Remark 2.2.12
In [4], the forwards morphisms are called ingressive and the backwards morphisms are called egressive.

Definition 2.2.13
Given a triple (C, C F , C B ) we define Span B,F (C) to be the simplicial subset of Span(C) containing only those simplices that correspond to maps σ : This is a consequence of the following simple observation: Proof For anyx in E over x ∈ B we have a commutative cube in the ∞-category of spaces. Here the bottom face is cartesian since a is a pullback, and the front and back faces are cartesian since the morphismsā →b andā →b are p-cartesian. Therefore the top face is also cartesian. Since this holds for allx ∈ E this meansā is the pullbackā ×bb , as required. In the remaining part of this subsection we give a reformulation of the results of [4, Sect. 12] that will be convenient for us.

Corollary 2.3.6 If E → B is as in Definition 2.3.1, then
Proof To prove (i) we must show that there exists a lift in every commutative square n k with 0 < k < n. This is equivalent to giving a lift in the corresponding commutative square Here

in E is locally p-cocartesian if and only if g is a locally p-cocartesian morphism in E.
Proof We first prove (i). Consider a 1-simplex φ of Span B,F (B), which corresponds to a We wish to show that the pullback φ * Span B,F (E) → 1 is a cocartesian fibration. Pick an object e of E lying over b. Then a 1-simplex of Span B,F (E) with source e lying over φ is a span ef ← − e ḡ − → e wheref is a cartesian morphism over f andḡ is any morphism over g. The space of maps from e to e in φ * Span B,F (E) can therefore be identified with the space Map E (e , e ) g of maps in E lying over g. From this it follows immediately that ifḡ : e → e is a locally cocartesian morphism from e over g then the span ef ← − e ḡ − → e is locally cocartesian, as required. This proves (i), from which (ii) follows by the pullback square of Proposition 2.3.2.

Day convolution for double ∞-categories
In this section we carry out the main technical construction of this paper: We show that for a certain class of double ∞-categories M, there exists a Day convolution double ∞-category M S such that for any non-symmetric ∞-operad O we have a natural equivalence In Sect. 3.1 we introduce an "unfolding" construction that we use to define M S in Sect. 3.2; we then establish the universal property in Sect. 3.3. Next we prove in Sect. 3.4 that we may view associative algebras in M as algebras in a family of monoidal ∞-categories. We also consider enriched variants of the Day convolution construction in Sect. 3.5, and in Sect. 3.6 we illustrate the theory by discussing the example of enriched ∞-categories.

An unfolding construction
Suppose we have a cocartesian fibration p : E → U and a cartesian fibration q : U → B.
Our goal in this subsection is to construct for every cocomplete ∞-category X a cocartesian fibration E X → B with the universal property that for any functor C → B there is a natural equivalence The universal property of E X is that of (qp) * (X × E), but it is not clear from the latter that E X will be a cocartesian fibration if X is cocomplete.
To define E X we first introduce an "unfolding construction" that uses p and q to construct is the functor given by the cartesian morphisms over f and the left square is a pullback; an object of f * E b then corresponds to a pair ( where the fibre product uses q ∨ and evaluation at 0 and the functor to B op uses evaluation at 1. Since q ∨ is a cocartesian fibration, the identity induces a functor over B op that preserves cocartesian morphisms. Dualizing again, we obtain a morphism of cartesian fibrations that preserves cartesian morphisms. The following lemma identifies the source of this functor with U ∨ × B op Tw r (B op ): to the free cocartesian fibration on f is equivalent to Proof We can write the cocartesian fibration C × B B 1 → B as the fibre product of cocartesian fibrations over B. Since dualization of fibrations is an equivalence of ∞categories, it preserves fibre products, hence we obtain an equivalence For E → U a cocartesian fibration, we define the unfolding Unf(E) as the fibre product is then a cocartesian fibration, since it decomposes as a composite where the first map is a pullback of the cocartesian fibration E → U and the second is a pullback of the cocartesian fibration U ∨ → B op .
In the top cube, the back and front faces are cartesian by definition of unfolding, and the right face is cartesian since the bottom right and right composite squares are cartesian. This implies that the left square in the top cube is cartesian. Moreover, since dualization of fibrations is compatible with pullbacks we have The bottom left face in the diagram is therefore cartesian, and so the left composite square is a pullback. Thus unfolding is compatible with base change, in the sense that we have a natural equivalence Proof Given morphisms a We can now define E X using the following construction: For any ∞-category X, let p X : F X → Cat ∞ denote the cartesian fibration correspoding to the functor Fun(-, X) : Cat op ∞ → Cat ∞ . If X is cocomplete, then this is also a cocartesian fibration (with cocartesian morphisms given by left Kan extensions). We then have a locally cocartesian fibration Span B,F (F X ) → Span(Cat ∞ ) by Proposition 2.3.7, where F X is equipped with the triple structure from Definition 2.3.1. Definition 3.1.8 Given a cocartesian fibration E → U and a cartesian fibration U → B, we let E X for a cocomplete ∞-category X be defined by the pullback Then E X → B is a locally cocartesian fibration. Lemma 3.1.9 Let X be a cocomplete ∞-category. The locally cocartesian fibration E X → B is a cocartesian fibration.
Proof We must show that the locally cocartesian morphisms are closed under composition.
For morphisms a as above, and we must show that the mate transformation , the mate transformation evaluates to the natural map of colimits It thus suffices to show that this functor is cofinal.
and g * f * E a are pulled back along U c → U b , we can rewrite this to see that there is a natural pullback square where π t denotes the projection E t → U t . In this square the right vertical functor is a cocartesian fibration, and the bottom horizontal functor is cofinal since both where the first functor is given by composition with f * E b → E b and the second by left Kan extension along and the functor f ! : f * E b → E b preserves cocartesian morphisms. The following lemma therefore implies that the left Kan extension along f ! can be computed fibrewise, i.e. for : given by cocartesian pushforward along the cartesian morphismf : f * u → u. We obtain the following description of the functor This ∞-category has as objects maps y → y over q(φ) together with commutative triangles where φ lies over id b . It therefore has an initial object, given by the cocartesian morphism Remark 3.1. 12 We can identify sections of the cocartesian fibration E X as follows: For any functor φ : C → B, we have By the pullback square in Proposition 2.3.2, the only condition for a point of the right-hand ∞-groupoid to lie in the image of to cartesian morphisms in F X . This amounts to the natural transformation being an equivalence. Since F X is by definition the cartesian fibration for the functor Fun(-, X) we can use [25,Proposition 7.3] and Remark 3.1.5 to obtain an equivalence over the morphism in Tw r (C) above, to equivalences in X.
corresponding to the cocartesian morphism u → f * u in U ∨ and the morphism

