Rectification of Algebras and Modules

Vladimir Hinich

Received: September 9, 2014 Revised: June 12, 2015

Communicated by Stefan Schwede

Abstract. Let 𝒪 be a topological (colored) operad. The Lurie ∞- category of 𝒪-algebras with values in (∞-category of) complexes is compared to the ∞-category underlying the model category of (classical) dg 𝒪-algebras. This can be interpreted as a ”rectification” result for Lurie operad algebras. A similar result is obtained for modules over operad algebras, as well as for algebras over topological PROPs.

2010 Mathematics Subject Classification: 18G55, 18D05, 18D50.

1. Introduction

1.1. Operad algebras. In this paper we compare two notions of operad algebras with values in complexes. Let 𝒪 be a topological colored symmetric operad. The functor of singular chains with coefficients in a commutative ring k converts 𝒪 into an operad in the category of complexes, so that one has the category Alg_{𝒪} (C(k)) of 𝒪-algebras in C(k) in the “conventional” sense: its objects are complexes A ∈ C(k) together with Σ_{n}-equivariant operations

C_{∗}(𝒪(n), k) ⊗ A^{⊗n} A

satisfying the standard compatibilities. The category Alg_{𝒪} (C(k)) has (sometimes) a model category structure with quasiisomorphisms as weak equivalences and surjective maps as fibrations. Sometimes it does not have such model structure. In any case, one can find a quasiisomorphism of dg operads ℛ → C_{∗}(𝒪) such that the category of ℛalgebras has a model structure; moreover, under a mild extra requirement, the model category Alg_{ℛ} (C(k)) is independent, up to Quillen equivalence, of the dg operad ℛ. The operad ℛ satisfying the above properties will be called homotopically sound, see 2.4.6 below.

A topological operad 𝒪 defines, on the other hand, an ∞-operad 𝒪^{⊗} in the sense of Lurie, [L.HA], Section 2, and an ∞-category of algebras Alg_{𝒪} (QC(k)) with values in the symmetric monoidal ∞-category QC(k) which is the ∞-category version of the derived category of k-modules. Our main result Theorem 4.1.1 claims that, given a quasiisomorphism of operads ℛ → C_{∗}(𝒪) with ℛ homotopically sound^{1} the ∞-category Alg_{𝒪} (QC(k)) is equivalent to the ∞-category underlying the “classical” model category Alg_{ℛ} (C(k)). This can be interpreted as a rectification result: any Lurie 𝒪-algebra with values in QC(k) can be presented by a strict ℛ-algebra. In good cases, when C_{∗}(𝒪) is homotopically sound, any Lurie 𝒪-algebra can be presented by a strict 𝒪-algebra with values in C(k). We feel this is an important (though not unexpected) result: the notion of operad algebra in Lurie theory is very flexible; however, an algebra over an ∞-operad is defined by a huge collection of coherence data which is difficult to specify. The description of Alg_{𝒪} (QC(k)) as a nerve of a model category allows one to present 𝒪^{⊗}-algebras in QC(k) and their diagrams by strict 𝒪-algebras in complexes.

1.1.1. The mere formulation of Theorem 4.1.1 requires a model category structure on the category of algebras over a colored dg operad. An account of the relevant theory is presented in Section 2. The results of this section are mostly well-known, at least for colorless operads. Our approach here is very close to the earlier colorless version [H]. Note that there exists a very general result by C. Berger and I. Moerdijk, [BM2], on model structure for algebras over color operads. Unfortunately, we were unable to deduce from their result that all dg operads are admissible in case k ⊃ ℚ. This is why we felt it important to present the colored version of the notion of Σ-splitness used in [H].

1.1.2. Dold-Kan. As we mentioned above, a simplicial operad can be converted, via the normalized chains functor, into a dg operad. This is due to the fact that the normalized chains functor

C_{∗} : sSet C(k)

is lax symmetric monoidal, via Eilenberg-MacLane map. Were it really symmetric monoidal, any strict operad algebra over 𝒪 with values in C(k) would automatically define an 𝒪-algebra in the sense of Lurie. In real life this is “almost so” — the functor C_{∗} induces an adjoint pair between the symmetric monoidal ∞-categories Cat_{∞} and dgCat. This “almost so” has to be explained, and we do so in Section 3 which precedes the rectification theorem.

1.1.3. The proof of Theorem 4.1.1 follows the idea of Lurie’s Theorem 4.1.4.4 of [L.HA] where a similar result for associative algebras with values in a combinatorial monoidal model category is proven.

1any operad is homotopically sound if (and only if) k is a field of characteristic zero.

1.2. Algebras over PROPs. Theorem 4.1.1 allows one to (partially) rectify algebras more general than algebras over operads, such as, for instance, associative bialgebras. These are algebras over PROPs, that is, symmetric monoidal functors from a certain symmetric monoidal category designed to describe the necessary structure (PROP), to the category of complexes. A topological PROP P gives rise to a SM ∞-category P^{⊗}; this leads to the notion of P -algebra with values in complexes as a SM functor P^{⊗} → QC(k)^{⊗}. We do not expect such algebras to be always presentable by strict P -algebras in complexes. One can, however, slightly generalize the notion of strict dg algebra over a PROP — allowing lax symmetric monoidal functors to complexes which are “homotopy SM”, see Definition 4.4.1^{2}. Our rectification theorem 4.1.1 implies easily the equivalence of two approaches, see Corollary 4.4.4.

1.3. Modules. The notion of module over an operad algebra is very straightforward in the “classical” theory. We describe a construction which assigns to a topological operad 𝒪 another operad M𝒪 whose algebras are pairs (A, M) where A is an 𝒪-algebra and M is an A-module. A similar construction can be easily defined in the world of ∞-operads as well. J. Lurie suggests another notion of module over an operad algebra. Similarly to the cases of commutative or associative algebras where modules (or bimodules) form a symmetric monoidal (or simply monoidal) category, his version of the ∞-category of modules over a fixed ∞-operad algebra A has an 𝒪-monoidal structure. To have such nicely behaved notion, one has to require some very special properties from 𝒪 — it has to be coherent, see [L.HA], Sect. 3. In Appendix B we prove that our definition of module coincides with the one suggested by Lurie (with discarded 𝒪-monoidal structure). Our rectification result easily implies the rectification for modules, see Corollary 5.2.3.

1.4. SM adjunction. Appendix A deals with adjunction between two (or more) symmetric monoidal (in what follows: SM) ∞-categories. It turns out that if any of the adjunctions is a lax SM functor, all others automatically acquire the same structure. This generalizes a well-known observation for conventional SM categories that the functor right adjoint to a SM functor, is necessarily lax. We believe that this is an important fact in its own. In this paper we use it to construct the functor from strict operad algebras to algebras over the corresponding ∞-operad. Appendix B contains some technical details of the comparison between two notions of module.

1.5. Acknowledgements. Parts of this paper were written during author’s visit to MIT and IHES. I am grateful to these institutions for hospitality and excellent working conditions. I am very grateful to J. Lurie for several useful conversations. I am also very grateful to the referee for numerous suggestions, as well as to F. Muro for pointing out to a wrong formula in the earler version of the manuscript.

2A very close notion was used by K. Costello in [C] (h-split symmetric monoidal functors).

2. Models for algebras

2.1. Introduction. In this section we present an account of the model category structure on algebras over colored operads. The results described in this section are mostly well-known, for the colorless case see [H, H.V, BM1], and for the colored case [BM2] (Berger and Moerdijk consider more general algebras with values in any model category). Having in mind applications to dg algebras, we wanted to make sure the important case of algebras in characteristic zero would be covered. The only proof we are aware of in the colorless case is via the notion of Σ-split operads presented in [H]. This is why we spend some time to give a colored version of this notion. We also use this section as an opportunity to fix notation for colored operads.

2.2. Colored operads. Fix a symmetric monoidal category 𝒞. A colored operad 𝒪 in 𝒞 consists of the following data.

1. A set [𝒪] (the set of colors of 𝒪) ^{3}.

2. An object 𝒪(c, d) of 𝒞 (of operations) given for each collection of colors c : I → [𝒪] (I is a finite set) and a color d ∈ [𝒪].

3. A composition map defined for each map of finite sets f : I → J, with collections of colors c : I → [𝒪], d : J → [𝒪], e ∈ [𝒪]. This is a map

(1)  𝒪(d, e) ⊗ ⨂_{j∈J} 𝒪(c_{j}, d(j))  𝒪(c, e),

where c_{j} denotes the restriction of c : I → [𝒪] to I_{j} = f^{−1}(j).

4. The unit maps 1_{c} : 1 𝒪({c}, c) for each c ∈ [𝒪], where 1 is the

unit object of 𝒞. The composition maps are required to satisfy the following associativity and unit conditions.

1. Associativity: for a pair of maps I ^{f} J ^{g} K, collections of colors c : I → [𝒪], d : J → [𝒪], e : K → [𝒪] and f ∈ [𝒪], two compositions

𝒪(e, f) ⊗ ⨂_{k∈K} 𝒪(d_{k}, e(k)) ⊗ ⨂_{j∈J} 𝒪(c_{j}, d(j)) 𝒪(c, f),

with c_{j} being the restriction of c to I_{j} = f^{−1}(j) and d_{k} being the restriction of d to J_{k} = g^{−1}(k), coincide.

2. Left unit: For any c : I → [𝒪], J = {j} consisting of one element and d(j) = e ∈ [𝒪], the map (1) induces the identity map

𝒪(c, e) = 1 ⊗ 𝒪(c, e) ^{1_{e}⊗id} 𝒪(d, e) ⊗ 𝒪(c, e) 𝒪(c, e).

3One can allow [𝒪] to be “big”, as the set of objects of a category.

3. Right unit: For f : I → J bijection with c = d ∘ f, the map (1) induces the identity map

𝒪(d, e) ^{id⊗⨂ 1_{d(j)}} 𝒪(d, e) ⊗ ⨂_{j∈J} 𝒪(c_{j}, d(j)) 𝒪(c, e).

Note that colored operads in our definition are “symmetric”: the right unit axiom defines a canonical action of the automorphism group of c : I → [𝒪] on all 𝒪(c, d). Colored operads are also known under other names as “multicategories” or “pseudo-tensor categories”. The first name is self-explanatory. The reason for the second one (due to Beilinson) is that colored operads can be assigned to symmetric monoidal categories. Thus, colored operads can be seen as the generalizations of symmetric monoidal categories. Here are the details. Let 𝒟 be a symmetric monoidal category enriched over 𝒞 (The case 𝒞 = Set is as interesting as any other). We can put [𝒟] = Ob(𝒟) and define 𝒟(c, d) = Hom_{𝒟} (⊗_{i∈I}c(i), d). This yields a colored operad in 𝒞. It is not difficult to understand when a colored operad 𝒪 comes in the above described manner from a symmetric monoidal category. First of all, any colored operad 𝒪 has an underlying category (also enriched over 𝒞) denoted 𝒪_{1} in the sequel. If we wish 𝒪 to come from a symmetric monoidal (SM) category, the functors d ↦ 𝒪(c, d) should be representable for each c : I → [𝒪]. This condition is not yet sufficient: assuming all functors above are representable by the objects denoted as ⊗_{i∈I}c(i), we obtain for each map f : I → J of finite sets a canonical map

(2)  ⊗_{i∈I} c(i)  ⊗_{j∈J} (⊗_{i∈I_{j}}c(i)).

If these maps are isomorphisms, our colored operad 𝒪 comes from a SM category (uniquely defined up to equivalence). ^{4}

2.2.1. Planar versions. If one replaces finite sets with totally ordered finite sets and the maps of finite sets with the monotone maps, we get a multicolor notion of planar (or asymmetric) operad. This notion generalizes the notion of monoidal category in the same way as the notion of colored operad generalizes the notion of SM category.

2.2.2. Maps of operads. Given two colored operads 𝒫 and 𝒬, a map of operads f : 𝒫 → 𝒬 is defined as a map f : [𝒫] → [𝒬] together with a compatible collection of maps

𝒫(c, d) → 𝒫(f ∘ c, f(d))

for each c : I → [M]. Compatibility means that the above maps preserve units and are compatible with compositions. If 𝒫 and 𝒬 are the operads corresponding to SM categories, a map of operads f : 𝒫 → 𝒬 is what is usually called a lax SM functor.

4By the way, defining a SM category as a colored operad satisfying the above properties, allows one not to care about associativity or commutativity constraints.

2.2.3. SM functors. In more detail, let 𝒫 and 𝒬 be SM categories and let f : 𝒫 → 𝒬 be a map of the corresponding colored operads. This means that a compatible collection

Hom(⊗_{i∈I}a_{i}, b) → Hom(⊗_{i∈I}f(a_{i}), f(b))

is given. By naturality, this is the same as a compatible collection of morphisms

(3)  ⊗_{i∈I} f(a_{i})  f(⊗_{i∈I}a_{i}).

This is what is usually called a lax SM functor. A map of operads f : 𝒫 𝒬 is called a SM functor if the maps (3) are isomorphisms.

2.2.4. Algebras. We assume that the base symmetric monoidal category 𝒞 admits colimits and the tensor product commutes with colimits along each one of the arguments. Let 𝒪 be a colored^{5} operad in 𝒞 and let 𝒟 be a SM category enriched over 𝒞. An 𝒪-algebra in 𝒟 is just a map of operads A : 𝒪 → 𝒟. The category of 𝒪-algebras in 𝒟 is denoted as Alg_{𝒪} (𝒟) or just Alg_{𝒪} if 𝒟 can be understood from the context. The following theorem is very standard, see, for instance, [BM2], 1.2.

2.2.5. Theorem. Let f : 𝒫 → 𝒬 be a map of (small) operads and let 𝒟 be a SM 𝒞-enriched category having colimits. There is a pair of adjoint functors

f_{!} : Alg_{𝒫} (𝒟) [figure] Alg_{𝒬} (𝒟) : f^{∗}

where f^{∗} is the forgetful functor, assigning to A : 𝒬 → 𝒟 the composition f^{∗}(X) = A ∘ f : 𝒫 → 𝒬 → 𝒟.

In the special case where 𝒫 = [𝒬] is the operad with the same colors as 𝒬 and with no nontrivial operations, the functor f_{!} is the free algebra functor which is worth of a more detailed description. Let V : [𝒪] → 𝒟 be a collection of objects of 𝒟 numbered by the colors. The free algebra 𝔽_{𝒪} (V) is the collection of objects 𝔽_{𝒪} (V)_{d}, d ∈ [𝒪], described as follows. Collections c : I → [𝒪] form a groupoid denoted Fin/[𝒪]. To each c ∈ Fin/[𝒪] we assign the object

(4)  𝒪(c, d) ⊗ ⨂_{i∈I} V_{c(i)}.

This gives rise to a functor 𝔉(V)_{d} : Fin/[𝒪] → 𝒟; its colimit is the component 𝔽_{𝒪} (V)_{d} of the free 𝒪-algebra generated by V. Note that Fin/[𝒪] is a groupoid. For c : I → [𝒪] ∈ Fin/[𝒪] denote Σ_{c} its automorphism group (this is a subgroup of the symmetric group Σ_{I}). Thus, the free 𝒪-algebra generated by V is given by the formula

(5)  𝔽_{𝒪} (V)_{d} = ⨁_{c∈π_{0}(Fin/[𝒪])} 𝒪(c, d) ⊗_{Σ_{c}} ⨂_{i∈I} V_{c(i)},

5In what follows we will say “operads” and “colorless operads” instead of “colored operads” and “operads”.

where the direct sum is over a set of representatives of isomorphism classes of objects in Fin/[𝒪]. The functor f_{!} carries the free 𝒫-algebra generated by a collection V = {V_{c}}_{c∈𝒫} to the free 𝒬-algebra generated by the collection d ↦ ∐_{c∈[𝒫]:f(c)=d} V_{d}.

