SSPACES OF OPERAD STRUCTURES
MARCY D. ROBERTSON
Abstract.
The purpose of this paper is to study the derived category of simplicial multicategories witharbitrary sets of objects (also known as, colored operads in simplicial sets). Our main result is a derivedMorita theory for operads–where we describe the derived mapping spaces between two multicategories P and Q in terms of the nerve of a certain category of P - Q -bimodules. As an application, we show that thederived category possesses internal Hom -objects.
Operads are combinatorial devices that encode families of algebras defined by multilinear operations andrelations. Common examples are the operads A , C and Lie whose algebras are associative, associative andcommutative, and Lie algebras, respectively. Morphisms between operads systematically encode relationsbetween different kinds of algebras. A well-studied example is the sequence
Lie −→ A −→ C which encodesthe property that any commutative algebra is an associative algebra and that commutators in an associativealgebra yield a Lie algebra. Multicategories, also known as colored operads, encode the laws of more com-plicated algebraic structures such as operadic modules, enriched categories, and even categories of operadsthemselves. In particular, multicategories provide a device for systematically studying morphisms betweenoperads.Operads are also a generalization of classical rings to a homotopy theoretic setting. Multicategories aresimply operads with “many objects,” analogous to pre-additive categories being rings with “many objects.”Taking this point of view, we have many tools available to study operads, including their representationtheory. The main purpose of this paper is to prove a type of Morita theory for multicategories. Wewill be more explicit in Section 4, but the general idea of Morita theory is that equivalences betweencategories of representations R Mod −→ S Mod correspond to some, geometrically meaningful, notion ofMorita equivalence between rings R −→ S . Moreover, these Morita equivalences are completely characterizedby families of R - S -bimodules and thus are easy to identify and study.To elaborate, recall that given an Abelian group G , we know that the group End ( G ) has a natural ringstructure. It is well knows that there exists a natural bijective correspondence between R -module structureson G and ring homomorphisms R −→ End ( G ). Similarly, we know that given R -modules M and N , the setof Abelian group homomorphisms Hom A b ( M, N ) has a natural structure as an R - R -bimodule, and thereexists a bijective correspondence between R -module homomorphisms M → N and R - R -bimodule maps R → Hom A b ( M, N ) . Morita theory proves that functors between R Mod and S Mod which commute withcolimits are in bijective correspondence with R - S -bimodules.All of these facts have a direct analogue in operadic algebra, namely that given any space X , we can definethe endomorphism operad End X characterized by the natural bijective correspondence between P -algebrastructures on X and operad homomorphisms P →
End X . Given two P -algebras A and B , there exists a P - P -bimodule End
A,B ( n ) := Hom C ( A ⊗ n , B ) such that P -algebra maps A → B are in bijective correspondencewith P - P -bimodule maps P →
End
A,B . Operads, however, are homotopy theoretic objects, and thus we want to add the phrase “up to homotopy”to every statement in this discussion. It follows that our main objects of study are the spaces of morphisms between two multicategories P to Q up to weak equivalence, i.e. the homotopy function complex Map h ( P , Q )(see Section 1). The main result of this paper is the following.A. Theorem.
The derived mapping space
Map h ( P , Q ) is weakly homotopy equivalent to the moduli space ofright quasi-free P - Q -bimodules, P M Q . a r X i v : . [ m a t h . A T ] N ov n particular, we can state that two objects are Morita equivalent if, and only if, they lie in the sameconnected component of the space Map h ( P , Q ). Theorem A is therefore an operadic version of the derivedMorita theory of To¨en [T]. This is different than, but entangled with, the theory which is called derivedMorita theory by Berger-Moerdijk in their paper [BM08]. More explicitly, Berger and Moerdijk provide a listof conditions that imply two operadic algebras have equivalent derived categories. These conditions makean appearance in this paper, in Lemma 4.6 and Proposition 4.5, but in a different form, as the desired endis not to study the derived category, but rather the associated simplicial category obtained via Dwyer-Kanlocalization.0.1. Related Work and Applications.
The main technical tool in this paper is a cofibrantly generatedmodel structure on the category of all small simplicial multicategories [R]. The weak equivalences are ablend of weak equivalences of operads (cf.[Rezk96, BM07]) and categorical equivalences. Many of the resultsin this paper can be extended to more operads which take values other monoidal model categories, as longas one makes additional some necessary technical adjustments.We also note that several results in this paper are generalizations of a recent preprint by Dwyer-Hess [DH].They restrict to non-symmetric operads with some connectivity, and use a description of the mapping spacesto prove that the space of tangentially straightened long knots is equivalent to the double loop space of amoduli space of bimodules.Our use of symmetric actions in this paper allows us to say a little more than [DH] about the structure ofthe derived mapping spaces between operads, in particular, we can describe the internal hom-objects of thehomotopy category of all small simplicial multicategories. In addition, we also prove a cosimplicial modelfor the mapping space, making precise an observation by Berger-Moerdijk in their paper [BM07, 6]. Thiscosimplicial model provides filtrations of the derived mapping space which are necessary for computationsin [R11].0.2.