Proof
The functor c U is by construction a map of cartesian fibrations over B that preserves cartesian morphisms. On fibres over b ∈ B, the functor The same argument also shows that U b is the localization at the larger class W U/B,b of morphisms  ] and its proof that if X → Y is a cocartesian fibration and η : Y → Y exhibits Y as the localization of Y at the morphisms in W , then the canonical functor X → X from the pullback X of X along η exhibits X as the localization of X at the cocartesian morphisms over W .
From this the universal property of E X now follows using Remark 3.1.14: Corollary 3.1.17 For any ∞-category X, there is a natural equivalence Remark 3.1. 18 We remark briefly on the naturality of the construction in the cocartesian fibration. Suppose then that we have a commutative triangle and a cartesian fibration q : U → B. We can replace the triangle by a cocartesian fibration p : E → U × 1 and then apply the construction to p and q × 1 : U × 1 → B × 1 to obtain for any cocomplete ∞-category X a cocartesian fibration E X → B× 1 . By naturality of the construction the fibres at i = 0, 1 identify with E i,X → B and so this cocartesian fibration corresponds to a commutative triangle where the horizontal functor preserves cocartesian morphisms. On the fibre over b ∈ B, this is the functor Fun(E 0,b , X) → Fun(E 1,b , X) given by left Kan extension along φ b : E 0,b → E 1,b . Making the same construction with 1 replaced by n for all n it is easy to see that we get a functor (-) X from (small) cocartesian fibrations over B to (large) cocartesian fibrations over B.

The day convolution double ∞-category
We now apply the construction of the previous subsection to obtain the Day convolution for a double ∞-category. First we need some notation:

Definition 3.2.1 Let˚n denote the partially ordered set of pairs of integers
This determines a functor˚• :´→ Cat by taking φ : [n] → [m] to the functor˚n →˚m that sends (i, j) to (φ(i), φ( j)); we write ˚→´o p for the cartesian fibration corresponding to this functor. We can also define a functor : ˚→´o p by sending  long the functor is a cocartesian fibration. Applying the unfolding construction of the previous subsection to this together with the cartesian fibration ˚→´o p , we get a new cocartesian fibration Unf(˚M) → Tw r (´o p ), corresponding to a functor U˚M :´o p → Span(Cat ∞ ), from which we obtain another cocartesian fibration where X is any cocomplete ∞-category.
where the functor M φ( j)−φ(i) → M j−i arises from the cocartesian morphisms over the restriction of φ to an (active) morphism . Note in particular that this is the identity if φ is an inert morphism.
In the first step, this is taken to the functor d * 1 (M ×´op˚2) → X that to (x, y) assigns the span which then in the second step is taken to the functor M ×´op˚1 → X that to x ∈ M 1 assigns where the colimit is over the fibre product M 2/x := M 2 × M 1 M 1/x defined using d 1,! : M 2 → M 1 . is simply given by restriction.
We now define a subobject of M + X that, in good cases, will be a double ∞-category: Definition 3.2.7 Let˜n be the full subcategory of˚n on the objects (i, j) such that j −i ≤ 1. We define˜M n to be the pullback˜M   ((i, j) ∈˚n, x i j ∈ M j−i ) and every integer k, i ≤ k ≤ j, the commutative square j) ∈˚n, x i j ∈ M j−i ), let˜M n,ξ/ denote the full subcategory of M n,ξ/ :=˜M n ×˚M n˚M n/ξ spanned by the cocartesian morphisms. Then˜M n,ξ/ ˜n (i, j)/ ˜j −i and the inclusion˜M n,ξ/ →˜M n,ξ/ is coinitial. It follows that F is a right Kan extension of its restriction to˜M n if and only if F(x i j ) is the limit over F( we see that the condition is that the map must be an equivalence. Inducting on j − i and using that limits commute, we see that this condition holds for all (i, j) if and only if the given commutative squares are all cartesian.
where x 0m → x i j is the cocartesian morphism over {i, . . . , j} → [m] and M φ( j)−φ(i)/x i j is defined using the restriction of φ to an active morphism Then define F 0m : M n/x 0m → X as the fibre product F 0k × F kk F km using these natural transformations and the equivalence M n/x 0m This induces a canonical distributivity morphism We say M is X-admissible if this morphism is always an equivalence, i.e. if colimits over these slices of M distribute over pullbacks.

Remark 3.2.10
Given a functor F :˚M n → X that is right Kan extended from˜M n , the condition of X-admissibility implies an equivalence

Example 3.2.11
If M 0 is an ∞-groupoid (in which case we may view M as an (∞, 2)category, in the sense of a (not necessarily complete) 2-fold Segal space), then (M 0 ) /x kk * , so M is X-admissible provided pullbacks in X preserve colimits, i.e. colimits in X are universal. In particular, M is X-admissible for any ∞-topos X.
is an equivalence. From the definition of M X it is immediate that we may identify M X,n with Fun(˜M n , X), where˜M n is equivalent to the pullback M ×´op˜n. Under this equivalence our functor is given by composition with Since M →´o p is a cocartesian fibration, pullback along it preserves colimits, hence this functor is equivalent to which is an equivalence by [33,Proposition 5.13].
In the case where X is S, we can give a more explicit description of the ∞-category This left fibration corresponds to the functor which we can rewrite as the desired expression.

Remark 3.2.16
Let us reformulate the description of the horizontal composition from Remark 3.2.5 in terms of our description of horizontal morphisms: Suppose is a horizontal morphism from F to G and is a horizontal morphism from G to H , so that we may view as a functor M 1,F,G → S and as a functor M 1,G,H → S, then their composite is the functor M 1,F,H → S given by (x, p ∈ F(x 00 ), q ∈ H (x 11 )) → colim Note that if the ∞-category M 0 is an ∞-groupoid, the formula above simplifies to (x, p ∈ F(x 00 ), q ∈ H (x 11 )) → colim y∈M 2/x colim r ∈H (y 11 ) (y 01 , p, r ) × (y 12 , r , q).

Remark 3.2.18
Let M ⊗ X, * denote the full subcategory of M X spanned by functors F :˚M n → X such that for all maps [0] → [n] in´the composite˚M 0 →˚M n → X is constant at the terminal object of X. For such F we have an equivalence giving a canonical morphism M ⊗ X, * is a monoidal ∞-category under the weaker hypothesis that this morphism is an equivalence. This holds, in particular, if X has finite products that commute with colimits in each variable, and the functors is an equivalence for all functors F : N 2 → S in N X . This happens, for instance, if N 0/x 11 is a contractible ∞-groupoid and colimits in X are universal, or if X is S and all ∞-categories of the form N 0/x 11 and N 1/x 01 admit cofinal functors from ∞-groupoids (since colimits indexed by spaces distribute over limits by [12,Corollary 7.17]).