2.3. DG version. From now on we fix a commutative ring k and we study operads and algebras with values in the category C(k) of complexes over k. First of all, the category of complexes C(k) admits a model structure, with quasiisomorphisms as weak equivalences and degree-wise surjective maps as fibrations, see, for example, [H]. For a large class of dg operads a model category structure on Alg_{𝒪} (C(k)) can be defined using the adjoint pair of functors

(6)  𝔽_{𝒪} : C(k)^{[𝒪]} [figure] Alg_{𝒪} (C(k)) : 𝔾,

where 𝔾 is the forgetful functor and 𝔽_{𝒪} is the free 𝒪-algebra functor. A map of 𝒪-algebras f : A → B is called a weak equivalence (resp., a fibration) if 𝔾(f) is a weak equivalence (resp., a fibration). In other words, f is a weak equivalence if for each color c ∈ [𝒪] the map A_{c} → B_{c} is a quasiisomorphism of complexes. It is a fibration if all maps A_{c} → B_{c} are surjective. It is called a cofibration if it satisfies the left lifting property with respect to all trivial fibrations.

2.3.1. Definition. An operad 𝒪 in C(k) is called admissible if the category of algebras Alg_{𝒪} (C(k)) admits a model category structure determined by weak equivalences and fibrations defined as above.

The model category structure on Alg_{𝒪} (C(k)) is cofibrantly generated as it is transferred from the cofibrantly generated model category structure on (collections of) complexes. In particular, any cofibration is a retract of a transfinite composition of maps of the form A → A⟨x⟩ where x is a free variable of a given color c, a given degree d, and a specified value of dx ∈ (A_{c})^{d+1}. The operads with a fixed collection of colors K can be described as the algebras over an appropriate operad whose colors are the finite collections of the elements of K. This allows one to define weak equivalence and fibration of a map of operads with a fixed collection of colors in a usual way. This allows one to define as well cofibrations for maps of operads via the left lifting property with respect to trivial fibrations. Note that we are not requiring or claiming here the existence of model structure for such category of dg operads. ^{6} One has the following

2.3.2. Proposition. A cofibrant operad is admissible.

The colorless case is proven in [H.V]. The same reasoning proves the colored case. □ Another class of admissible operads (Σ-split operads) is described in Subsection 2.5. It includes, for instance, all planar operads, or all operads over k ⊃ ℚ. Note the following criterion of admissibility.

6In this we follow [BM2].

2.3.3. Theorem. An operad 𝒪 in C(k) is admissible if and only if for any 𝒪algebra A and for any collection of contractible cofibrant complexes M = {M_{c}}, c ∈ [𝒪], the natural map

(7)  A  A ∐ 𝔽_{𝒪} (M)

is a weak equivalence.

The proof for colorless operads is given in [H]. The same reasoning proves the colored case. □ One immediately sees that it is sufficient to check that (7) is a weak equivalence for M = {M_{c}} with M_{c} = 0 for c ≠ c_{0}, and M_{c_{0}} contractible cofibrant.

2.4. Change of operad. Recall that a map f : 𝒫 𝒬 of operads gives rise to a pair of adjoint functors

(8)  f_{!} : Alg_{𝒫} [figure] Alg_{𝒬} : f^{∗}

where f^{∗} forgets a part of the structure. One has the following

2.4.1. Theorem. Assume 𝒫 and 𝒬 are admissible. Then the pair of adjoint functors (f_{!}, f^{∗}) is a Quillen pair.

Proof. The forgetful functor obviously preserves fibrations (surjective maps) and trivial fibrations (surjective quasiisomorphisms). □

One can expect the pair (f_{!}, f^{∗}) to be a Quillen equivalence under some favorable conditions. Recall that for a dg operad 𝒪 we denote by 𝒪_{1} the underlying dg category which remembers only unary operations of 𝒪. Passing to the zeroth cohomology of all Hom complexes, we get a category H^{0}(𝒪_{1}).

2.4.2. Definition. 1. A map f : 𝒫 → 𝒬 is called a weak equivalence if

a. For each c : I → [𝒫] and d ∈ [𝒫] the morphism 𝒫(c, d) → 𝒬(f ∘

c, f(d)) is a quasiisomorphism. ^{7} b. The functor H^{0}(f_{1}) : H^{0}(𝒫_{1}) → H^{0}(𝒬_{1}) is an equivalence of cat-

egories.

2. A map f : 𝒫 → 𝒬 is a strong equivalence if instead of (a) the following stronger condition is fulfilled. a′. Let c : I → [𝒫] be a collection of colors and d ∈ [𝒫]. Choose a

decomposition c = c′ ∘ p

(9)  I ^{p} J ^{c′} [𝒫]

with p surjective, and let G be the subgroup of automorphisms of I over J. The map

(10)  𝒫(c, d) ⊗_{G} k  𝒬(f ∘ c, f(d)) ⊗_{G} k

is a quasiisomorphism for all c, d and p.

7In case f induces bijection on the colors, our notion of weak equivalence coincides with the one mentioned in 2.3.

2.4.3. Definition. An operad 𝒪 in C(k) is called Σ-cofibrant if for each c : I → [𝒪] and d ∈ [𝒪] with G the group of automorphisms of c, the complex 𝒪(c, d) is a (projectively) cofibrant complex of G-modules.

2.4.4. Remark. Weak equivalence of operads implies their strong equivalence in case they are Σ-cofibrant.

2.4.5. Theorem. A strong equivalence of admissible operads f : 𝒫 𝒬 gives rise to a Quillen equivalence (f_{!}, f^{∗}).

The colorless case is proven in [H.V]. The proof of [H.V] directly generalizes to the colored case. Theorem 4.1 of [BM2] proves that weak equivalence of admissible Σ-cofibrant operads gives rise to a Quillen equivalence. □ The above observations lead us to the following definition.

2.4.6. Definition. An operad 𝒪 is called homotopically sound if it is admissible and Σ-cofibrant.

One can define therefore homotopy 𝒪-algebras as algebras over an operad which is a homotopically sound replacement of 𝒪.

2.5. Σ-split operads. In this subsection we present another class of admissible operads in C(k).

2.5.1. There is an obvious forgetful functor 𝒪 ↦ 𝒪^{♯} assigning to an operad its planar counterpart. The functor ♯ admits a left adjoint functor which we denote 𝒪 ↦ 𝒪^{Σ}; if 𝒪 is a planar operad, the operad 𝒪^{Σ} has the same colors; it is defined by the formula

(11)  𝒪^{Σ}(c, d) = ⨁_{θ:I≃⟨n⟩} 𝒪((c, θ), d),

where ⟨n⟩ = {1, …, n} is the standard (totally ordered) n-element set, the direct sum is over all bijections θ and a pair (c, θ) describes a colored collection c numbered by the totally ordered set (I, θ). Let 𝒪 be an operad. Applying the adjoint pair of functors described above, we get a new operad which we will denote 𝒪^{Σ}. It has the same colors as 𝒪 and its complexes of operations are defined by the formulas

(12)  𝒪^{Σ}(c, d) = ⨁_{θ:I≃⟨n⟩} 𝒪(c, d),

in the previously explained notation. The composition is defined as follows. Let f : I → J be a map of sets and let c : I → [𝒪] and d : J → [𝒪] be collections. The map

(13)  𝒪^{Σ}(d, e) ⊗ ⨂_{j} 𝒪^{Σ}(c_{j}, d(j))  𝒪^{Σ}(c, e)

is defined as follows. Choice of a total order on J together with a choice of total orders on each fiber f^{−1}(j) defines a lexicographical total order on I: if two elements of I belong to different fibers, we compare the fibers, and if they belong to the same fiber, we compare them inside the fiber. With the described above choice of the orderings, the corresponding component of the map (13) is given by the composition (1) for 𝒪. For example, if 𝒪 is the operad for commutative algebras, 𝒪^{Σ} is the operad for associative algebras.

2.5.2. One has a canonical map

π : 𝒪^{Σ} 𝒪

summing up the components corresponding to different orderings (this is just the the counit of the adjunction). One defines Σ-splitting as a collection of splittings t = t^{c,d} : 𝒪(c, d) 𝒪^{Σ}(c, d) of the canonical map π described above, satisfying the properties (SPL), (INV), (COM) which will be specified later on. We will usually omit the superscript (c, d) from the notation. A Σ-splitting t is defined by a collection of its components t_{θ} : 𝒪(c, d) → 𝒪(c, d) numbered by different orderings of I. The first two requirements for Σ-splitting are

(SPL) The map t splits π, that is ∑_{θ} t_{θ} = id. (INV) For any isomorphism f : c′ → c (that is, a bijection f : I′ → I satisfying

c′ = c ∘ f) the induced isomorphism f^{∗} : 𝒪(c, d) → 𝒪(c′, d) commutes with t. The latter means that

f^{∗} ∘ t_{θ} = t_{θf} ∘ f^{∗}.

The last requirement of Σ-splitness is a weak form of compatibility of the splitting with the compositions. Let c : I → [𝒪], d : J → [𝒪], a : K → [𝒪], a′ : K′ → [𝒪] be finite collections in 𝒪. Let f : I → J be a map of finite sets and let ϕ : a → a′ be an isomorphism of collections (that is, a bijection ϕ : K → K′ such that a = a′ ∘ ϕ). Gluing the above data, one gets collections c ⊔ a : I ⊔ K → [𝒪] and d ⊔ a′ : J ⊔ K′ → [𝒪], as well as a map of finite sets f ⊔ ϕ : I ⊔ K → J ⊔ K′. The requirement (COM) describes a compatibility of the splitting with the composition in 𝒪

(14)  𝒪(d ⊔ a′, e) ⊗ ⨂_{j∈J} 𝒪(c_{j}, d(j))  𝒪(c ⊔ a, e)

induced by the morphism f ⊔ ϕ. We are now able to formulate the third requirement of Σ-splittings. (COM) The following diagram is commutative.

(15)

𝒪(d ⊔ a′, e) ⊗ ⨂_{j∈J} 𝒪(c_{j}, d(j))  \overline{}𝒪(c ⊔ a, e)

[figure: ⨁_{η:J⊔K′≃⟨|J|+|K|⟩} 𝒪(d ⊔ a′, e) ⊗ ⨂_{j∈J} 𝒪(c_{j}, d(j)); t; ⨁_{θ:I⊔K≃⟨|I|+|K|⟩}; t; 𝒪(c ⊔ a, e)]

[figure: ⨁_{k∈K′} 𝒪(d ⊔ a′, e) ⊗ ⨂_{j∈J} 𝒪(c_{j}, d(j)); q; ⨁_{k∈K} 𝒪(c ⊔ a, e); q]

The upper vertical arrows in the diagram are defined by splitting of 𝒪(d⊔a′, d) and of 𝒪(c ⊔ a, e) respectively. In order to define the lower vertical arrows we will introduce the following notation. For each ordering η of the set J ⊔ K′ we denote by min_{K′}(η) the smallest element of the subset K′ of J ⊔ K′. In the same manner we define min_{K}(θ). Now the maps q send each η-component (resp., θ-component) to the corresponding min_{K′}(η)-component (resp., min_{K}(θ)-component).

2.5.3. Remark. There is another (stronger) version of Σ-splitness where q is replaced with a projection to the sum over orderings of K′ (resp., of K). It seems more satisfactory aesthetically; in this formulation the condition (INV) is its special case for I = J = ∅. This stronger version was used in the definition given in [H] for the colorless case.

2.5.4. Example. In the case k ⊃ ℚ the map

t_{θ}(m) = \frac{1}{n!}m

defines a Σ-splitting.

2.5.5. Example. Let 𝒫 be a planar colored operad and let 𝒪 = 𝒫^{Σ}. The canonical map of planar operads 𝒫 → 𝒪^{♯} defines a map of operads t : 𝒪 → 𝒪^{Σ} splitting the canonical map 𝒪^{Σ} → 𝒪. This map satisfies obviously the conditions (SPL), (INV), (COM).

2.6. Admissibility of Σ-split operads. One has

2.6.1. Theorem.

• Σ-split operads in C(k) are admissible.

• If the components 𝒪(c, d) of a Σ-split operad 𝒪 are cofibrant complexes, 𝒪 is homotopically sound.

The second claim of the theorem immediately follows from the first one, as 𝒪(c, d) is a direct summand of 𝒪^{Σ}(c, d) which is cofibrant Σ_{c}-complex, where Σ_{c} denotes the group of automorphisms of c : I → [𝒪]. The forgetful functor commutes with filtered colimits. Thus, it is sufficient to check that the map A → A ∐ 𝔽(H_{a}) is a quasiisomorphism for H_{a} standard contractible complex Cone(id_{k})[d] concentrated at a color a ∈ [𝒪]. The proof of the theorem is given in 2.6.2—2.6.5 below.

2.6.2. Extending homotopy to a free algebra. Let V = {V_{d}|d ∈ [𝒪]} be a collection of complexes, α : V → V an endomorphism and h a homotopy of α with id_{V}, that is a degree −1 map satisfying the condition

dh = id_{V} − α.

The endomorphism α induces an endomorphism 𝔽_{𝒪} (α) : 𝔽_{𝒪} (V) → 𝔽_{𝒪} (V); we will present an explicit homotopy between id_{𝔽_{𝒪} (V)} and 𝔽_{𝒪} (α) which we will denote 𝔽_{𝒪} (h). The homotopy 𝔽_{𝒪} (h) will be based on a Σ-splitting of 𝒪. Recall that one has a morphism of operads π : 𝒪^{Σ} → 𝒪 identical on the colors, as well as a Σ-splitting t : 𝒪(c, d) → 𝒪^{Σ}(c, d). We are now ready to define a homotopy H on 𝔽_{𝒪} (V). Recall that 𝔽_{𝒪} (V)_{d} is the direct limit of the functor 𝔉(V)_{d} carrying a collection c : I → [𝒪] to

𝔉(V)_{d}(c) = 𝒪(c, d) ⊗ ⨂_{i} V_{c(i)}.

We will define a degree −1 endomorphism H of each separate 𝔉(V)_{d}(c) compatible with the isomorphisms c → c′ of collections. It is given by the composition

(16) 𝒪(c, d) ⊗ ⨂_{i} V_{c(i)} ^{t}  ⨁_{θ:I≃⟨n⟩} 𝒪(c, d) ⊗ ⨂_{i} V_{c(i)} ^{S}

⨁_{θ:I≃⟨n⟩} 𝒪(c, d) ⊗ ⨂_{i} V_{c(i)} ^{π} 𝒪(c, d) ⊗ ⨂_{i} V_{c(i)},

with the map S being defined at the θ-component as

S_{θ} = ∑_{i} id_{𝒪(c,d)} ⊗ α^{i−1} ⊗ h ⊗ id^{n−i}.

2.6.3. In order to check that the morphism A → A ∐ 𝔽_{𝒪} (H_{a}) is a quasiisomorphism for H_{a} = Cone(id_{k})[d], one proceeds as follows. Let A′ = A⊕H_{a}. Then A ∐ 𝔽_{𝒪} (H_{a}) can be described as the quotient of 𝔽_{𝒪} (A′) by the ideal generated by the kernel of the natural map 𝔽_{𝒪} (A) A. Let α : A′ → A′ be zero on H_{a} and id_{A} on A. Let h : A′ → A′ be the degree −1 map vanishing on A such that dh = id − α. Then h defines a homotopy 𝔽_{𝒪} (h) on 𝔽_{𝒪} (A′) extending h. Let ℐ the the kernel of the natural projection 𝔽_{𝒪} (A) → A and let 𝒥 be the ideal in 𝔽_{𝒪} (A′) generated by ℐ. We check below that H(𝒥) ⊂ 𝒥 and this induces a homotopy on the quotient 𝔽_{𝒪} (A′)/J = A ∐ 𝔽_{𝒪} (H_{a}).