Notation and Conventions.
Multicategories are frequently referred to as colored operads , or simply operads in the literature. There are times in this paper where the author uses the word operad or multicat-egory (interchangeably) without explicitly mentioning sets of objects. When it plays an important role, setsof objects will always be specified. At all other times, the result holds for general sets of objects.We will use interchangeably the notation for a category and its nerve. As such, we follow the conventionthat a functor will be a weak equivalence if it induces a weak homotopy equivalence on the respective nerves.We will always use the phrase weak homotopy equivalence to refer to a weak equivalence of simplicial setsin the standard (Kan) model structure. Given a model category C it makes sense to consider the (not full)subcategory w C of C which is the category with the same objects as C and morphisms the weak equivalencesbetween objects in C . We call w C the moduli category of C . The moduli space of C will be the nerve of w C .An adjoint pair E : M (cid:28) B : U of functors between model categories is a Quillen pair if E preservescofibrations and trivial cofibrations (equivalently, if U preserves fibrations and trivial fibrations). The pair( E, U ) forms a Quillen equivalence if for all cofibrant B ∈ B and fibrant M ∈ M , a map EM −→ B is aweak equivalence in B if, and only if, the adjoint M −→ U B is a weak equivalence in M .Given that the right adjoint U : B −→ M preserves all weak equivalences, we have an induced functor onthe moduli categories wU : w B → w M , and consequently an induced map of moduli spaces wU : w B −→ w M . If U is part of a Quillen pair then the induced morphism wU is a weak homotopy equivalence.If F : C → D is a functor and X is an object of D , F (cid:38) X denotes the over category of F with respectto X . Objects of this category are pairs ( Y, g ) where Y ∈ C and g is a map F ( Y ) → X in D . A morphism( Y, g ) → ( Y (cid:48) , g (cid:48) ) is a map Y → Y (cid:48) in C rendering the appropriate diagram commutative. The dual notion of under category is denoted X (cid:38) F . If F is the identity functor on D , we write D (cid:38) X , respectively, X (cid:38) D .We take the following argument to be standard. Suppose that F : C → D is a functor such that for everymorphism h : X → X (cid:48) in D the map F (cid:38) X → F (cid:38) X (cid:48) induced by composition with h is a weak homotopyequivalence. Then one can apply Quillen’s Theorem B to show that for any X ∈ D the homotopy fiber of the nerve of) F over the vertex of D represented by X is naturally weakly homotopy equivalent to F (cid:38) X .A similar result holds with over categories replaced by under categories (see [GJ, 5.2]).1. Homotopy Function Complexes
Given two objects X and Y in a model category C , there is an associated simplicial set Map h C ( X, Y ) calleda homotopy function complex or derived mapping space from X to Y . In [DK1, DK2, DK3] Dwyer andKan show that Map h ( X, Y ) can be computed as the nerve of a category of “zig-zags,” i.e. a category whoseobjects are zig-zags [ X ∼ (cid:17) U −→ V ∼ (cid:27) Y ] and where the maps are natural transformations of diagrams whichare the identity on X and on Y . There are many variations of these zig-zag categories, including those withobjects [ X ∼ ←− U −→ V ∼ ←− Y ] , or zig-zags [ X −→ U ∼ (cid:17) V −→ Y ]. As it happens, all of these variationswill have homotopy equivalent nerves, and thus all of these variations have nerves homotopy equivalent toMap h ( X, Y ). We will also require several, more rigid, models for Map h ( X, Y ).Let c C denote the Reedy model structure on the category of cosimplicial objects in C [Hir03, Chapter 15].For any object X in C we write cX for the constant cosimplicial object consisting of X in every dimensionwith identity maps for all co-face and co-degeneracies. A cosimplicial resolution of X in C is a Reedycofibrant replacement QX • → cX in c C . Given such a cosimplicial resolution and an object Y in C we formthe simplicial set C ( Q • X, Y ) given by [ n ] (cid:55)→ C ( Q n X, Y ). If Y → Y (cid:48) is a weak equivalence between fibrantobjects then the induced map C ( Q • X, Y ) −→ C ( Q • X, Y (cid:48) ) is a weak equivalence of simplicial sets.Now, for a fixed object X ∈ C , let Q ( X ) denote the category whose objects are pairs [ Q, Q → X ] where Q is some cofibrant object in C and Q → X is a weak equivalence. For any object Y in C , we have a functor C ( − , Y ) : Q ( X ) op −→ S et which sends [ Q, Q → X ] to C ( Q, Y ). We can regard this