The universal property
Suppose M is an X-admissible double ∞-category and O is a generalized non-symmetric ∞-operad. Our goal in this subsection is to show that there is a natural equivalence

Remark 3.3.1
We already know from Corollary 3.1.17 that M + X →´o p has the universal property that there is a natural equivalence Map /´o p (I, M + X ) Map(I ×´op˚M, X). Our first goal is to reduce the right-hand side to functors from I ×´op M by a further localization.   , which exhibits I ×´op M as the localization at the set I I,M of morphisms whose image iń op is inert, whose image in I is cocartesian, and whose image in ˚is cartesian (and whose image in M is therefore an equivalence).
Proof As the proof of [34, Proposition 3.9], using the lifts defined in the proof of the previous proposition. Restricting further to functors with value in M X , we get:

Corollary 3.3.6 Suppose f : I →´o p is any functor such that I has f -cocartesian morphisms over inert maps in´o p . Then there is an equivalence
Then there is an equivalence

Day convolution monoidal structures
Our goal in this subsection is to show that M X induces a family of monoidal ∞-categories, and in some cases (including for associative algebras) algebras in M X are algebras in these monoidal ∞-categories. This will follow from a general observation about framed double ∞categories, which we will consider after some simple observations about algebras in double ∞-categories:  This is a pullback of cocartesian fibrations over´o p along functors that preserve cocartesian morphisms, hence M ⊗ →´o p is again a cocartesian fibration.

Proposition 3.4.3 Suppose O is a generalized non-symmetric ∞-operad such that the inclu-
is an equivalence.
Proof The double ∞-category M ⊗ is defined by a pullback square of generalized nonsymmetric ∞-operads, so we have a pullback square Here the right vertical map is equivalent to the iterated fibre product and so is a cartesian fibration since M is framed. It follows that M ⊗ n → M 0 is a cartesian fibration, and hence we can conclude that M ⊗ → M 0 is a cartesian fibration and that cartesian morphisms lie over equivalences in´o p .
The fibre M ⊗ X →´o p at X ∈ M 0 is a monoidal ∞-category, so it remains to check that the functor f * : M ⊗ Y → M ⊗ X corresponding to the cartesian morphisms over f : X → Y in M 0 is lax monoidal, i.e. preserves those cocartesian morphisms that lie over inert morphisms in´o p .
Since the cartesian morphisms lie over identities in´o p , it is equivalent to check that for every inert morphism φ :

Remark 3.4.9
The underlying ∞-category of the monoidal ∞-category M ⊗ X is the ∞category M(X , X ) of horizontal endomorphisms of X , and the monoidal structure is given by horizontal composition. Since M is framed, a vertical morphism f : X → Y gives rise to two horizontal morphisms, say f : from Y to X , and f is left adjoint to f . The underlying functor of the lax monoidal functor where we use Y for horizontal composition over Y as in Notation 2.1.7. The lax monoidal structure comes from the unit transformation (note the non-standard order of composition) First observe that i * has a left adjoint i ! , given by left Kan extension along i. Note that for X ∈ M 1 ⊆˚M 1 the ∞-category (M 0 M 0 ) /X is empty, and so for F : M 0 M 0 → X we have i ! F(X ) ∅. Given G :˚M 1 → X and φ : i * G → F consider the pushout square Since pushouts in Fun(˚M 1 , X) are computed pointwise, we see that F which is what we need in order to apply [32,Corollary 4.52] to conclude that i * is a cocartesian fibration (with G → G being the cocartesian morphism over i * G → F).
It follows that the cartesian pullback of : M 1,F ,G → S is given by composition with this functor.
Applying Corollary 3.4.10 to M X , we now get:

Enriched day convolution
In this subsection we generalize our construction slightly by showing that if M is a double ∞-category and V ⊗ is an M-monoidal ∞-category then in good cases there exists a double ∞-category M V such that we have a natural equivalence More generally, an object M V with this property will exist as a generalized non-symmetric ∞-operad that is not a double ∞-category. If V ⊗ is given by Day convolution, we can obtain M V from the following observation: Suppose M is an X-admissible double ∞-category where pullbacks in X preserve colimits, and U ⊗ → M is a small M-monoidal ∞-category. Then U ⊗ is also an X-admissible double ∞-category.
Proof Given an active morphism φ : [m] → [n] in´and an object u 0m ∈ U ⊗ m over x 0m ∈ M m , the functor U ⊗ n/u 0m → M n/x 0m is a cocartesian fibration, whose fibre at y 0n andφ : y 0n → x 0m over φ we can identify with the slice (U ⊗ n,y 0n ) /u 0m , defined using the cocartesian morphisms overφ. For 0 ≤ k ≤ m we can decompose this as a product Given functors where the second equivalence uses that fibre products in X preserve colimits in each variable and the third equivalence uses the X-admissibility of M.
Moreover, any M-monoidal functor U ⊗ → V ⊗ induces a morphism of double ∞-categories Proof The only part that is not an immediate consequence of Proposition 3.5.1 and our results in the previous subsections is the claim about M-monoidal functors, which follows using the criterion of Remark 3.2.19, where we can compute the colimits by decomposing them as in the proof of Proposition 3.5.1.
where α ! : U ⊗ S,1,y 01 × U ⊗ S,1,y 12 U ⊗ S,2,y → U ⊗ S,1,x is the cocartesian pushforward along α (given by a left Kan extension along the corresponding operation for U ⊗ ). More generally, we can obtain M V by passing to a larger universe: (i) M V is a generalized non-symmetric ∞-operad.