2.6.4. Action of 𝔽_{𝒪} (h) on 𝔽_{𝒪} (A′). Some relevant notation. For c : I → [𝒪] and n ≥ 0 we define c^{∗n} : I ⊔ ⟨n⟩ → [𝒪] by the formula

c^{∗n}(i) = c(i) for i ∈ I; c^{∗n}(k) = a for k ∈ ⟨n⟩;

The e-component of the free algebra 𝔽_{𝒪} (A′) with A′ = A ⊕ H_{a} is the colimit of the complexes

𝒪(c^{∗n}, e) ⊗ ⨂_{i} A_{c(i)} ⊗ H_{a}^{⊗n}

The homotopy 𝔽_{𝒪} (h) is defined by the components S_{θ} numbered by the total ordering θ of the set I ⊔ ⟨n⟩. Since h vanishes on A and α is identity on A and vanishes on H_{a}, the map S_{θ} has form

S_{θ} = id_{𝒪(c^{∗n},e)} ⊗ id_{A} ⊗ id^{⊗k−1} ⊗ h ⊗ id^{n−k}

where the homotopy h is applied to the k-th component of H_{a}, with k := min_{⟨n⟩}(θ).

2.6.5. End of the proof. We keep the notation of 2.6.3. The ideal 𝒥 in 𝔽_{𝒪} (A′) generated by ℐ, is spanned by the expressions

(17)  u ⊗ δ ⊗ ⨂_{i∈I−{0}} b_{i} ⊗ ⨂_{k∈⟨n⟩} x_{k}

where c : I → [𝒪], 0 ∈ I, c(0) = c_{0}, δ ∈ ℐ_{c_{0}}, u ∈ 𝒪(c^{∗n}, e), b_{i} ∈ A_{c(i)} and x_{k} ∈ H_{a}. We will now explicitly calculate the image of (17) under the homotopy 𝔽_{𝒪} (h) = ∑_{θ} S_{θ} ∘ t_{θ} to make sure it belongs to 𝒥. Let

t(u) = ∑_{θ:I^{∗n}→⟨|I|+n⟩} t_{θ}(u).

We claim that 𝔽_{𝒪} (h) carries (17) to the sum

(18)  ∑_{θ} u_{θ} ⊗ δ ⊗ ⨂_{i∈I−{0}} b_{i} ⊗ ⨂_{k∈⟨n⟩} x_{θ,k},

where

(19)  x_{θ,k} = \{ \begin{matrix}x_{k}, & k ≠ min_{⟨n⟩}(θ) \\ h(x_{k}), & k = min_{⟨n⟩}(θ) \end{matrix}

It is sufficient to check the formula (18) in case δ is a monomial in 𝔽_{𝒪} (A):

(20)  δ = m ⊗ ⨂_{j∈J} a_{j}

with m ∈ 𝒪(d, c_{0}), d : J → [𝒪], a_{j} ∈ A_{d(j)}. Replace δ in (17) with the expression (20). We get a monomial

(21)  z := u ∘ m ⊗ ⨂_{j∈J} a_{j} ⊗ ⨂_{i∈I−{0}} b_{i} ⊗ ⨂_{k∈⟨n⟩} x_{k},

where u ∘ m denotes the composition of u and m belonging to 𝒪(c ∘ d, e) where c ∘ d : I − {0} ⊔ J → [𝒪] is the restriction of c ⊔ d, whose image under 𝔽_{𝒪} (h) is given by the formula

(22)  𝔽_{𝒪} (h)(z) = ∑_{η:I−{0}⊔J≃⟨|I|+|J|−1⟩} S_{η} ∘ t_{η}.

By the axiom (COM) of Σ-splitness applied to the surjection I −{0}⊔J I sending the elements of J to 0 and the elements of I − {0} to themselves, we deduce that 𝔽_{𝒪} (h)(z) is equal to (18).

2.7. Simplicial structure in characteristic zero. All operads are Σsplit when k ⊃ ℚ, so in this case the category of algebras Alg_{𝒪} (C(k)) has a model structure described in Theorem 2.3.3. Moreover, polynomial differential forms allow one to define a simplicial structure on the category Alg_{𝒪} (C(k)) which is (partly) compatible with the model category structure. We will present the definitions and formulate the theorem. The proof is identical to the colorless case described in [H], 4.8. For k ⊃ ℚ and n ≥ 0 one defines a dg commutative algebra Ω_{n} by the formula

Ω_{n} = k[x_{0}, …, x_{n}, dx_{0}, …, dx_{n}]/(∑ x_{i} − 1, ∑ dx_{i}).

The assignment n ↦ Ω_{n} defines a simplicial object in the category of commutative dg algebras over k. It is canonically extended to a contravariant functor

                     Ω : sSet     Alg_{Com}(C(k))

carrying colimits to limits. For A, B ∈ Alg_{𝒪} (C(k)) the simplicial set Map(A, B) is defined by the formula

Map(A, B)_{n} = Hom(A, Ω_{n} ⊗ B).

The compatibility of the simplicial structure on Alg_{𝒪} (C(k)) with the model category structure is described in the following theorem.

2.7.1. Theorem. Assume k ⊃ ℚ and let 𝒪 be an operad in C(k). The category Alg_{𝒪} (C(k)) of 𝒪-algebras with values in C(k) has a structure of model category with quasiisomorphisms as weak equivalences and componentwise surjective maps as fibrations. The category Alg_{𝒪} (C(k)) has a “weak simplicial model category structure” (see [H.L], 1.4.2), that is a simplicial structure such that the axioms (M7) and the half of the axiom (M6), see [Hir], 9.1.6, are satisfied. (\frac{1}{2}M6) For every finite simplicial set K and A ∈ Alg_{𝒪} (C(k)) the “weak path

object” (see [H.L], 1.4.1) A^{K} exists and is defined by the formula

A^{K} = Ω(K) ⊗ A.^{8}

8It has an 𝒪-algebra structure as the tensor product of a commutative algebra with an 𝒪-algebra.

(M7) For a cofibration i : A → B and a fibration p : X → Y in Alg_{𝒪} (C(k)) the map of simplicial sets

(23)  Map(B, X)  Map(A, X) ×_{Map(A,Y)} Map(B, Y)

is a fibration which is trivial if either i or p is trivial.

2.8. A pushout of algebras. Let 𝒪 be an admissible operad. In this subsection we will present, for a later use, explicit formulas for a specific type of cofibrations of operad algebras. This is a generalization of formula presented in [SS00] in the proof of Lemma 6.2, see also F. Muro’s very detailed account of the planar case [Mu], Lemmas 8.2, 8.5. Recall that the model category structure on Alg_{𝒪} (C(k)) is transported from the projective model structure on C(k) via adjunction (6). Given a cofibration f : V → W of cofibrant objects in C(k)^{[𝒪]} , that is a collection of cofibrations of cofibrant compexes f_{c} : V_{c} → W_{c} numbered by c ∈ [𝒪], and a pushout diagram in Alg_{𝒪} (C(k))

(24)  [figure: 𝔽_{𝒪} (V); 𝔽_{𝒪} (f); 𝔽_{𝒪} (W); A; B]

with cofibrant A, we will explicitly describe the morphism 𝔾(A) 𝔾(B) as a colimit of a sequence of cofibrations

(25)  𝔾(A) = B_{0}  B_{1}  …  B_{k}  …

in C(k)^{[𝒪]} .

2.8.1. Enveloping operad. In order to describe the precise formula for B_{i}, we need a colored version of enveloping operads, as defined and used by Berger and Moerdijk [BM3]. Let 𝒪 be a colored operad and let A be a 𝒪-algebra. The enveloping operad 𝒪_{A} can be defined as the operad governing 𝒪-algebras X endowed with a morphism of 𝒪-algebras A X. Admissibility criterion 2.3.3 immediately implies that, if 𝒪 is admissible, then for any 𝒪-algebra A the operad 𝒪_{A} is as well admissible. Furthermore, if 𝒪 is Σ-cofibrant and A is a cofibrant, the enveloping operad 𝒪_{A} is also Σ-cofibrant. The latter is proven in [BM1], 5.4, for colorless operads, but the colored version can be proven in the same way.

2.8.2. In the formula (27) below we will use the following notation. Given a collection of maps ϕ_{i} : X_{i} → Y_{i}, i ∈ I in C(k), we can form a functor from the standard |I|-cube, considered as a poset of subsets S ⊂ I, to C(k), carrying S to Z(S) := ⨂_{i∈I} Z_{i} where Z_{i} = X_{i} for i ∉ S and Y_{i} for i ∈ S. This yields a map

(26)  ⋀_{i∈I} ϕ_{i} : colim_{S≠I} Z(S)  Z(I) = ⊗_{i∈I}Y_{i}.

We will need some more notation. Recall that for c : I → [𝒪] we denote Σ_{c} the automorphism group of c. Now, fix d ∈ [𝒪]. The d-component of the map B_{k−1} B_{k} is defined as the pushout of the map

(27)  ⨁_{{c∈π_{0}(Fin/[𝒪]) \atop |I|=k}} 𝒪_{A}(c, d) ⊗_{Σ_{c}} ⋀_{i∈I} f_{c(i)},

where f_{c(i)} : V_{c(i)} → W_{c(i)} is a component of the map f : V → W. We have to specify the map from the source of (27) to the d-component of B_{k−1}. Fixing c : I → [𝒪] with |I| = k, the formula (27) is obtained from a cubic diagram whose vertices are numbered by subsets T ⊂ I so that T -th vertex has |T | times the factor W and k − |T | times the factor V. We have to specify a map from all such vertices corresponding to T ≠ I, to B_{k−1}. This is done by replacing factors of V_{c(i)} with respective factors of A_{c(i)} = 𝒪_{A}(∅, c(i)), composing the components of 𝒪_{A} via

𝒪_{A}(c, d) ⊗ ⨂_{i∈I−T} 𝒪_{A}(∅, c(i)) 𝒪_{A}(c|_{T} , d),

and, finally, composing the result with the map

𝒪_{A}(c|_{T}) ⊗ ⨂_{i∈T} W_{c(i)} B_{k−1}.

We now have

2.8.3. Proposition. Let 𝒪-algebra B be given by the pushout diagram (24). Then the object 𝔾(B) ∈ C(k)^{|𝒪|} is a colimit of the sequence of cofibrations (25) so that B_{k−1} → B_{k} is defined by the pushout diagram (27) in C(k)^{|𝒪|} .

Proof. The 𝒪-algebra B can be presented as a split coequalizer of

A ⊔ 𝔽_{𝒪} (V ⊕ W) [figure: α; β] A ⊔ 𝔽_{𝒪} (W),

where α is determined by the map (f, id_{W}) : V ⊕ W → W, and β is id_{W} on W and is defined by V → A on V. The above coequalizer can be more conveniently rewritten as the coequalizer of

𝔽_{𝒪_{A}} (V ⊕ W) [figure: α; β] 𝔽_{𝒪_{A}} (W).

We will apply the functor 𝔾 before calculating the colimit. The free algebras considered as collection of complexes, are direct sums of components 𝔽^{k} corresponding to collections of colors c : I → [𝒪] with |I| = k. We define B_{k} as the image of the map 𝔽_{𝒪_{A}}^{≤k}(W) B. Then it is obvious that the image of 𝔽_{𝒪_{A}}^{k} (W) and B_{k−1} generate the whole B_{k} and that the d-component of the fiber product 𝔽^{k}(W) ×_{B_{k}} B_{k−1} coincides with the source of the map (27). □

2.8.4. Corollary. Assume 𝒪 is homotopically sound. Let A be a cofibrant 𝒪-algebra. Then for each d ∈ [𝒪] the complex 𝔾(A)_{d} is cofibrant.

Proof. Any cofibrant 𝒪-algebra is a retract of a transfinite sequence of cofibrations as in (24). Formulas (27) show that each step is a cofibration in C(k)^{[𝒪]} , so the limit is as well a cofibration. Finally, retract of a cofibration is a cofibration. □

3. SM ∞-categories

In this section we will construct certain SM ∞-categories (of ∞-categories, of dg categories and others) by Dwyer-Kan localization. We construct an adjoint pair of functors

                  ℭ_{dg} : Cat_{∞}^{×} [figure] N(dgCat)^{⊗} : 𝔑_{dg}

between the symmetric monoidal ∞-categories of infinity-categories and of dg categories, induced by Dold-Kan equivalence. The functor ℭ_{dg} is symmetric monoidal, whereas 𝔑_{dg} is lax symmetric monoidal (that is, a morphism of ∞- operads). We will denote Cat_{∞}^{SM} := Alg_{Com}(Cat_{∞}) and dgCat^{SM} = Alg_{Com}(N(dgCat)). The functor 𝔑_{dg} induces a functor

                     𝔑_{dg} : dgCat^{SM}  Cat_{∞}^{SM}.

Furthermore, we will see that both ∞-categories dgCat^{SM}(k) and Cat_{∞}^{SM} are enriched over Cat_{∞}, so that 𝔑_{dg} preserves this enrichment. The latter means that for a pair of symmetric monoidal dg categories C, D one has a functor

Fun^{⊗}(C, D) Fun^{⊗}(𝔑_{dg}(C), 𝔑_{dg}(D)).

3.1. Localization. Given a symmetric monoidal ∞-category 𝒞^{⊗} and a collection of arrows W, we would like to be able to define a SM structure on the localization ℒ(𝒞, W). This is easy if the tensor product preserves W, see [H.L], 3.2 or [L.HA], 4.1.3.4. In this case the localization of the total category 𝒞^{⊗} with respect to the collection of arrows in 𝒞^{⊗} generated by W, yields what we call a strict SM localization: this is a SM functor

(28)  𝒞^{⊗}  ℒ(𝒞^{⊗}, W^{⊗})

universal among SM functors 𝒞^{⊗} → 𝒟^{⊗} carrying W to equivalences. Moreover, the underlying ∞-category of the strict SM localization is the localization ℒ(𝒞, W) and the localization functor is also universal among lax monoidal functors 𝒞^{⊗} → 𝒟^{⊗}, see [H.L], 3.2. Strict SM localizations seldom exist: tensor product does not always preserve weak equivalences. In this paper we will use the following ad hoc construction. Given a SM ∞-category 𝒞^{⊗} and a collection of arrows W in 𝒞, we will present a full subcategory 𝒞_{0}^{⊗} such that

• The pair (𝒞_{0}^{⊗}, W_{0} = W ∩ 𝒞_{0}) admits a strict SM localization.

• The embedding 𝒞_{0} 𝒞 induces an equivalence of the localizations ℒ(𝒞_{0}, W_{0}) → ℒ(𝒞, W).

We will call ℒ(𝒞_{0}^{⊗}, W_{0}^{⊗}) the SM localization of (𝒞^{⊗}, W). This construction depends, in general, on the choice of 𝒞_{0}. We believe that in the examples below, it satisfies a universal property which makes it right SM localization as defined in [H.L], 3.3. By [H.L], 3.3.3, this is so in Example 3.1.1.

3.1.1. Example: QC(k)^{⊗}. Let k be a commutative ring, 𝒞 := C(k) the category of complexes of k-modules and let W be the collection of quasiisomorphisms. We choose 𝒞_{0} to be the full subcategory of cofibrant complexes. As a result, we get a SM ∞-category denoted as QC(k). This is the SM ∞-category of k-modules; its homotopy category is the derived category of k.

3.1.2. Example: categories of enriched categories. We will use a similar construction to define SM ∞-categories of certain enriched categories. The corresponding model categories were defined by G. Tabuada, see [T1, T2]. These are

• dgCat(k), the category of categories enriched over C(k).

• dg^{≤0}Cat(k), that of categories enriched over C^{≤0}(k).

• sMod-Cat(k), that of categories enriched over the simplicial k-modules. Symmetric monoidal structure on all these categories is induced by the symmetric monoidal structure on C(k), C^{≤0}(k) and sMod(k) respectively. In each one of the cases 𝒞 := dgCat, dg^{≤0}Cat or sMod-Cat, the full subcategory 𝒞_{0} is spanned by the categories whose Hom-objects are cofibrant. In all three cases tensor product preserves weak equivalence of categories belonging to 𝒞_{0}. It remains to check that in all three cases the embedding 𝒞_{0} 𝒞 induces an equivalence of DK localizations. This is routinely done using Key Lemma 1.3.6 of [H.L]^{9}. This yields symmetric monoidal ∞-categories which we denote N(dgCat)^{⊗}, N(dg^{≤0}Cat)^{⊗} and N(sMod-Cat)^{⊗}.

3.2. Dold-Kan correspondence.

3.2.1. Classical Dold-Kan equivalence. Here we will fix some notation. The functor of normalized chains

C_{∗} : sMod(k) C^{≤0}(k)

from simplicial k-modules to nonpositively graded complexes of k-modules is well-known to be an equivalence, with the inverse functor

N_{∗} : C^{≤0}(k) sMod(k)

defined by the formula N_{n}(X) = Hom(C_{∗}(Δ^{n}), X). The functor C_{∗} is not symmetric monoidal, but it is very close to be one. One has functorial maps

(29)  ▽_{X,Y} : C_{∗}(X) ⊗ C_{∗}(Y)  C_{∗}(X ⊗ Y)

9see 1.3.7 of [H.L] for the routine.

(Eilenberg-MacLane, or shuffle, map), and

(30)  △_{X,Y} : C_{∗}(X ⊗ Y)  C_{∗}(X) ⊗ C_{∗}(Y)

(Alexander-Whitney map) such that

• The functor C_{∗} is lax symmetric monoidal via ▽.

• It is also colax monoidal ^{10} via △ (equivalently, N_{∗} is lax monoidal via △).

• Both △ and ▽ are homotopy equivalences and △ ∘ ▽ = id.

3.2.2. Enriched categories. Any lax monoidal functor F : M N induces a functor

                        F : Cat_{M}  Cat_{N}

between the respective enriched categories. Therefore, one has a pair of functors

(31)  C̄ : sMod-Cat(k) [figure] dg^{≤0}Cat(k) : N̄,

where C̄ = (C_{∗}, ▽) and N̄ = (N_{∗}, △), with a natural isomorphism N̄ ∘ C̄ = id. Note that the functors C̄, N̄ do not form an adjoint pair. The functor C̄ is lax symmetric monoidal. Developing the ideas of [SS03], Tabuada proved in [T2] that the functor C̄ has a left adjoint and this pair defines a Quillen equivalence. Therefore, an equivalence

(32)  N(C̄) : N(sMod-Cat(k))  N(dg^{≤0}Cat(k))

is induced. Since it is lax SM, it is a symmetric monoidal equivalence. Since N(N̄) is left inverse, it is an inverse symmetric monoidal equivalence.

3.2.3. We can now define an adjoint pair

(33)  ℭ_{dg} : Cat_{∞}^{×} [figure] N(dgCat(k))^{⊗} : 𝔑_{dg}

as a composition (34) N(sCat^{×}) [figure] N(sMod-Cat(k))^{⊗} [figure] N(dg^{≤0}Cat(k))^{⊗} [figure] N(dgCat(k))^{⊗}.

The adjoint pair in the middle is a SM equivalence. In two other adjoint pairs the left adjoint is symmetric monoidal, therefore the right adjoint is a map of operads, see Appendix A. Thus, ℭ_{dg} is symmetric monoidal and 𝔑_{dg} is a map of operads.

10but not colax symmetric monoidal!

3.2.4. We will now show that the functor 𝔑_{dg} carries the dg category C(k) to QC(k) ∈ Cat_{∞}. Moreover, 𝔑_{dg} carries the commutative algebra object C(k)^{⊗} to QC(k)^{⊗}. Let C_{♯}^{c}(k) denote the category of cofibrant complexes of k-modules. It is klinear and so is enriched in a “trivial” way over C(k):

(35)  ℋom_{♯}(X, Y) := Z^{0}(ℋom(X, Y)).

The functor 𝔑_{dg} carries C_{♯}^{c}(k) into the category whose Hom object are discrete (as simplicial objects) k-modules. We identify 𝔑_{dg}(C_{♯}^{c}(k)) with C_{♯}^{c}(k) for the obvious reason. The functor 𝔑_{dg} applied to the map C_{♯}^{c}(k) → C^{c}(k) yields a map

C_{♯}^{c}(k) 𝔑_{dg}(C^{c}(k))

which carries quasiisomorphisms to equivalences. Therefore, a map

QC(k) 𝔑_{dg}(C^{c}(k))

is induced. It is an equivalence by [H.L], 1.4.3. Let us show that 𝔑_{dg} also preserves the symmetric monoidal structure of C^{c}(k). The adjoint pair

ℭ_{dg} : Cat_{∞}^{×} [figure] N(dgCat(k))^{⊗} : 𝔑_{dg}

gives rise to an adjoint pair of functors between the ∞-categories of commutative algebras in respective categories,

(36)  ℭ_{dg} : Cat_{∞}^{SM} [figure] dgCat^{SM}(k) : 𝔑_{dg},

that is between symmetric monoidal ∞-categories and (weak) symmetric monoidal dg categories. We claim that 𝔑_{dg} carries the symmetric monoidal dg category C^{c}(k)^{⊗} to QC(k)^{⊗} as constructed in 3.1.1. The dg category C_{♯}^{c}(k) has a symmetric monoidal structure and the map C_{♯}^{c}(k) C^{c}(k) is a symmetric monoidal functor. Therefore, the induced arrow

C_{♯}^{c}(k)^{⊗} 𝔑_{dg}(C^{c}(k)^{⊗})

is also a symmetric monoidal functor. By universality of symmetric monoidal localization we get a symmetric monoidal functor

(37)  QC(k)^{⊗}  𝔑_{dg}(C^{c}(k)^{⊗}).

Since we already know that the induced functor QC(k) → 𝔑_{dg}(C^{c}(k)) is an equivalence, it is an equivalence of symmetric monoidal ∞-categories.

3.3. 𝔑_{dg}, enriched. We will now show that the ∞-categories Cat_{∞}^{SM} and dgCat^{SM}(k) are enriched over Cat_{∞} and the functor 𝔑_{dg} defined in (36) preserves this enrichment. More precisely, we will present, for a pair A, B of symmetric monoidal dg categories, a map of ∞-categories

(38)  Fun^{⊗}(A, B)  Fun^{⊗}(𝔑_{dg}(A), 𝔑_{dg}(B))

of respective symmetric monoidal functors extending the map of of spaces of morphisms defined by the functor 𝔑_{dg}. We will first explain the construction in the setup of conventional categories, and then will provide the ∞-categorical generalization, using the formalism of SM adjunction (see Appendix A).

3.3.1. A general setup (conventional categories). Let

(39)  λ : 𝒞 [figure] 𝒟 : ρ

be an adjoint pair of functors between symmetric monoidal categories, so that λ is symmetric monoidal (and therefore ρ is lax symmetric monoidal). We assume that 𝒟 is cotensored over 𝒞, which means that there exists a functor η : 𝒞^{op} ×𝒟 𝒟, (X, A) ↦ A^{X}, right adjoint to the bifunctor 𝒞×𝒟 𝒟 carrying the pair (X, A) to λ(X)⊗A. One can easily see that η is lax symmetric monoidal. Assume now that 𝒞 is cartesian. Then any object X ∈ 𝒞 has an obvious coalgebra structure defined by the diagonal. This implies that for any commutative algebra A in 𝒟 and any object X in 𝒞 the power object A^{X} has a commutative algebra structure. The multiplication in A^{X} is given by the composition

A^{X} ⊗ A^{X}  (A ⊗ A)^{X×X}  A^{X×X}  A^{X}.

We can therefore define inner hom on Alg_{Com}(𝒟) by the formula

(40)  Hom(X, Fun^{⊗}(A, B)) = Hom_{Alg_{Com}(𝒟)} (A, B^{X}),

provided the right-hand side is representable. The 𝒞-enrichment on Alg_{Com}(𝒟) so defined is functorial in the following sense. Given a sequence of adjoint pairs

𝒞 [figure: λ_{1}; ρ_{1}] 𝒟_{1} [figure: λ_{2}; ρ_{2}] 𝒟_{2}

between SM categories, satisfying the above properties, one has a natural isomorphism

(41)  ρ_{2}(B^{X}) = ρ_{2}(B)^{X},

which induces a canonical map

(42)  Fun^{⊗}(A, B)  Fun^{⊗}(ρ_{2}(A), ρ_{2}(B)).

3.3.2. Construction for SM ∞-categories. The only claim requiring a special attention when extending the above construction to ∞-categories is the structure of lax symmetric monoidal functor on η : 𝒞^{op} × 𝒟 𝒟 induced by the adjunction (39). The functor λ leads to a 𝒞-left-tensored structure on 𝒟 given by a SM functor

𝒞 × 𝒟 𝒟

defined by the formula (a, x) ↦ λ(a) ⊗ x.

The corresponding functor 𝒞^{op} × 𝒟^{op} × 𝒟 𝒮 carrying the triple (a, x, y) to Map(a ⊗ x, y) is then lax monoidal by A.5.1. Existence of 𝒞-cotensor structure on 𝒟 is equivalent to {1, 3}-representability of this functor. Once more, according to A.5.1, this implies that the functor

(43)  η : 𝒞^{op} × 𝒟  𝒟

is lax SM, see A.5.3. Now, a lax SM functor induces a functor between respective ∞-categories of commutative algebras. A commutative algebra in 𝒞^{op} × 𝒟 is a pair (X, A) where X ∈ 𝒞 and A ∈ Alg_{Com}(𝒟). This yields a required functor

(44)  η^{SM} : 𝒞^{op} × Alg_{Com}(𝒟)  Alg_{Com}(𝒟),

which allows one to define 𝒞-valued inner Hom on Alg_{Com}(𝒟) by the formula (40).

3.3.3. We wish to apply the above construction to 𝒞 = 𝒟_{1} = Cat_{∞}, 𝒟_{2} = N(dgCat(k)). The ∞-categories Cat_{∞} and N(dgCat(k)) can be described as ∞-categories underlying combinatorial model categories, see [L.T], 2.2.5.1 and [T1], Thm 1.8. Therefore, the corresponding underlying ∞-categories are presentable. The tensor products in these ∞-categories commute with colimits, so by Corollary 3.2.3.5 of [L.HA] the categories of commutative algebras in Cat_{∞} and N(dgCat(k)) are as well presentable. Furthermore, the functors η and η^{SM} preserve limits in each of the arguments. This implies presentability of the 𝒞-valued inner Hom given by (44).

We now have to make sure that the Cat_{∞}-enrichment of Cat_{∞}^{SM} defined by the above universal construction, coincides with the standard one, see [L.HA], Definition 2.1.3.7.

3.3.4. Inner Hom for SM ∞-categories. In case 𝒞 = 𝒟 has products, the functor η^{SM} defined in (44) can be described much easier: the functor η (43) preserves products in the second variable; therefore, it carries algebras to algebras. Here is an explicit description of η^{SM} for 𝒞 = Cat_{∞}, where commutative algebras in Cat_{∞} are presented as ∞-categories cocartesian over NFin_{∗}, see [L.HA], Section 2. Given a simplicial set X and a SM ∞-category p : B NFin_{∗}, we define a simplicial set B^{X} with a map q : B^{X} NFin_{∗} as follows. The n-simplices of B^{X} over σ : Δ^{n} NFin_{∗} are the commutative diagrams

(45)  [figure: Δ^{n} × X; σ̃; B; Δ^{n}; pr_{1}; σ; NFin_{∗}; p] .

A diagram (45) with n = 1 represents a cocartesian lifting of σ iff the restriction of σ̃ to each vertex of X is cocartesian. This implies that q is a cocartesian fibration; the fiber of q at ⟨n⟩ is B_{⟨n⟩}^{X}, so q : B^{X} NFin_{∗} is a SM ∞- category. Now the identity

(46)  Map(X, Fun^{⊗}(A, B)) = Map_{Cat}{SM \atop ∞}(A, B^{X})

can be easily verified, which proves that Cat_{∞}-valued function space defined be our general construction is the conventional one for 𝒞 = Cat_{∞}.

4. Rectification of algebras

4.1. Introduction. Let 𝒪 be a topological operad (that is, a fibrant simplicial operad) with the set of colors [𝒪]. We denote 𝒪^{⊗} the corresponding ∞-operad in the sense of Lurie [L.HA] which is defined as follows. Let Fin_{∗} denote the category of finite pointed sets. Its objects are finite pointed sets I_{∗} = I ⊔ {∗} and the maps f : I_{∗} → J_{∗} satisfy f(∗) = ∗. We will define first of all a simplicial category 𝒪̃^{⊗} over Fin_{∗}, and then will put 𝒪^{⊗} to be the (homotopy coherent) nerve of the simplicial category 𝒪̃^{⊗}, see [L.T], 1.1.5.5. Here is the definition of 𝒪̃^{⊗}. Its objects over I_{∗} ∈ Fin_{∗} are maps c : I → [𝒪] and the simplicial sets of morphisms over f : I_{∗} → J_{∗} defined by the formula

Mapf ; 𝒪̃^{⊗} (c, d) = ∏_{j∈J} 𝒪(c|_{f^{−1}(j)}, d(j)).

The composition in 𝒪̃^{⊗} is determined by the composition in 𝒪, see the details in [L.HA], 2.1.1.22. Fix a commutative ring k. We are mostly interested in algebras over 𝒪^{⊗} with values in the SM ∞-category QC(k) of complexes of k-modules described in detail in 3.1.1. We want to compare the ∞-category Alg_{𝒪} (QC(k)), as defined in Lurie’s book [L.HA], 2.1.3.1 (this is just the ∞-category of operad maps 𝒪^{⊗} QC(k)), with the category of “strict” 𝒪-algebras Alg_{𝒪}^{st}(C(k)) defined as in Section 2. ^{11} Assume now we are given a quasiisomorphism of operads ℛ → C_{∗}(𝒪, k) with ℛ homotopically sound. In this case, as we know, the category Alg_{ℛ}^{st}(C(k)) admits a model structure with quasiisomorphisms as weak equivalences and surjective maps as fibrations. Applying the nerve construction (see [H.L], 1.3) to Alg_{ℛ}^{st}(C(k)), we get an ∞- category. A certain effort is required in order to be able to interpret a strict ℛ-algebra as an object of Alg_{𝒪} (QC(k)). Unexpectedly, the problem exists even if ℛ = C_{∗}(𝒪). The reason is that the singular chains functor C_{∗} : sSet C(k) is not symmetric monoidal.

11Note that we have changed the notation in order to distinguish two notions of 𝒪-algebra!

The construction of the functor Alg_{ℛ}^{st}(C(k)) Alg_{𝒪} (QC(k)) is explained in Subsection 4.2 below. Once we have this functor, the universal property of the ∞-localization yields an ∞-functor

(47)  Φ : N(Alg_{ℛ}^{st}(C(k))  Alg_{𝒪} (QC(k)).

Here is the central result of this paper.

4.1.1. Theorem. Let 𝒪 be a topological operad and let ℛ → C_{∗}(𝒪) be a homotopically sound replacement. Then the functor Φ (47) is an equivalence.

4.1.2. The proof follows the idea of the proof of [L.HA] 4.1.4.4 dealing with rectification of associative algebras. An ∞-categorical version of Barr-Beck theory [L.HA], 6.2, allows one to present an 𝒪-algebra A in QC(k) as a colimit of its monadic Bar-resolution. The latter consists of free algebras which can be easily lifted to Alg_{ℛ}^{st}(C(k)). The homotopy colimit of this simplicial object in Alg_{ℛ}^{st}(C(k)) gives the lifting of A. Further details of the proof are given in 4.3 below. But first of all we have to define the map Φ in a greater detail.