(ii) For any generalized non-symmetric ∞-operad O we have a natural equivalence
is a cartesian and cocartesian fibration. From the description of the cartesian morphisms in Remark 3.4.12 it follows that these restrict to M V , which proves (iv). (i) Suppose that V ⊗ is a locally small M-monoidal ∞-category such that: (1) For every x ∈ M 1 the ∞-category V x has colimits indexed by ∞-groupoids and by the ∞-categories M n/y for y ∈ M 1 , (2) These colimits are preserved by the functors f ! : V x → V x induced by the cocartesian morphisms over f : Then M V →´o p is a locally cocartesian fibration. (ii) Suppose V ⊗ and U ⊗ are locally small M-monoidal ∞-categories satisfying conditions (1) and (2) in (i), and U ⊗ → V ⊗ is an M-monoidal functor such that: (3) For all x the functor U x → V x preserves colimits indexed by ∞-groupoids and by the ∞-categories M n/y .
Then the induced morphism M U → M V of generalized non-symmetric ∞-operads from Proposition 3.5.6(iii) preserves locally cocartesian morphisms.
For an arbitrary S-admissible double ∞-category M it seems extremely awkward to formulate a condition on V such that M V is a double ∞-category. We therefore content ourselves with the following observation:  Proof We know from Proposition 3.5.7(i) that M V →´o p in (i) is a locally cocartesian fibration. If we can show this is actually a cocartesian fibration, then (ii) also follows since Proposition 3.5.7(ii) implies the functor preserves locally cocartesian morphisms. By Lemma 2.1.25 to show we have a cocartesian fibration it suffices to check that for every active map φ : [2] → [n] and for ∈ M V,n the canonical map For over F 0 , . . . , F n : M 0 → S, the object α n,! is given at (x, p, q) by y 01 , t 0 , t 1 ), . . . , n (y (n−1)n , t n−1 , t n )), Since V ⊗ is compatible with these colimits, we can pass these colimits past ζ ! in the expression for d 1,! φ ! (x, p, q), obtaining an expression for this object as an iterated colimit of terms of the form . . . , n (v (n−1)n , t n−1 , t n ))).
Note that we can rewrite this as since V ⊗ is cocartesian. Thus we can rewrite the expression for d 1,!
On the other hand, we can evaluate the colimit over M n/x in the expression for α n,! by first taking a left Kan extension along the functor φ ! : M n/x → M 2/x given by the cocartesian morphisms over φ. For a functor f out of M n/x this gives colim y→x∈M n/x f colim 12 , which is equivalent to M φ(1)/z 01 × M n−φ(1)/z 12 since M 0 is an ∞-groupoid. Rewriting our expression for α n,! using this, we get exactly our last formula for d 1,! φ ! , as required. Remark 3.5. 9 We can describe the double ∞-category M V as follows: • its objects are functors M 0 → S, and its vertical morphisms are natural transformations of these, • its horizontal morphisms from F to G are functors M 1,F,G → V ⊗ 1 over M 1 , • the composite of horizontal morphisms : M 1,F,G → V ⊗ 1 and : ( (y 01 , p, t), (y 12 , t, q)).