4.2. Construction of Φ. Let 𝒪 be a fibrant simplicial operad and let ℛ → C_{∗}(𝒪, k) be a quasiisomorphism of dg operads (bijective on colors) with ℛ homotopically sound. The simplicial operad 𝒪 generates a simplicial PROP P_{𝒪} defined as follows. Let Fin be the category of finite sets. It can be considered as a subcategory of Fin_{∗} via the functor Fin → Fin_{∗} carrying a finite set I to the pointed set I_{∗}. One defines the simplicial category P_{𝒪} as the fiber product 𝒪̃^{⊗} ×_{Fin_{∗}} Fin. The objects of P_{𝒪} are finite collections of colors of 𝒪, and the arrows are composed from the operations in 𝒪. The symmetric monoidal structure is defined by disjoint union of collections. The image of P_{𝒪} under ℭ_{dg}, see formula (36), is presented by the SM dg category P_{C_{∗}𝒪} which is the k-linear PROP generated by dg operad C_{∗}(𝒪). We will replace P_{C_{∗}𝒪} with an equivalent SM dg category P_{ℛ} , the PROP generated by the dg operad ℛ. Any cofibrant ℛ-algebra A in C(k) gives rise to a symmetric monoidal dg functor

A : P_{ℛ} C(k)

such that all A(x), x ∈ P_{ℛ} , are cofibrant, see 2.8.4. This yields, in particular, an arrow in dgCat^{SM}(k). Applying to it the functor 𝔑_{dg}, and composing with the unit map, we get

P_{𝒪}  𝔑_{dg}(P_{ℛ} )  𝔑_{dg}(C(k)) = QC(k)^{⊗}.

The above construction defines a composition

(48) Alg_{ℛ}^{st}(C(k))^{c} → Fun^{⊗}(P_{ℛ} , C(k)) ≃ Fun^{⊗}(P_{C_{∗}𝒪} , C(k)) → → Fun^{⊗}(𝔑_{dg}(P_{C_{∗}𝒪} ), QC(k)^{⊗}) → Fun^{⊗}(P_{𝒪} , QC^{⊗}) = Alg_{𝒪} (QC(k)).

Thus, we have a functor

ϕ : Alg_{ℛ}^{st}(C(k))^{c} Alg_{𝒪} (QC(k)).

It remains to check that ϕ carries weak equivalences of cofibrant ℛ-algebras to an equivalence. This is really easy: the functor ϕ constructed above commutes with the forgetful functors G^{st} ^{12} and G in the following diagram

(49)  [figure: Alg_{ℛ}^{st}(C(k))^{c}; ϕ; Alg_{𝒪} (QC(k)); (C(k)^{c})^{[𝒪]}; G^{st}; ϕ^{triv}; QC(k)^{[𝒪]}; G],

where ϕ^{triv} is defined by localization. Since weak equivalences in Alg_{ℛ}^{st} are detected by G^{st} and since the functor G is conservative (see [L.HA], Lemma 3.2.2.6), the assertion follows.

4.3. Proof of 4.1.1.

4.3.1. Look at the commutative diagram obtained from (49) by application of the nerve functor to G^{st}.

(50)  [figure: N(Alg_{ℛ}^{st}(C(k)); Φ; Alg_{𝒪} (QC(k)); QC(k)^{[𝒪]}; G; NG^{st}].

The reasoning briefly explained in 4.1.2 is formalized in Corollary 6.2.2.14 of [L.HA]. It claims that the map Φ in (47) is an equivalence, provided the following properties are verified.

1. The functor G is conservative.

1^{st}. The functor NG^{st} is conservative.

2. The functor G admits a left adjoint functor F.

2^{st}. The functor NG^{st} admits a left adjoint functor F^{st}.

3. Any G-split simplicial object in Alg_{𝒪} (QC(k)) has a colimit and G preserves this colimit.

3^{st}. The same for NG^{st}.

4. The unit map X → NG^{st}F^{st}(X) = GΦ(F^{st}(X)) induces an equiva-

lence

F (X) Φ(F^{st}(X)).

12we denoted it 𝔾 in Section 2.

Let us check the assertions 1–4. The properties 1,2,3 is are proven in [L.HA], see 3.2.2.6, 3.1.3.5 and 3.2.3.1. ^{13} The functor F^{st} is obtained by application of the the nerve construction (see Proposition 1.5.1, [H.L]) to the functor 𝔽_{ℛ} which is left adjoint to the forgetful functor G^{st} : Alg_{ℛ}^{st}(C(k)) → C(k)^{[𝒪]} . This proves 2^{st}. The functor NG^{st} is conservative as weak equivalences in Alg_{ℛ}^{st} are detected by G^{st}. Thus, it remains to verify the assertions 3^{st} and 4. Assertion 3^{st}. According to [H.L], 1.5.2, colimits in an ∞-category underlying a combinatorial model category can be calculated via derived colimits in the model category. This is applicable to both C(k) and Alg_{ℛ}^{st}(C(k)). A simplicial complex X ∈ C(k)^{Δ^{op}} will be called colim-adapted if

• The canonical map L colim X colim X is a quasiisomorphism.

• For all n ∈ Δ^{op} the components X_{n} ∈ C(k) are cofibrant.

• The complex colim X is cofibrant. Simplicial objects in Alg_{ℛ}^{st}(C(k)) are algebras over a certain operad which we will denote ℛ^{Δ^{op}}. In Lemma 4.3.2 below we check that the operad ℛ^{Δ^{op}} is also homotopically sound. Since the category Δ^{op} is sifted, the colimit over Δ^{op} commutes with the forgetful functor G. Therefore, in order to deduce Assertion 3^{st}, it remains to prove that the forgetful functor

Alg_{ℛ^{Δ^{op}}} (C(k)) = Alg_{ℛ} (C(k))^{Δ^{op}} C(k)^{[ℛ]×Δ^{op}}

carries cofibrant simplicial algebras to [ℛ]-collections of colim-adapted simplicial complexes. This is also proven in Lemma 4.3.2 below. Let C be a (small) category and ℛ a dg operad. We will now describe a dg operad ℛ^{C} such that ℛ^{C}-algebras are precisely the functors C → Alg_{ℛ} (C(k)). The colors of ℛ^{C} are pairs (c, m) where c is a color of ℛ and m ∈ C. A collection of colors I → [ℛ^{C}] is given by a pair (c, m) where c : I → [ℛ] is a collection of colors in ℛ and m a function I → Ob(C). The complex ℛ^{C}((c, m), (d, n)) is defined as ∏_{i} Hom_{C}(m_{i}, n) × ℛ(c, d), and the composition in ℛ^{C} is defined by the compositions in ℛ and in C.

4.3.2. Lemma. Assume ℛ is a homotopically sound operad. Then a) ℛ^{Δ^{op}} is also homotopically sound.

b) the forgetful functor

G : Alg_{ℛ^{Δ^{op}}} (C(k)) C(k)^{[ℛ]×Δ^{op}}

carries cofibrant algebras to collections of colim-adapted simplicial complexes.

13Note that 3.1.3.5 is applicable since tensor product in QC(k) commutes with colimits along each one of the arguments.

Proof. Assertion (a). We prove that ℛ^{C} is homotopically sound for any category C. First of all, let us prove admissibility of ℛ^{C}. Let A be an ℛ^{C} algebra and M a cofibrant contractible complex. Choose a color (c, m) of ℛ^{C}. We have to prove that the map A → B is a weak equivalence, where B is obtained from A by freely joining M at color (c, m). In other words, we have to check that for each n ∈ C the map A(n) → B(n) is a weak equivalence. Note that B(n) is freely generated over A(n) as ℛ-algebra by Hom_{C}(m, n) × M which is also cofibrant and contractible. Therefore, ℛ^{C} is admissible by the criterion 2.3.3. It remains to make sure ℛ^{C} is Σ-cofibrant. The automorphism group of a collection (c, m) of colors in 𝒫^{C} is a subgroup of the automorphism group of the collection c in ℛ. Thus, if ℛ is Σ-cofibrant, ℛ^{C} is Σ-cofibrant as well. This proves the assertion a). Assertion (b). Here we follow, with minor amendments, Lurie [L.HA], 4.1.4.13 14. A morphism of simplicial complexes X Y will be called colim-adapted, if

• X, Y are colim-adapted.

• For each n ∈ Δ^{op} the map X_{n} Y_{n} is a cofibration in C(k).

• The map colim X colim Y is a cofibration in C(k). The collection of colim-adapted morphisms satisfies the following properties (proven in [L.HA], 4.1.4.13).

• Let X be a cofibrant complex. Then the constant simplicial complex defined by X is colim-adapted.

• Let f, g be two colim-adapted maps. Then f ∧ g defined as in (26) is colim-adapted.

• All cofibrations in C(k)^{Δ^{op}} are colim-adapted.

• The collection of colim-adapted morphisms is closed under transfinite composition.

• Base change: if f : X → Y and Z in a pushout diagram

(51)  [figure: X; f; Y; Z; f′; T]

are colim-adapted, then f′ is also colim-adapted. Let us now prove assertion (b). Obviously, a retract of a colim-adapted simplicial complex is colim-adapted, so we may assume that the cofibrant algebra A is a colimit of a transfinite sequence of cofibrations A = colim A_{α} where A^{0} is the initial ℛ^{Δ^{op}}-algebra, A^{α} ≃ colim_{β<α} A_{β} for α limit ordinal, and such that A_{α+1} is obtained from A_{α} via a pushout diagram

14We use the term ”colim-adapted” instead of Lurie’s ”good”.

(52)  [figure: 𝔽_{ℛ^{Δ^{op}}} (V); 𝔽_{ℛ^{Δ^{op}}} (W); A_{α}; A_{α+1}],

where V → W is a cofibration of cofibrant objects in C(k)^{[ℛ]×Δ^{op}} . According to 2.8.3, A_{α+1}, considered as an object of C(k)^{[ℛ]×Δ^{op}} , is obtained as a colimit of arrows obtained by pushforward along the maps (27). Since the components of A_{c(i)} are cofibrant complexes and ℛ is homotopically sound, the properties of colim-adapted maps listed above imply that the map 𝔾(A_{α}) → 𝔾(A_{α+1}) is colim-adapted. This proves the assertion.

□

4.3.3. To prove assertion 4, we will need a minor generalization of [L.HA], 3.1.3.11, describing free algebras generated by a collection of objects corresponding to different colors. Let 𝒪^{⊗} be an ∞-operad and a : I → [𝒪] a collection of objects in 𝒪. Let Θ ⊂ Fin_{∗} denote the subcategory defined by the inert arrows in Fin_{∗}, so that Θ is the “trivial ∞-operad”. We identify Θ^{I} with the category whose objects are finite sets over I and whose morphisms are inert partial maps over I. There is an (essentially unique) extension of the map a : I → 𝒪 to a map θ : Θ^{I} → 𝒪^{⊗} of ∞-operads. Let q : 𝒞^{⊗} → 𝒪^{⊗} be an 𝒪-monoidal ∞-category. A collection of objects X = {X_{i} ∈ 𝒞_{i}}, i ∈ I defines (essentially uniquely) a Θ^{I}-algebra X̄ in 𝒞 such that q ∘ X̄ = θ : Θ^{I} → 𝒪^{⊗}. Let F ∈ Alg_{𝒪} (𝒞). The lemma below allows one to check whether a given morphism X̄ → θ^{∗}(F) exhibits F as free 𝒪-algebra generated by the collection {X_{i}}, i ∈ I. Denote Θ_{iso}^{I} the maximal subgroupoid of Θ^{I}. In other words, this is the groupoid of finite sets over I. For each y ∈ 𝒪 we define a Kan simplicial set 𝒫_{I,y} as the full sub(∞-)category of Θ_{iso}^{I} ×_{𝒪^{⊗}} 𝒪_{/y}^{⊗} spanned by the objects whose component in 𝒪_{/y}^{⊗} is given by an active arrow. One has a canonical map h : 𝒫_{I,y} × Δ^{1} 𝒪^{⊗} defined as follows. Its restriction h_{0} to 𝒫_{I,y} × {0} is the composition 𝒫_{I,y} → Θ_{iso}^{I} → 𝒪^{⊗} whereas the restriction h_{1} to 𝒫_{I,y} × {1} carries everything to {y} ∈ 𝒪. In general, if π is a k-simplex of 𝒫_{I,y} and σ_{i} is a k-simplex of Δ^{1} having i times value 1 (i = 0, …, k + 1), then the image of (π, σ_{i}) is defined by the formula

(53)  h(π, σ_{i}) = \{ \begin{matrix}d_{k+1}τ, & i = 0 \\ s_{k−i+1}^{i−1}d_{k−i+1}^{i}τ, & i > 0 \end{matrix}

where τ is the k + 1-simplex of 𝒪^{⊗} defining the projection of π to 𝒪_{/y}^{⊗}. Now, the map q : 𝒞^{⊗} → 𝒪^{⊗} being cocartesian fibration, the map h : 𝒫_{I,y} ×

Δ^{1}  𝒪^{⊗} can be lifted to a map H : 𝒫_{I,y}×Δ^{1}  𝒞^{⊗} so that the restriction

H_{0} to 𝒫_{I,y} × {0} is the composition

𝒫_{I,y} Θ_{iso}^{I} ^{X̄} 𝒞^{⊗}

and such that for each π ∈ 𝒫_{I,y} the arrows H(π, Δ^{1}) is cocartesian. This yields a map H_{1} : 𝒫_{I,y} 𝒞_{y}.

Definition. The colimit of H_{1} constructed above, if exists, is denoted Sym_{𝒪} (X̄)_{y}.

Let now, X̄ be as above, and let A be a a 𝒪-algebra in 𝒞. A choice of a map of Θ^{I}-algebras f : X̄ → θ^{∗}(A), which is given essentially by a collection of maps f_{i} : X_{i} A(i), i ∈ I, defines a canonical collection of maps

(54)  F_{y} : Sym_{𝒪} (X̄)_{y}  A_{y}

as follows. The map f : X̄ → θ^{∗}(A) induces a map

H_{0}  A ∘ h_{0} : 𝒫_{I,y}  𝒞^{⊗}.

By construction of Sym_{𝒪} (X̄)_{y}, one obtains a canonical map

H  A ∘ h : 𝒫_{I,y} × Δ^{1}  𝒞^{⊗}

whose restriction to 𝒫_{I,y} × {1} yields a map F_{y} : Sym_{𝒪} (X̄)_{y} A_{y}. The following lemma is a straightforward generalization of [L.HA], 3.1.3.11.

4.3.4. Lemma. A collection of maps f_{i} : X_{i} → θ^{∗}(A)_{i}, i ∈ I, exhibits A as a free algebra generated by X iff for all y ∈ 𝒪 the natural map

F_{y} : Sym_{𝒪} (X̄)_{y} A_{y}

is an equivalence.