Example: enriched ∞-categories as associative algebras
In this subsection we illustrate our results on Day convolution by considering a simple example of our construction: we will give a description of enriched ∞-categories as associative algebras in a family of monoidal ∞-categories. An alternative construction of these monoidal ∞-categories is given in [39], where this perspective on enriched ∞-categories is developed extensively.
Remark 3.6.1 Our construction will extend the following description of ordinary enriched categories: If S is a set and V is a monoidal category where the tensor product preserves coproducts in each variable, then there is a monoidal structure on Fun(S × S, V) given by with unit 1 the functor This is sometimes known as the "matrix multiplication" tensor product, since the formula is a "categorified" version of that for multiplication of matrices. An associative algebra A in the category Fun(S × S, V) with this tensor product is the same thing as a V-category with set of objects S: Let us consider first the result of applying Day convolution to the simplest double ∞category, namely´o p . This is trivially X-admissible for any ∞-category X with pullbacks, and so by Proposition 3.2.12 there is a double ∞-category ´o p X which by Corollary 3.3.7 has the universal property that for any generalized non-symmetric ∞-operad O there is a natural equivalence By construction, the fibre ´o p X,n is the ∞-category of functors˚n → X that are right Kan extended from˜n. We thus see that: • objects of ´o p X are objects of X, • vertical morphisms (morphisms in ´o p X,0 ) are morphisms in X, • horizontal morphisms (objects in ´o p X,1 ) are spans in X, i.e. diagrams of shape • ← • → •, • squares (morphisms in ´o p X,1 ) are diagrams of shape • composition of horizontal morphisms is given by taking pullbacks.
Indeed, the double ∞-category ´o p X,n is precisely the double ∞-category SPAN + (X) of spans constructed in [33], and its universal property is that established in [34]. In particular, we have an equivalence identifying associative algebras in the double ∞-category ´o p X with category objects in X. Specializing to the ∞-category S of spaces, this says that associative algebras in ´o p S are equivalent to Segal spaces, which describe the algebraic structure of ∞-categories.
By Corollary 3.4.13, the restriction Alg´op ( ´o p S ) → S is a cartesian fibration, with fibre at a space X given by Alg´op ( ´o p S (X , X )). Here ´o p S (X , X ) is equivalent to S /X ×X Fun(X × X , S). The monoidal structure is given by pullback of spans, which in terms of functors to S admits the following description: Proposition 3.6.2 For any space X there is a monoidal structure on the ∞-category Fun(X × X , S) such that (i) the tensor product of F and G is given by (ii) the unit 1 is given by where Map X (x, y) is the mapping space in the ∞-groupoid X , i.e. the space of paths from x to y in X .
In other words, ∞-categories with space of objects X are associative algebras in Fun(X × X , S) with this monoidal structure. Now we want to consider the analogue of this result for enriched ∞-categories. Propositions 3.5.6 and 3.5.8 specialize to give the following: Proposition 3.6.3 Let C ⊗ →´o p be a monoidal ∞-category compatible with colimits indexed by ∞-groupoids. Then there is a framed double ∞-category ´o p C such that for any generalized non-symmetric ∞-operad O there is an equivalence A monoidal functor C ⊗ → D ⊗ induces a natural morphism of generalized non-symmetric ∞-operads ´o p C → ´o p D , and this preserves cocartesian morphisms if the monoidal functor preserves colimits indexed by ∞-groupoids.
In particular, we get an equivalence where as in Example 2.1.21 the right-hand side is the model of enriched ∞-categories considered in [24]. Specializing Remark 3.5.9 gives the following description of the double ∞-category ´o p C : • its objects are spaces, and its vertical morphisms are morphisms of spaces, • a horizontal morphism from X to Y is a functor X × Y → C, • the composite of the horizontal morphisms : X × Y → C and : Y × Z → C is the functor X × Z → C given by From Corollary 3.4.10 we know that the restriction Alg´op ( ´o p C ) → S is a cartesian fibration, with fibre at a space X given by Alg´op Corollary 3.6.4 Let C be a monoidal ∞-category compatible with colimits indexed by ∞groupoids. Then there is a monoidal structure on the ∞-category Fun(X × X , C) such that (i) the tensor product of F and G is given by (ii) the unit 1 is given by

The composition product and ∞-operads
In this section we apply our results on Day convolutions to describe ∞-operads as associative algebras in double ∞-categories. We first consider ordinary ∞-operads (in spaces) in Sect. 4.1, and then enriched ∞-operads in Sect. 4.2. We also briefly observe, in Sect. 4.3, that a version of the bar-cobar adjunction between ∞-operads and ∞-cooperads follows from this description of ∞-operads.

∞-operads as associative algebras
In this subsection we will see that ∞-operads are given by associative algebras in a double ∞-category of symmetric collections (or coloured symmetric sequences) in S. For this we use Barwick's model of ∞-operads from [3]; this is known to be equivalent to other models of ∞-operads thanks to the results of [3,[13][14][15]. Before we recall Barwick's definition we first introduce some notation: is an equivalence, (2) for every object I = ( [1], k → l), the natural map is an equivalence, (3) for every object I = ([0], k), the natural map is an equivalence.
We write Seg is an equivalence.
Segal presheaves on´F describe the algebraic structure of ∞-operads: If we write e := ([0], 1) and c n := ( [1], n → 1), then the Segal conditions describe how F(I ) decomposes as a limit of F(e) and F(c n ). We can think of an object of´F as a forest with levels; then e correponds to a plain edge and c n to a corolla with n leaves, while the Segal condition corresponds to the decomposition of a forest into its edges and corollas. If F is viewed as an ∞-operad, the value F(e) is the space of objects of F, while F(c n ) is the space of n-ary morphisms.
The following is the starting point for our construction of the composition product on symmetric sequences: Proposition 4.1.5 The projection´o p F →´o p is an X-admissible double ∞-category for any cocomplete ∞-category X with pullbacks where colimits are universal.
For the proof we need some notation and a lemma: is an active map in´and A = (a 0 → · · · → a n ) is an object of (´F) [n] . An object of the slice (´F) [m],A/ , defined using φ, is an object (b 0 → · · · → b m ) of (´F) [m] together with injective maps a i → b φ(i) such that the squares ,A/ denote the full subcategory of (´F) [m],A/ containing those objects where the map a n → b φ(m) is an isomorphism. be an active morphism in´, and let A = (a 0 → · · · → a n ) be an object of (´F) [n] . For 0 ≤ i ≤ j ≤ n let A i j := (a i → a i+1 → · · · → a j ).

(i) The inclusion (´F) iso
[m],A/ → (´F) [ • an object is a sequence a 0 → · · · → a n of morphisms in F, • a morphism is a commutative diagram a 0 a 1 · · · a n−1 a n b 0 b 1 · · · b n−1 b n where the squares are cartesian and the maps a i → b i are injective.
This clearly satisfies the Segal condition, i.e. (´F) [n] (´F) [1] × (´F) [0] · · · × (´F) [0] (´F) [1] . It follows that´o p F →´o p is a cocartesian fibration corresponding to the functor´o p → Cat ∞ taking [n] to (´F) op [n] and so is also a double ∞-category. Suppose φ : [n] → [m] is an active map in´. If A = (a 0 → · · · → a n ) and A = (a 0 → · · · → a k ) and A = (a k → · · · → a n ) then we must show that the natural map is an equivalence for any appropriate functor F.
We have a commutative square which is an equivalence since colimits in X are universal.   • Squares are natural transformations of such diagrams.
• If is a horizontal morphism from F to G and is a horizontal morphism from G to To see this it suffices to check that the horizontal morphisms in COLL(S) are closed under composition. If is a horizontal morphism from F to G and is one from G to H , and both lie in COLL(S), then we have i.e. G also lies in COLL(S), as required.
The double ∞-category COLL(S) can be described as follows: • Its objects can be identified with spaces (since functors F op inj → S in COLL(S) 0 are determined by their value at 1).
• Its vertical morphisms are maps of spaces.
• A horizontal morphism from X to Y is determined by assigning to ([1], n → 1) a span where (n) has a n -action compatible with permuting the factors of X ×n . • A square is a natural transformation of such diagrams.
• If is a horizontal morphism from X to Y and is one from Y to Z , then their composite assigns to ( [1], n → 1) the space over X ×n × Z given by Restricting Corollary 4.1.9 to COLL(S), we get:

(S).
In other words, ∞-operads can be described as associative algebras in the double ∞-category COLL(S). Moreover, applying Corollary 3.4.13 we see that the restriction Alg´op (COLL(S)) → S is a cartesian fibration. If we write for the ∞-category of X -collections in S, the fibre at X ∈ S is given by Alg´op (Coll X (S)).
To describe the monoidal structure on Coll X (S) we first need to introduce some notation: Notation 4.1.13 Let F denote the maximal subgroupoid of F, and write j : F → (´o p F ) [1] for the fully faithful functor taking n to n → 1. Given X ∈ S, we write (´o for the left fibration corresponding to the unique product-preserving functor (´o ,Y . If we define F X := × n=0 X ×n h n to be the free commutative monoid on the ∞-groupoid X , then we have a pullback [1] .  [1],X ,Y → S, and under this identification it is easy to see that the functors that lie in COLL(S)(X , Y ) are precisely those that are right Kan extended from F X × Y .

Remark 4.1.15
In particular, we may identify the ∞-category Coll * (S) of horizontal endomorphisms of the point with the ∞-category Fun(F , S) of symmetric sequences in S. More generally, the ∞-category Coll X (S) is equivalent to Fun(F X × X , S), the ∞-category of X -collections, or X -coloured symmetric sequences, in S. Notation 4.1.16 For a functor F : F X ×Y → C, we will denote its value at ((x 1 , . . . , x n ) Remark 4.1. 18 In particular, the ∞-category Fun(F , S) of symmetric sequences has a monoidal structure with tensor product given by This formula is easily seen to agree with the usual formula for the composition product of symmetric sequences by expanding out Fact(n → 1) as a coproduct of its components, cf. [18,Lemma A.4].

Remark 4.1.19
In [3], Barwick defines -∞-operads for operator categories as Segal presheaves on categories´ , of which´F is a special case. The proof of Corollary 4.1.17 works for any operator category, giving a monoidal structure on the ∞-category Fun( X , S) of "X -coloured -symmetric sequences" where associative algebras are -∞-operads with X as their space of objects. In particular, replacing F with the category O of ordered finite sets we obtain the analogous results for non-symmetric ∞-operads.