□

We will now apply the above lemma to prove Assertion 4. In our context 𝒪^{⊗} is the ∞-operad constructed from a topological operad 𝒪. We put I = [𝒪] and we represent the collection of X_{i} ∈ QC(k) by their cofibrant representatives Y_{i}. Let 𝔽_{ℛ} (Y) be the free ℛ-algebra on Y = {Y_{i}}_{i∈[𝒪]} . We denote 𝔽 to be the 𝒪-algebra in QC(k) corresponding to 𝔽_{ℛ} (Y) as explained in 4.2. We have canonical maps Y_{i} → θ^{∗}(𝔽)_{i}, so we can apply the above lemma. It remains to check that the maps F_{y} : Sym_{𝒪} (X̄)_{y} → 𝔽_{y} are equivalences. Recall that Θ_{iso}^{I} identifies with the groupoid of collections Fin/[𝒪] used in the description of the (classical) free algebra, see 2.2.5. One has a canonical projection 𝒫_{I,y} → NFin/[𝒪] and n-simplices of 𝒫_{I,y} over σ : c_{0} → … → c_{n} in NFin/[𝒪] correspond to n-simplices of 𝒪(c_{0}, y). The map Sym_{𝒪} (X̄)_{y} 𝔽_{y} is constructed as follows. The map H_{1} : 𝒫_{I,y} QC is the composition

(55)  𝒫_{I,y}  NFin/[𝒪]  QC,

where the second arrow carries a collection c : J → [𝒪] to ⊗_{j∈J}Y_{c(j)}. The canonical maps Y_{i} θ^{∗}(𝔽)_{i} allow one to extend H_{1} to a functor

H_{1}^{▷} : 𝒫_{I,y}^{▷} QC

carrying the vertex on the left to 𝔽_{y} = colim 𝔉(Y)_{y}, where the functor 𝔉(Y)_{y} : Fin/[𝒪] C(k) is given by the formula 𝔉(Y)_{y}(c) = ℛ(c, y) ⊗ ⨂_{j∈J} Y_{c(j)}, see formula (4). The extension H_{1}^{▷} is constructed as follows. One chooses a section s of the projection N_{∗}(ℛ(c, y)) N_{∗}(C_{∗}(𝒪(c, y))). We will denote by the same letter the composition s : 𝒪(c, y) N_{∗}(C_{∗}(𝒪(c, y))) N_{∗}(ℛ(c, y)). Now, given an n-simplex (σ, τ) of 𝒫_{I,y}, where σ : c_{0} → … → c_{n} in NFin/[𝒪] and τ ∈ 𝒪(c_{0}, y)_{n}, one extends it with the map

C_{∗}(Δ^{n}) ⊗ ⨂_{j∈J_{0}} Y_{c_{0}(j)} 𝔽_{y}

defined by the map C_{∗}(Δ^{n}) ℛ(c_{0}, y) determined by s(τ). It remains to check that H_{1}^{▷} : 𝒫_{I,y}^{▷} QC(k) is a colimit diagram. In fact, since the functor H_{1} factors through NFin/[𝒪], see (55), colim H_{1} can be calculated as the colimit of the left Kan extension of H_{1} via the projection

𝒫_{I,y} NFin/[𝒪].

This functor is a Kan fibration, so by [L.T], 4.3.3.1 the left Kan extension of H_{1} is precisely the functor 𝔉(Y)_{y}. Since ℛ is Σ-cofibrant, its (naive) colimit calculates as well the required homotopy colimit.

□

4.4. Algebras over a PROP. Theorem 4.1.1 allows one to get a certain rectification result for algebras over a topological PROP, or, more generally, over any SM topological category. A topological SM category 𝒫 determines a SM ∞-category 𝒫^{⊗}, see 4.1, and the ∞-category of algebras Fun^{⊗}(𝒫^{⊗}, QC(k)^{⊗}). We are going to give a “classical” description of this notion of ∞-algebra.

4.4.1. Definition. Let ℛ be a symmetric monoidal dg category. A homotopy ℛ-algebra in C(k) is a lax SM functor A : 𝒫 C(k) such that the natural map

A(x) ⊗ A(y) A(x ⊗ y)

induces a quasiisomorphism A(x) ⊗^{L} A(y) A(x ⊗ y) for all x, y ∈ 𝒫.

We denote by 𝒫^{o} the dg operad defined by 𝒫. A lax SM functor 𝒫 → C(k) is just a strict 𝒫^{o}-algebra in C(k). We need a minor generalization of the above definition. A dg operad ℛ will be called weak SM category if the corresponding operad enriched over the derived category of k is an (enriched) SM category. The only example of weak SM category we need is the following.

4.4.2. Lemma. Let 𝒫 be a SM dg category and let ℛ → 𝒫^{o} be a homotopically sound replacement of dg operads. Then ℛ is a weak SM category.

□

If ℛ is a weak SM category and x, y ∈ [ℛ] two objects, tensor product x ⊗ y is defined uniquely up to equivalence. Then Definition 4.4.1 is applicable also for weak SM categories. We will repeat it once more.

4.4.3. Definition. Let ℛ be a weak SM dg category. A homotopy ℛ-algebra in C(k) is an algebra over the operad ℛ such that the natural map

A(x) ⊗ A(y) A(x ⊗ y)

induces a quasiisomorphism A(x) ⊗^{L} A(y) A(x ⊗ y) for all x, y ∈ [ℛ].

Let 𝒫 be a topological SM category and let ℛ C_{∗}(𝒫^{o}, k) be a homotopically sound replacement. The ∞-category of homotopy ℛ-algebras is defined as the full subcategory of N(Alg_{ℛ}^{st}(C(k))) consisting of homotopy ℛ-algebras. It can be otherwise described as the DK localization of the category of cofibrant homotopy ℛalgebras, with respect to weak equivalences. Theorem 4.1.1 immediately implies the following result.

4.4.4. Corollary. Let 𝒫 be a topological SM category, and let ℛ → C_{∗}(𝒫, k) be a homotopically sound replacement of C_{∗}(𝒫, k) considered as an operad. Let 𝒫^{⊗} be the SM ∞-category defined by 𝒫. Then the equivalence of ∞-categories

Φ : N(Alg_{ℛ}^{st}(C(k)) Alg_{P^{⊗}}(QC(k))

induces an equivalence of the subcategory of homotopy ℛ-algebras with Fun^{⊗}(𝒫^{⊗}, QC(k)^{⊗}).

□

5. Modules

In this section we deduce from Theorem 4.1.1 the rectification for modules over operad algebras. Our definition of module over an 𝒪-operad algebra is very straightforward. For any ∞-operad 𝒪 we define a new ∞-operad denoted M𝒪 such that algebras over M𝒪 are pairs (A, M) where A is an 𝒪-algebra and M is an A-module, see Subsection 5.2. Theorem 4.1.1 implies the main result of this section Theorem 5.2.3 saying that the ∞-category of modules over an operad algebra can be described as the infinity category underlying the corresponding model category. The precise formulation is given in 5.2.3. The proof is based on the result on localization of families of ∞-categories given in [H.L], Sect. 2. In his foundational book [L.HA] J. Lurie gives another definition which, under some restrictions, yields for any 𝒪-algebra A an 𝒪-monoidal category of Amodules. In Appendix B we show that our definition is equivalent to the one suggested by Lurie, with discarded 𝒪-monoidal structure.

5.1. Classical setting. Let 𝒪 be a topological operad. We define a new topological operad M𝒪 as follows. We double each color, defining

[M𝒪] = [𝒪] × {a, m}.

A collection of colors in M𝒪 is given by a pair of maps c̃ = (c, X_{c}) where c : I → [𝒪], X_{c} : I → {a, m}. The space of operations M𝒪(c̃, d̃) for c̃: I → [M𝒪], d̃∈ [M𝒪] is nonempty in two cases described below.

• X_{c}(i) = a for all i ∈ I, X_{d} = a.

• X_{c}(i) = m for precisely one i ∈ I, X_{d} = m.

In these cases M𝒪(c̃, d̃) = 𝒪(c, d). Note that the same construction makes sense for operads enriched over any SM category. One has a map 𝒪 → M𝒪 carrying any c ∈ [𝒪] to (c, a). In the opposite direction, a map M𝒪 → 𝒪 erases the X-marking of a color. Algebras over M𝒪 are pairs (A, M) where A is an 𝒪-algebra and M is a Amodule.

5.2. A similar construction makes sense in the context of ∞-operads. Given an ∞-operad 𝒪^{⊗} we define a new operad M𝒪^{⊗} by the formula

M𝒪^{⊗} = CM^{⊗} ×_{NFin_{∗}} 𝒪^{⊗},

where CM is the two-color operad governing pairs (A, M) with A a commutative algebra and M an A-module, and CM^{⊗} is the corresponding ∞-operad.

5.2.1. Definition. Let 𝒪^{⊗} be an ∞-operad and 𝒞^{⊗} a SM ∞-category. Let A ∈ Alg_{𝒪} (𝒞). The ∞-category Mod_{A}^{𝒪}(𝒞) is defined as the fiber product

Mod_{A}^{𝒪}(𝒞) = Alg_{M𝒪} (𝒞) ×_{Alg_{𝒪} (𝒞)} {A}

Let 𝒪 be a topological operad and let ℛ C_{∗}(𝒪) be a homotopically sound replacement. We are not sure that Mℛ is always homotopically sound. The following, however, is very easy.

5.2.2. Lemma. Let ℛ be a Σ-split operad in C(k). Then Mℛ is also Σ-split. If ℛ is Σ-cofibrant, then Mℛ is Σ-cofibrant.

Proof. If the collection of maps t_{θ} : ℛ(c, d) → ℛ(c, d), θ : I → ⟨|I|⟩, provides a Σ-splitting for ℛ, the same maps provide a Σ-splitting for nonzero components of Mℛ in the notation of 5.1 and 2.5. The second claim is also obvious. □

In any case, we can choose a homotopically sound replacement

ℳ C_{∗}(M𝒪)

and define ℛ as the full suboperad spanned by the original colors of 𝒪. Then by 2.3.3 ℛ is a homotopicaly sound replacement of C_{∗}(𝒪).

5.2.3. Theorem. Let 𝒪 be a topological operad. Let ℳ be a homotopically sound replacement of C_{∗}(M𝒪) and let ℛ be the full suboperad of ℳ spanned by the original colors of 𝒪. Let A be an ℛ-algebra in C(k). There is a canonical equivalence

(56)  N(Mod_{A}^{ℛ}(C(k))) → Mod_{A}^{𝒪}(QC(k)),

where we denote by the same letter A the corresponding 𝒪-algebra in QC(k).

Proof. According to Theorem 4.1.1 one has a commutative diagram of ∞- categories whose horizontal maps are equivalences.

(57)

N(Alg_{ℳ}^{st} (C(k))) \overline{}Alg_{M𝒪} (QC(k))

[figure: N(Alg_{ℛ}^{st}(C(k))); ϕ^{st}; Alg_{𝒪} (QC(k)); ϕ]

This yields an equivalence of the homotopy fibers of the vertical maps. It remains to identify the map of homotopy fibers with the map (56). The forgetful functor

(58)  Alg_{ℳ}^{st} (C(k))  Alg_{ℛ}^{st}(C(k))

inducing the left vertical arrow ϕ^{st}, preserves cofibrant algebras. This map, restricted to the subcategories spanned by the cofibrant objects, and considered as marked ∞-categories (quasiisomorphisms being the marked arrows), is a marked cocartesian fibration in the sense of Definition 2.1.1, [H.L]: this is a cocartesian fibration of categories (a map f : A → A′ of algebras gives rise to base change map f^{∗} : Mod_{A} → Mod_{A′}), the base change preserves quasiisomorphisms of cofibrant modules, and weak equivalence of cofibrant algebras gives rise to equivalence of the corresponding categories of modules). Then, according to Proposition 2.1.4, [H.L], the (homotopy) fibers of ϕ^{st} identify with the DK localizations of the fiber of the functor

(59)  ϕ^{st} : Alg_{ℳ}^{st} (C(k))^{c}  Alg_{ℛ}^{st}(C(k))^{c}

at a cofibrant algebra A. If we had ℳ = Mℛ, the fiber would be precisely the category of cofibrant Amodules. In general one has to add a few lines.

We will now present a mini-theory, generalizing to algebras over colored operads the notion of enveloping algebra. Let ℛ be a full suboperad of an operad ℳ such that the following “linearity” condition holds. Let c : I → [ℳ] and d ∈ [ℳ] satisfy the condition ℳ(c, d) ≠ 0. Then either both d and the image of c belong to [ℛ], or d ∉ [ℛ] and there is precisely one i ∈ I such that c(i) ∉ [ℛ]. Given a pair of operads ℳ ⊃ ℛ satisfying the linearity condition, and an ℛ-algebra A in C(k), we can look at the fiber of the functor Alg_{ℳ} (C(k)) → Alg_{ℛ} (C(k)) at A as the category of generalized A-modules. In case ℳ = Mℛ this fiber is the category of A-modules. Keeping in mind this analogy, we will denote it Mod_{A}^{ℳ} .

The category Mod_{A}^{ℳ} can be easily described as the category of representations (that is, C(k)-enriched functors to C(k)) of a category enriched over C(k) which we will call the enveloping category of A and will denote U_{ℛ}^{ℳ} (A). Its objects are the elements of [ℳ]−[ℛ]. The complex of maps from x to y, x, y ∈ [ℳ]−[ℛ], is a certain colimit of tensor products of ℳ(c, d) along a category of marked trees. In case ℛ is a colorless operad and ℳ = Mℛ, we recover the classical notion of universal enveloping algebra. In case ℛ is a full suboperad of two operads ℳ and ℳ′, a map of operads f : ℳ → ℳ′ is said to be over ℛ if f|_{ℛ} = id. If A is a cofibrant ℛ-algebra and f : ℳ → ℳ′ is a quasiisomorphism of operads over ℛ, the induced map

U_{ℛ}^{ℳ} (A) U_{ℛ}^{ℳ′} (A)

is an equivalence of dg categories. This can be checked precisely as in the colorless case, see [H], 5.3.3. We can now complete our proof applying the above claim to ℛ, ℳ as in the theorem and ℳ′ = Mℛ. □

Appendix A. Symmetric monoidal adjunction

Let F : C [figure] D : G be an adjoint pair of symmetric monoidal categories, so that F is a symmetric monoidal functor. Then it is easy to see that G is automatically lax symmetric monoidal. In this subsection we study the above phenomenon and its generalizations in the context of ∞-categories.

A.1. Fibrations in Cat_{∞}. We are going to use the notions of left or cocartesian fibration applied to arrows of Cat_{∞} rather than of sSet as in [L.T], 2.1 and 2.4. Here are the appropriate definitions. A map f : X → Y in Cat_{∞} is called a left fibration if the diagram below defined by f

(60)  [figure: X^{Δ^{1}}; Y^{Δ^{1}}; X^{{0}}; f; Y^{{0}}]

is cartesian. Equivalently, this means that a map f is equivalent to one represented by a left fibration in sSet. Similarly, one defines a cocartesian fibration in Cat_{∞} as a map equivalent to one represented by a cocartesian fibration in sSet. Let X, Y be ∞-categories. A map f : X → Y in sSet represents a cocartesian fibration in Cat_{∞} if it can be embedded into a homotopy commutative triangle of ∞-categories

(61)  [figure: X; h; Z; Y; g; f],

where g is a cocartesian fibration in sSet and h is an equivalence of ∞- categories. Given an ∞-category C, we denote as Coc_{C} the subcategory of (Cat_{∞})_{/C} spanned by cocartesian fibrations, with arrows preserving the cocartesian liftings. The category Left_{C} is the full subcategory of (Cat_{∞})_{/C} spanned by the left fibrations. One also defines the ∞-category Left as the full subcategory of Fun(Δ^{1}, Cat_{∞}) spanned by the left fibrations. The ∞-cateory Left_{C} is the fiber at C of the cartesian fibration e_{1} : Fun(Δ^{1}, Cat_{∞}) → Cat_{∞} assigning to each arrow in Cat_{∞} its target.

A.2. SM Grothendieck construction. Recall that for an ∞-category C there is an equivalence

(62)  Coc_{C}  \overline{}^{∼} Fun(C, Cat_{∞}).

In this subsection we will describe symmetric monoidal versions of this correspondence. A map of SM ∞-categories is called SM cocartesian fibration, see [L.HA], 2.1.2.13, if it is presented by a cocartesian fibration of the corresponding ∞- categories over NFin_{∗}. In the following proposition Coc_{C^{⊗}}^{SM} denotes the category of SM cocartesian fibrations over C^{⊗}.

A.2.1. Proposition. Let C^{⊗} be a SM ∞-category. There is an equivalence

(63)  Coc_{C^{⊗}}^{SM}  \overline{}^{∼} Fun^{lax}(C^{⊗}, Cat_{∞}),

compatible with the equivalence (62).

Proof. Since Cat_{∞} is cartesian closed, the right hand side identifies with the full subcategory of Fun(C^{⊗}, Cat_{∞}) spanned by the functors F : C^{⊗} Cat_{∞} which are lax cartesian structures in the sense of [L.HA], 2.4.1.1: any object X = X_{1} ⊕ … ⊕ X_{n} with X_{i} ∈ C exhibits F (X) as the product of F (X_{i}). Now the claim follows from the equivalence of two definitions of operad cocartesian fibration, see [L.HA], 2.1.2.12. □

The following result is an immediate consequence of the above.

A.2.2. Corollary. Let C^{⊗} be a SM ∞-category. There is an equivalence

(64)  Left_{C^{⊗}}^{SM}  \overline{}^{∼} Fun^{lax}(C^{⊗}, 𝒮)

between the ∞-category of left fibrations M^{⊗} C^{⊗} which are SM functors and lax SM functors C^{⊗} 𝒮, compatible with the Grothendieck construction.

The ∞-category Left_{C^{⊗}}^{SM} has a simple interpretation in terms of Left. The latter ∞-category has a cartesian SM structure. One has

A.2.3. Lemma. Let C^{⊗} be a SM ∞-category. The obvious functor

(65)  Left_{C^{⊗}}^{SM}  Alg_{Com}(Left)

identifies the left-hand side with the fiber of the forgetful functor

                      Alg_{Com}(Left)  Cat_{∞}^{SM}

at C^{⊗}.

Proof. We have to check that if a SM functor f : D^{⊗} C^{⊗} induces a left fibration D C of the respective ∞-categories, then f itself is a left fibration. Applying Proposition 2.4.2.11 of [L.T], we get that f is a locally cocartesian fibration. Moreover, the same proposition gives a description of locally cocartesian arrows: these are arrows α : d d′ embeddable into a commutative triangle

(66)  [figure: d; α; d′; d′′; γ; β] ,

where β is π_{D}-cocartesian for π : D^{⊗} NFin_{∗}, and π(γ) = id. Obviously, these are all arrows in D^{⊗}. Therefore, f is a left fibration. □

A.3. SM pairings. A pairing of ∞-categories is a pair of maps C M D, such that the induced map M C × D is a left fibration ^{15}. The ∞-category of pairings, Pair, is defined as the full subcategory of the category Fun(Λ_{0}^{2}, Cat_{∞}), spanned by the diagrams giving rise to a left fibration. Equivalently, Pair can be defined by a cartesian diagram

(67)  [figure: Pair; Left; Cat_{∞} × Cat_{∞}; ×; Cat_{∞}].

A pairing is uniquely defined, up to a usual ambiguity, by a corresponding functor to the category of spaces C × D 𝒮. The forgetful functor Pair Cat_{∞} × Cat_{∞} induces a functor

                  Alg_{Com}(Pair)  Cat_{∞}^{SM} × Cat_{∞}^{SM}.

For a pair (C, D) of SM ∞-categories we denote as Alg_{Com}(Pair)_{(C,D)} the fiber of this functor at (C, D).

15Note: Lurie [L.X], 3.1 and 4.2, uses right fibrations instead.

According to Corollary A.2.2 and Lemma A.2.3, one has an equivalence

(68)  Alg_{Com}(Pair)_{(C,D)}  \overline{}^{∼} Fun^{lax}(C × D, 𝒮).

A.4. Representability. A pairing (p, q) : M C × D is called leftrepresentable if for any x ∈ C the fiber p^{−1}(x) has an initial object. We define Pair^{ℓ} as the full subcategory of Pair spanned by the leftrepresentable pairings ^{16}. A left pairing p : M → C × D corresponds to a functor C × D 𝒮 which can be equivalently converted into a functor p̃: C P (D^{op}). ^{17} Then p is left representable iff p̃ factors through Yoneda embedding D^{op} P (D^{op}). More precisely, one has the following.

A.4.1. Lemma. The equivalence

Pair_{(C,D)}  \overline{}^{∼} Fun(C × D, 𝒮)

identifies the full subcategory Pair_{(C,D)}^{ℓ} on the left with Fun(C, D^{op}) on the right.

Proof. We have to verify that the natural map Fun(C, D^{op}) Fun(C×D, 𝒮) is fully faithful. This is the composition of the equivalence

Fun(C, P (D^{op})) Fun(C × D, 𝒮)

with the map

Fun(C, D^{op}) Fun(C, P (D^{op}))

which is fully faithful by Yoneda lemma. □

Pair^{ℓ} is closed under direct products. Therefore, commutative algebras in Pair^{ℓ} form a full subcategory of Alg_{Com}(Pair). Our aim is to prove the following SM version of Lemma A.4.1.

A.4.2. Proposition. The equivalence (68)

Alg_{Com}(Pair)_{(C,D)}  \overline{}^{∼} Fun^{lax}(C × D, 𝒮)

identifies the full subcategory Alg_{Com}(Pair^{ℓ})_{(C,D)} on the left with Fun^{lax}(C, D^{op}) on the right.

The proof is given in A.4.3–A.4.4 below.

16This differs from the category of left-representable pairings considered in [L.X], 4.2.7 where the arrows are required to be left-representable.

17Here P (X) denotes the presehaves on X.

A.4.3. Opposite S-family. Given a cocartesian fibration p : C S corresponding to p′ : C Cat_{∞}, we define a cocartesian fibration p^{∘} : C^{∘} S as the one corresponding to the composition of p′ with the functor X ↦ X^{op}. More explicitly, if p is presented by a cocartesian fibration C S in sSet, the cocartesian fibration C^{∘} S is defined by the formula

(69) Hom_{S}(T, C^{∘}) = Hom^{ℓ}(T ×_{S} C, 𝒮) :=

{f ∈ Hom(T ×_{S} C, 𝒮)|∀t ∈ T f_{t} : {t} ×_{S} C → 𝒮 is corepresentable }.

Note that the formula (69) yields immediately

(70)  Fun_{S}(T, C^{∘}) = Fun^{ℓ}(T ×_{S} C, 𝒮),

where Fun^{ℓ}(T ×_{S} C, 𝒮) is defined as

Fun^{ℓ}(T ×_{S} C, 𝒮)_{n} = Hom^{ℓ}((T × Δ^{n}) ×_{S} C, 𝒮).

This is a full subcategory of Fun(T ×_{S} C, 𝒮) consisting of left-representable functors.

A.4.4. Proof of A.4.2. We will apply the formula (70) to S := NFin_{∗}, T := C^{⊗}, C := D^{⊗}. The right-hand side of (70) contains the full subcategory

Fun^{ℓ,lax}(C^{⊗} ×_{N Fun_{∗}} D^{⊗}, 𝒮)

spanned by the left-representable lax cartesian structures on C^{⊗} ×_{N Fun_{∗}} D^{⊗} in the sense of [L.HA], 2.4.4.1. We will now check that the corresponding full subcategory of Fun_{NFin_{∗}}(C^{⊗}, D^{∘⊗}) coincides with Fun^{lax}(C, D^{∘}). In fact, let f : C^{⊗} ×_{NFin_{∗}} D^{⊗} 𝒮 be lax cartesian and left-representable. Recall that f is lax cartesian if for c = ⊕_{i=1}^{n}c_{i}, d = ⊕_{i=1}^{n}d_{i} the natural map f(c, d) ∏ f(c_{i}, d_{i}) defined by the inert maps ρ^{i} : ⟨n⟩ → ⟨1⟩, is an equivalence. In this case left representability can be checked at ⟨1⟩ ∈ NFin_{∗} only; it will follow automatically for all ⟨n⟩ ∈ NFin_{∗}. Left representability of f yields a map f̃ : C^{⊗} D^{∘⊗} over NFin_{∗}. If c = ⊗c_{i} and d = f̃(c), we have immediately d = ⊕f̃(c_{i}) which is equivalent to preservation of inert arrows. This proves Proposition A.4.2.

A.5. A generalization. Proposition A.4.2 has an obvious (and important) generalization to multi-variable adjunction. Let K be a subset of {1, …, n}. A functor F : C_{1} × … × C_{n} 𝒮 will be called K-representable if the corresponding functor F′ : ∏_{i∈K} C_{i} P (∏_{i∉K} C_{i}^{op}) factors through the fully faithful embedding

∏_{i∉K} C_{i}^{op} P ( ∏_{i∉K} C_{i}^{op}).

Let now C_{i} be symmetric monoidal. The claim below directly follows from Proposition A.4.2.

A.5.1. Corollary. There is an equivalence between the following ∞- categories.

1. Fun^{lax}(∏_{i∈K} C_{i}, ∏_{i∉K} C_{i}^{∘}),

2. The full subcategory of Fun^{lax}(∏_{i=1}^{n} C_{i}, 𝒮) spanned by the lax functors which are K-representable, once the SM structure is discarded.

A.5.2. Example. In particular, if a SM functor f : C D admits a right adjoint as a functor between ∞-categories, its right adjoint has a canonical lax SM structure, see also [H.L], 3.1.1.

A.5.3. Example. A symmetric monoidal functor f : C D determines on D a C-tensored structure, defined by a functor

C × D D, (x, y) ↦ f(x) ⊗ y

which is also symmetric monoidal. This implies that, if f induces also a Ccotensored structure on D

C^{op} × D D,

it is automatically lax symmetric monoidal.

Appendix B. Comparison of two notions of module

B.1. In this appendix we assume that the operad 𝒪^{⊗} is unital (see [L.HA], 2.3.1), that is that the space Map(∅, x) is contractible for any x ∈ 𝒪. Here ∅ belongs to the contractible space 𝒪_{⟨0⟩}^{⊗}. Denote 𝒮_{𝒪} the full subcategory of Fun(Δ^{1}, 𝒪^{⊗}) spanned by the semi-inert arrows (see [L.HA], 3.3.1) x → y in 𝒪^{⊗} with p(x) = ⟨1⟩ ∈ Fin_{∗} ^{18}. The maps s, t : 𝒮_{𝒪} 𝒪^{⊗} assign to an edge its source and its target, respectively.

B.1.1. Lemma. The map t : 𝒮_{𝒪} 𝒪^{⊗} is a categorical fibration.

Proof. The map is the composition 𝒮_{𝒪} → Fun(Δ^{1}, 𝒪^{⊗}) ^{e_{1}} 𝒪^{⊗}. The second map is a cartesian fibration by [L.T], 2.4.7.11 and 2.4.7.5. In particular, it is a categorical fibration. The first map is an embedding as a full subcategory, so is an inner fibration. Now Joyal’s criterion [L.T], 2.4.6.5 immediately shows it is also a categorical fibration.

□

An edge α in 𝒮_{𝒪} will be called inert if both s(α) and t(α) are inert edges in 𝒪^{⊗}. Note that s(α) has to be an equivalence since it lives over an inert endomorphism of ⟨1⟩ ∈ Fin_{∗} which has to be identity.

18In the notation of Lurie [L.HA], 3.3.2.1, 𝒮_{𝒪} is the fiber of the composition p∘e_{0} : 𝒦_{𝒪}^{⊗} → 𝒪^{⊗} → NFin_{∗} at ⟨1⟩.

The assignment 𝒪^{⊗} ↦ 𝒮_{𝒪} is functorial. One can identify 𝒮_{Com} with a subpreoperad of CM^{⊗♮} via the map

(71)  ι : 𝒮_{Com}        CM^{⊗♮}

carrying an arrow α : ⟨1⟩ → I_{∗} to the characteristic function h : I → {a, m} of the image of α: h has value m on the image and a otherwise.

The commutative diagram

(72)

                    [figure: 𝒮_{𝒪}; 𝒮_{Com}; CM^{⊗}]

[figure: 𝒪^{⊗}; t; NFin_{∗}; t; = = = =; NFin_{∗}]

defines the maps

(73)  𝒮_{𝒪}  ^{π} 𝒮_{Com} ×_{NFin_{∗}} 𝒪^{⊗} ^{ι} CM^{⊗} ×_{NFin_{∗}} 𝒪^{⊗}.

The following result shows that Definition 5.2.1 of the category of modules over an ∞-operad algebra is equivalent to the one given by Lurie in [L.HA], 3.3.3.8. In particular, Corollary 5.2.3 is applicable to Lurie modules (with 𝒪-monoidal structure discarded).

B.1.2. Proposition. Assume that 𝒪^{⊗} is unital and 𝒪 is Kan (this is so if 𝒪^{⊗} is coherent [L.HA], 3.3.1.9). Then the maps π and ι in (73) are weak equivalences in Pop_{∞}.

The proof of the proposition is given in B.2.1–B.2.7. We will prove that the ι and ι ∘ π are weak equivalences in Pop_{∞}. This will be done using Lurie’s notion of approximation of operads. We check that both ι and ι ∘ π are approximations in the sense of [L.HA], 2.3.3.6. Then, using [L.HA], 2.3.3.23(1), we deduce that the maps ι and ι ∘ π are weak equivalences.

B.2. Proof of B.1.2.

B.2.1. ι is an approximation. The map ι is obtained from the embedding ι_{Com} : 𝒮_{Com} → CM^{⊗} by a base change along fibration p : 𝒪^{⊗} → NFin_{∗}. Therefore, by Remark 2.3.3.9 of [L.HA], in order to prove ι is an approximation, it suffices to check that the map ι_{Com} is an approximation. 𝒮_{Com} is a full subcategory of CM^{⊗} spanned by the objects having at most one appearance of m. Thus, if an arrow α : x → y in CM^{⊗} is inert and x ∈ 𝒮_{Com} then y ∈ 𝒮_{Com}. Similarly, if α is active and y ∈ 𝒮_{Com} then x ∈ 𝒮_{Com}. This implies ι_{Com} is an approximation.

We will now prove that ι ∘ π is also an approximation. This is done in B.2.2– B.2.6.

B.2.2. ι ∘ π is a categorical fibration. Let A → B be a trivial cofibration of simplicial sets in the Joyal model structure. Given a pair of compatible maps a : A → 𝒮_{𝒪} and b : B → CM^{⊗} ×_{NFin_{∗}} 𝒪^{⊗}, we have to find a lifting c : B → 𝒮_{𝒪} making two triangles commutative. We proceed as follows. Look at the commutative square with vertical arrows A → B and 𝒮_{𝒪} → 𝒪^{⊗}, as shown in the diagram.

(74)  [figure: A; 𝒮_{𝒪}; B; c; CM^{⊗} ×_{NFin_{∗}} 𝒪^{⊗}; ι∘π; 𝒪^{⊗}; t]

By Lemma B.1.1 there is a lifting c : B → 𝒮_{𝒪} making two triangles AB𝒮_{𝒪} and B𝒮_{𝒪} 𝒪^{⊗} commutative. We claim c makes also commutative the triangle we need. In fact, the commutative diagram

(75)  [figure: A; CM^{⊗}; B; NFin_{∗}]

has a unique lifting (as CM^{⊗} and NFin_{∗} are both nerves of categories, and the functor between them admits lifting of isomorphisms). This implies that the lifting c automatically satisfies the required property.

B.2.3. Property (1) of [L.HA], 2.3.3.6. If a : x → y is an object in 𝒮_{𝒪} and b̄ : p(y) → z̄ is an inert arrow in Fin_{∗}, we can lift b̄ to an inert edge b : y → z in 𝒪^{⊗} (so that p(b) = b̄). One has a triangle x ^{a} y ^{b} z in 𝒪^{⊗} that determines an edge in 𝒮_{𝒪} . Its image is obviously inert in CM^{⊗} ×_{NFin_{∗}} 𝒪^{⊗}.