Enriched ∞-operads as associative algebras
In this subsection we extend the results of the previous subsection to ∞-operads enriched in a symmetric monoidal ∞-category. The starting point is the following analogue of Proposition 3.5.8 for´o p F -monoidal ∞-categories: Proposition 4.2.1 Let U ⊗ be a´o p F -monoidal ∞-category that is compatible with colimits indexed by ∞-groupoids. Then ´o p F,U is a framed double ∞-category. If U ⊗ → V ⊗ is a´o p F -monoidal functor between such´o p F -monoidal ∞-categories such that each functor U X → V X for X ∈ (´F) [1] preserves colimits indexed by ∞-groupoids, then the natural morphism of generalized non-symmetric ∞-operads ´o p F,U → ´o p F,V preserves cocartesian morphisms.
Proof Follows as in the proof of Proposition 3.5.8, using Lemma 4.1.7 to restrict to colimits indexed by ∞-groupoids.
We now recall some definitions from [10]; we refer the reader there for motivation for these definitions. , a 0 → · · · → a n ) to ( n i=1 a i ) + , and a morphism ([n], a 0 → · · · → a n ) → ([m], b 0 → · · · → b m ) over φ : [n] → [m] in´o p to the map ( m i=1 a i ) + → ( n j=1 b j ) + given on the component a i by the map a i → ( n j=1 b j ) + taking x ∈ a i to an object y ∈ b j if φ( j − 1) < i ≤ φ( j) and the map a i → a φ( j) takes x to the image of y under the map b j → a φ( j) , and to the base point * otherwise. The functor V assigns to a forest its set of vertices with an added basepoint. Note that V assigns every morphism in´o p F that lies over an identity morphism in´o p to an inert morphism in F * .

Definition 4.2.3
If V ⊗ → F * is a symmetric monoidal ∞-category, and V ⊗ → F op * is the corresponding cartesian fibration, then we define the ∞-category´V F by the pullback squaré , a 0 → · · · → a n ) ∼ = V ( [1], a 0 → a 1 ) + · · · + V ([1], a n−1 → a n ), which implies that´V ,op F is a´F-monoidal ∞-category. F op inj → S that take coproducts of finite sets to products, (2) whose inert restrictions to [1] correspond to functors  op [1] × F * V ⊗ is also a right Kan extension of its restriction to the full subcategory F , and this restriction is the constant functor with value V. Thus a horizontal morphism from X to Y in COLL(V) is uniquely determined by its restriction to a functor F X × Y → V. Moreover, if φ : V → W is a symmetric monoidal functor that preserves colimits indexed by ∞-groupoids, then composition with φ gives a monoidal functor Fun(F X × X , V) → Fun(F X × X , W).

∞-cooperads and a bar-cobar adjunction
In this subsection we will apply Lurie's bar-cobar adjunction for associative algebras [44,Sect. 5.2.2] to obtain a version of the bar-cobar adjunction between ∞-operads and ∞cooperads with a fixed space of objects. We first spell out the variant of ∞-cooperads that this applies to: and Cobar(Q) is given by Proof Apply [44, Theorem 5.2.2.17] to the monoidal ∞-category Fun(F X × X , V) 1 X //1 X .

Remark 4.3.3
Here we have defined ∞-cooperads as coalgebras in symmetric sequences, following the definition proposed in, for instance, [20]. However, the notion of cooperad in V that is relevant in bar-cobar duality for operads often seems to be that of operads enriched in V op (as for example used by Ching to define the bar-cobar adjunction for operads in spectra However, if we make some assumptions on both the ∞-operads we consider and on the ∞-categories we enrich in, then the two notions do agree: First suppose V is a semiadditive ∞category, meaning it has a zero object and finite biproducts (i.e. finite products and coproducts coincide). (For example, this holds in any stable ∞-category, such as those of spectra or chain complexes.) If we then restrict ourselves to consider only reduced ∞-operads O ∈ Opd * (V), meaning ∞-operads such that O(0) 0, then the coproducts in O • O are finite and hence are equivalent to products. Moreover, for such reduced symmetric sequences we can rewrite the formula for the composition product without taking any homotopy orbits: 6 where i j = |n j |. This is easy to see in the coordinate-free description as discussed in Sect. 1.1: passing to reduced symmetric sequences means only surjective maps of sets appear, and these have no automorphisms in F [2], .
Therefore where n = i 1 + · · · + i k . This is precisely the structure of an ∞-operad enriched in V op with the same n-ary operations as O.
For reduced ∞-operads enriched in semiadditive ∞-categories, we therefore expect that the bar-cobar adjunction arising from the composition product is the correct one for understanding bar-cobar duality for enriched ∞-operads. One might wonder if there exists some more general version of a bar-cobar adjunction without these restrictions, but this setting does in fact seem to cover all the cases of bar-cobar duality for operads in the literature that we are aware of.

Funding Open access funding provided by NTNU Norwegian University of Science and Technology (incl St. Olavs Hospital -Trondheim University Hospital)
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.