B.2.4. Property (2) of [L.HA], 2.3.3.6. Here the assumptions of B.1.2 on 𝒪^{⊗} will be used. Let a : x → y be an object of 𝒮_{𝒪} and let (α, y) ∈ CM^{⊗} ×_{NFin_{∗}} 𝒪^{⊗} denote the image ι ∘ π(a). An active edge β : (γ, z) (α, y) in CM^{⊗} ×_{NFin_{∗}} 𝒪^{⊗} is uniquely determined by an active edge b : z → y in 𝒪^{⊗}, together with an element γ : p(x) → p(z) in 𝒮_{Com} such that p(a) = p(b) ∘ γ. We have to find a cartesian lifting for β, that is a 2-simplex in 𝒪^{⊗}

(76)  \begin{aligned}& z \\ x & \overline{^{a}} & ^{^{c}} & y & ^{^{b}} \end{aligned} ,

Documenta Mathematica 20 (2015) 879–926

such that p(c) = γ and satisfying a certain universal property. In case a is null, the map γ is null and we choose c to be the null map from x to z. The required 2-simplex is now essentially unique as Map(0, y) is contractible. In case a is not null let y = ⊕y_{i} with y_{i} ∈ 𝒪, z = ⊕z_{i} with z_{i} ∈ 𝒪^{⊗}, b = ⊕b_{i} : z_{i} → y_{i}. Let, furthermore, a be defined by an equivalence a_{k} : x → y_{k}. Write z_{k} = ⊕z^{j} where z^{j} ∈ 𝒪. The map γ : p(x) → p(z) factors through γ′ : p(x) → p(z_{k}) which singles out an element j in p(z_{k}). This allows one to produce an arrow z^{j} → y_{i} in 𝒪 obtained from b_{i} : z_{i} → y_{i} by precomposing with units. It should be equivalence as 𝒪 is Kan. Choose a two-simplex x → z^{j} → y_{k} with the described above edges x → y_{k} and z^{j} → y_{k}. Adding to it an essentially unique triangle

0 z ⊖ z^{j} ^{0} y ⊖ y_{k},

we get a triangle with required properties ^{19}. We claim that the edge in 𝒮_{𝒪} defined by the above two-simplex, is a ι ∘ πcartesian lifting of β : (γ, z) (p(a), y). The map f : 𝒮_{𝒪} CM^{⊗} ×_{NFin_{∗}} 𝒪^{⊗} is a categorical fibration, so the criterion 2.4.4.3 of [L.T] can be applied. We have to check that for any d : s → w in 𝒮_{𝒪} the following homotopy commutative diagram

(77)

Map_{𝒮_{𝒪}} (d, c)  \overline{}Map_{𝒮_{𝒪}} (d, a)

[figure: Map_{CM^{⊗}×_{NFin∗} 𝒪^{⊗}} (f(d), f(c)); Map_{CM^{⊗}×_{NFin∗} 𝒪^{⊗}} (f(d), f(a))]

is homotopy cartesian^{20}. Since 𝒮_{𝒪} is a full subcategory of Fun(Δ^{1}, 𝒪^{⊗}), we can replace Map_{𝒮_{𝒪}} in the above diagram with Map_{Fun(Δ^{1},𝒪^{⊗})} . Here the following easy lemma is very convenient.

B.2.5. Lemma. Let 𝒞 be an ∞-category and 𝒟 = Fun(Δ^{1}, 𝒞). Let a : x → y and a′ : x′ → y′ be two objects in 𝒟. Then one has a homotopy cartesian diagram

(78)

Map_{𝒟} (a, a′) \overline{}Map_{𝒞} (y, y′)

[figure: Map_{𝒞} (x, x′); Map_{𝒞} (x, y′)]

19Of course y ⊖ y_{k} = ⊕_{i≠k}y_{i} and similarly for z ⊖ z^{j}.

20To make this formulation precise, one has to replace the map spaces with their explicit representatives by Kan simplicial sets, so that the diagram (77) is commutative.

Proof. The formulation of the lemma is imprecise: the diagram (78) has to be replaced with an explicit commutative diagram of spaces, similarly to the diagram (77). It is described below. By definition, Map_{𝒞} (x, y′) is realized as the fiber of the categorical fibration ^{21}

Fun(Δ^{1}, 𝒞) Fun(∂Δ^{1}, 𝒞) = 𝒞^{2}

at the point (x, y′). Similarly, Map_{𝒟} (a, a′) is realized as the fiber of

Fun(Δ^{1} × Δ^{1}, 𝒞) = Fun(Δ^{1}, 𝒟) → Fun(∂Δ^{1}, 𝒟) = Fun(Δ^{1}, 𝒞)^{2}

at the point (a, a′). Furthermore, we replace the space Map_{𝒞} (y, y′) with Map′_{𝒞} (y, y′) defined as the fiber of the map

Fun(Δ^{2}, 𝒞) Fun(Δ^{1}, 𝒞) × 𝒞,

induced by ∂^{2} : Δ^{1} → Δ^{2} and ∂^{0}∂^{1} : Δ^{0} → Δ^{2}, at the point (a, y′). Similarly, we replace Map_{𝒞} (x, x′) with Map′_{𝒞} (x, x′) defined as the fiber of the map

Fun(Δ^{2}, 𝒞) 𝒞 × Fun(Δ^{1}, 𝒞),

induced by ∂^{0} : Δ^{1} → Δ^{2} and ∂^{1}∂^{2} : Δ^{0} → Δ^{2}, at the point (x, a′). The canonical maps Map′_{𝒞} (y, y′) → Map_{𝒞} (y, y′) and Map′_{𝒞} (x, x′) → Map_{𝒞} (x, x′) are trivial Kan fibrations as they can be obtained by base change from the trivial fibration

(79)  Fun(Δ^{2}, 𝒞)  Fun(Λ_{1}^{2}, 𝒞).

This, in particular, proves that all Map′-spaces are Kan. The commutative square

(80)

Map_{𝒟} (a, a′) \overline{}Map′_{𝒞} (y, y′)

[figure: Map′_{𝒞} (x, x′); Map_{𝒞} (x, y′)]

replacing the diagram (78), is now obtained from the following commutative cube (81)

21See Lurie, [L.T], 4.2.1, this is the realization via “alternative join”

Fun(Δ^{1} × Δ^{1}, 𝒞)  \overline{}Fun(Δ^{2}, 𝒞)

[figure: Fun(Δ^{2}, 𝒞); Fun(Δ^{1}, 𝒞)]

[figure: Fun(Δ^{1}, 𝒞)^{2}; Fun(Δ^{1}, 𝒞) × 𝒞]

[figure: 𝒞 × Fun(Δ^{1}, 𝒞); 𝒞^{2}]

as the fiber along the vertical edges at (a, a′) ∈ Fun(Δ^{1}, 𝒞)^{2} and at its images. Here the upper face is defined by the presentation of Δ^{1} ×Δ^{1} as a union of two 2-simplices glued along an edge. The upper and the lower faces are cartesian and homotopy cartesian. Therefore, the commutative square (80) of the fibers is also homotopy cartesian. □

B.2.6. End of the proof of Property (2). Recall that we have to verify that the diagram (77) is homotopy cartesian. Note that

Map_{CM^{⊗}×_{NFin∗} 𝒪^{⊗}} (f(d), f(c)) = (82)

Map_{𝒪^{⊗}} (w, z) ×_{Hom_{_{Fin}_{∗}} (p(w),p(z))} Hom_{𝒮_{Com}} (p(d), p(c)) =
Map_{𝒪^{⊗}} (w, z) ×_{Hom_{_{Fin}_{∗}} (p(s),p(z))} Hom_{Fin_{∗}}(p(s), p(x)),

and similarly

Map_{CM^{⊗}×_{NFin∗} 𝒪^{⊗}} (f(d), f(a)) = (83)

Map_{𝒪^{⊗}} (w, y) ×_{Hom_{_{Fin}_{∗}} (p(s),p(y))} Hom_{Fin_{∗}}(p(s), p(x)).

Let us first replace the diagram (77) with a commutative diagram so that the claim become formally meaningful. Similarly to what we did in the proof of Lemma B.2.5, we replace in (77) Map_{𝒮_{𝒪}} (d, c) and

Map_{CM^{⊗}×_{NFin∗} 𝒪^{⊗}} (f(d), f(c)) =

= Map_{𝒪^{⊗}} (w, z) ×_{Hom_{_{Fin}_{∗}} (p(s),p(z))} Hom_{Fin_{∗}}(p(s), p(x))

with homotopy equivalent versions, Map′_{𝒮_{𝒪}} (d, c) and

Map′_{𝒪^{⊗}} (w, z) ×_{Hom_{_{Fin}_{∗}} (p(s),p(z))} Hom_{Fin_{∗}}(p(s), p(x))

where Map′_{𝒮_{𝒪}} (d, c) is the fiber of the map

(84)  Fun(Δ^{2}, 𝒮_{𝒪} )  𝒮_{𝒪} × Fun(Δ^{1}, 𝒮_{𝒪} )

at (d, β : c → a) whereas Map′_{𝒪^{⊗}} (w, z) is the fiber of

(85)  Fun(Δ^{2}, 𝒪^{⊗})  𝒪^{⊗} × Fun(Δ^{1}, 𝒪^{⊗})

at (w, b : z → y). The maps (84) and (85) are both induced by the embedding Δ^{0} ⊔ Δ^{1} Δ^{2} defined by ∂^{1}∂^{2} and by ∂^{0}. The diagram (77)^{22} is now replaced with a commutative diagram

(86)

Map_{𝒮_{𝒪}} (d, c) \overline{^{π_{c}}} Map_{𝒪^{⊗}} (w, z) ×_{Hom_{_{Fin}_{∗}} (p(s),p(z))} Hom_{Fin_{∗}}(p(s), p(x))

[figure: Map′_{𝒮_{𝒪}} (d, c); ≃; π′_{c}; Map′_{𝒪^{⊗}} (w, z) ×_{Hom_{_{Fin}_{∗}} (p(s),p(z))} Hom_{Fin_{∗}}(p(s), p(x)); ≃]

[figure: Map_{𝒮_{𝒪}} (d, a); π_{a}; Map_{𝒪^{⊗}} (w, y) ×_{Hom_{_{Fin}_{∗}} (p(s),p(y))} Hom_{Fin_{∗}}(p(s), p(x))]

The upwards arrows are weak equivalences; we will prove that the lower commutative square is homotopy cartesian, considering separately the cases a = 0 and a ≠ 0 ^{23}. In case a is non-null, the edge c is also non-null. In this case we will check that the horizontal arrows π_{c} and π_{a} of (86) are equivalences. The case a = 0 = c will be verified separately. Case a ≠ 0. We will prove that if d : s → w and a : x → y are in 𝒮_{𝒪} so that a ≠ 0, then the natural map

π_{a} : Map_{𝒮_{𝒪}} (d, a) Map_{𝒪^{⊗}} (w, y) ×_{Hom_{_{Fin}_{∗}} (p(s),p(y))} Hom_{Fin_{∗}}(p(s), p(x))

is an equivalence. The proof goes as follows. We replace Map_{𝒮_{𝒪}} with Map_{Fun(Δ^{1},𝒪)} and use Lemma B.2.5 to express it as a fiber product. We have to check therefore that the map

(87) Map′_{𝒪^{⊗}} (w, y) ×_{Map_{𝒪^{⊗}} (s,y)} Map′_{𝒪^{⊗}} (s, x)

Map_{𝒪^{⊗}} (w, y) ×_{Hom_{_{Fin}_{∗}} (p(s),p(y))} Hom_{Fin_{∗}}(p(s), p(x))

is an equivalence, where the notation for Map′ is as in Lemma B.2.5. Since the map

Map′_{𝒪^{⊗}} (w, y) Map_{𝒪^{⊗}} (w, y)

22rotated by 90^{∘} to fit on the page

23we denote by 0 a null map which exists and is essentially unique for any choice of source and target.

is a trivial fibration, it is sufficient to check that the diagram

(88)

Map′_{𝒪} (s, x)  \overline{}Map_{𝒪^{⊗}} (s, y)

[figure: Hom_{Fin_{∗}}(p(s), p(x)); Hom_{Fin_{∗}}(p(s), p(y))]

is cartesian if a ≠ 0. In other words, we have to check that for any map e : p(s) → p(x) (there are two such maps as p(s) = p(x) = ⟨1⟩) the fiber of the left vertical map at e is equivalent to the fiber of the right vertical map at p(a) ∘ e. If e = 0, both fibers are contractible. Otherwise write y = ⊕y_{i} with y_{i} ∈ 𝒪, so that a is determined by an equivalence a_{k} : x → y_{k} for some k. Then the fibers are equivalent respectively to Map′_{𝒪} (s, x) and to Map_{𝒪} (s, y_{k}), that is, equivalent to each other. Case a = 0. We will show that the lower commutative square in (86) is equivalent to the following diagram

(89)

Map_{𝒪} (s, x) × F_{c} \overline{}Hom_{Fin_{∗}}(p(s), p(x)) × F_{c}

[figure: Map_{𝒪} (s, x) × F_{a}; Hom_{Fin_{∗}}(p(s), p(x)) × F_{a}]

for appropriately chosen F_{c} and F_{a}. This will imply the claim. We proceed as follows. Define F̄_{a} as the fiber of the map Map_{𝒪^{⊗}} (w, y) Hom_{Fin_{∗}}(p(s), p(y)) at zero. Then the target of π_{a} in the diagram (86) identifies with F̄_{a} × Hom_{Fin_{∗}}(p(s), p(x)). Similarly, we define F̄_{c} as the fiber of the map Map′_{𝒪^{⊗}} (w, z) Hom_{Fin_{∗}}(p(s), p(z)) at zero. This will identify the target of π′_{c} in the diagram (86) with F̄_{c} × Hom_{Fin_{∗}}(p(s), p(x)). The projections s, t : 𝒮_{𝒪} → 𝒪^{⊗} yield the maps (90)

s_{a} : Map_{𝒮_{𝒪}} (d, a)  Map_{𝒪} (s, x), t_{a} : Map_{𝒮_{𝒪}} (d, a)  Map′_{𝒪^{⊗}} (w, y),

where Map′_{𝒪^{⊗}} (w, y) is defined as the fiber of the map

Fun(Δ^{2}, 𝒪^{⊗}) Fun(Δ^{1}, 𝒪^{⊗}) × 𝒪^{⊗}

defined by ∂^{2} : Δ^{1} → Δ^{2} and ∂^{0}∂^{0} : Δ^{0} → Δ^{2}, at (d, y). The composition

Map_{𝒮_{𝒪}} (d, a) ^{t_{a}} Map′_{𝒪^{⊗}} (w, y) Map_{𝒪^{⊗}} (s, y)

is zero. We can therefore define F_{a} as the fiber of

(91)  Map′_{𝒪^{⊗}} (w, y)  Map_{𝒪^{⊗}} (s, y)

at 0 (at the contractible space of null maps), and get a canonical map

(92)  Map_{𝒮_{𝒪}} (d, a)  Map_{𝒪} (s, x) × F_{a}.

One easily sees this is an equivalence. Note that one has a canonical map F_{a} → F̄_{a} which is a trivial Kan fibration. Similarly, one has a pair of maps (93)

s_{c} : Map′_{𝒮_{𝒪}} (d, c)  Map_{𝒪} (s, x), t_{c} : Map_{𝒮_{𝒪}} (d, c)  Map′′_{𝒪^{⊗}} (w, z),

with Map′′_{𝒪^{⊗}} (w, z) defined as the fiber of the map

(94)  Fun(Δ^{3}, 𝒪^{⊗})  Fun(Δ^{1}, 𝒪^{⊗})^{2}

given by ∂^{2}∂^{3} : Δ^{1} → Δ^{3}, ∂^{0}∂^{0} : Δ^{1} → Δ^{3}, at (d, b). Once more the composition

Map′_{𝒮_{𝒪}} (d, c) ^{t_{c}} Map′′_{𝒪^{⊗}} (w, z) Map_{𝒪^{⊗}} (s, z)

is zero, so we define F_{c} as the fiber of

(95)  Map′′_{𝒪^{⊗}} (w, z)  Map_{𝒪^{⊗}} (s, z)

at 0 and get a canonical map

(96)  Map′_{𝒮_{𝒪}} (d, c)  Map_{𝒪} (s, x) × F_{c}.

It is also an equivalence and the map F_{c} → F̄_{c} is a trivial Kan fibration. We are done.

B.2.7. Having checked that ι and ι∘π are approximations, we can now deduce from Theorem 2.3.3.23(2) of [L.HA] that both ι and ι∘π are weak equivalences.

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Vladimir Hinich

Department of Mathematics University of Haifa Mount Carmel Haifa 3498838 Israel hinich@math.haifa.ac.il