functor as taking values in s S et by composingwith the embedding S et → s S et . We can now consider the simplicial set hocolim Q ( X ) op C ( − , Y ), where herewe model the hocolim functor by first taking the simplicial replacement of a diagram and then applyinggeometric realization. In dimension n the simplicial replacement consists of diagrams of weak equivalences Q ← Q ← ... ← Q n over X, where each Q i → X is in Q ( X ), together with a map Q → Y . In otherwords, the simplicial replacement is the same as the nerve of the category for which an object is a zig-zag[ X ∼ ←− Q → Y ], where Q is cofibrant and Q → X is a weak equivalence. The maps from [ X ∼ ←− Q → Y ] to[ X ∼ ←− Q (cid:48) → Y ] are just maps Q (cid:48) → Q which fit into the usual commutative diagram. If Y is fibrant, it is wellknown that this simplicial set is weakly equivalent to Map h ( X, Y ) (see [D1, DK3]). A dual argument showsthat, if X is cofibrant, Map h ( X, Y ) is weakly equivalent to the nerve of the zig-zag category [ X → Q ∼ ←− Y ].So that we do not have to limit ourselves to only studying mapping spaces with cofibrant source or fibranttarget we make use of the following proposition.1.1. Proposition. [DH, 2.6]
Let C be a left proper model category, and let X and Y be objects in C such that X c (cid:96) Y −→ X (cid:96) Y is a weak equivalence. Then M ap h ( X, Y ) is weakly homotopy equivalent to the nerve ofthe zig-zag category [ X → Q ∼ ←− Y ] . We will only make brief use of the cosimplicial model for Map h ( X, Y ) in this paper, but this is the moreconvenient model for computation. Let Q • X −→ X be a cosimplicial resolution of X in c C . We want torelate the simplicial set C ( Q • X, Y ) to the zig-zags of categories considered above. For a given simplicial set K , let ∆ K be the category of simplices of K , i.e. the over category ( S ↓ K ), where S : ∆ → s S et is thefunctor [ n ] (cid:55)→ ∆[ n ]. The nerve of ∆ K is naturally weakly equivalent to K (see [D1, text prior to Prop.2.4]). There is a functor sending ∆ C ( Q • X, Y ) to another zig-zag category, which sends ([ n ] , Q n X → Y ) to[ X ∼ (cid:17) Q n X −→ Y ].1.2. Proposition.
Let Q • X → X be a Reedy cofibrant resolution of X and let Y be a fibrant object of C .Then ∆ C ( Q • X, Y ) is weakly equivalent to Map h ( X, Y ) .Proof. The result is proven in [DK3], but see also [D1, Thm. 2.4]. (cid:3) . Operads and Multicategories
The basic idea of a multicategory is very like the idea of a category, it has objects and morphisms, but ina multicategory the source of a morphism can be an arbitrary finite sequence of objects rather than just asingle object.A multicategory , P , consists of the following data: • a set of objects obj( P ); • for each n ≥ x , ..., x n , x a set P ( x , ..., x n ; x ) of operations whichtake n inputs ( x , ..., x n ) to a single output (the object x ).These operations are equipped with structure maps for units and composition. Specifically, if I = {∗} denotes the one-point set, then for each object x there exists a unit map η x : I → P ( x ; x ) taking ∗ to 1 x , where 1 denotes the unit of the symmetric monoidal structure on the category S et . The compositionoperations are given by maps P ( x , ..., x n ; x ) × P ( y , ..., y k ; x ) × · · · × P ( y n , ..., y nk n ; x n ) −→ P ( y , ..., y nk n ; x )which we denote by p, q , ..., q n (cid:55)→ p ( q , ..., q n ) . The structure maps satisfy the associativity and unitary coherence conditions of monoids. A symmetricmulticategory is a multicategory with the additional property that the operations are equivariant underthe permutation of the inputs. Explicitly, for σ ∈ Σ n and each sequence of objects x , ..., x n , x we have aright action of Σ n , i.e., a morphism σ ∗ : P ( x , · · · , x n ; x ) → P ( x σ (1) , ..., x σ ( n ) ; x ). The action maps are wellbehaved, in the sense that all composition operations are invariant under the Σ n -actions, and ( στ ) ∗ = τ ∗ σ ∗ .In practice, one often uses the following, equivalent, definition of the composition operations, given by: P ( c , · · · , c n ; c ) × P ( d , · · · , d k ; c i ) ◦ i (cid:47) (cid:47) P ( c , · · · , c i − , d , · · · , d k , c i +1 , · · · , c n ; c ) . All of our definitions will still make sense if we ask that the k -morphisms P n ( x , . . . , x n ; x ) take valuesin a symmetric monoidal category other than sets; the examples we are interested in take values in eithercategories, symmetric spectra or simplicial sets. Multicategories whose operations take values in C are called multicategories enriched in C or C -multicategories . In particular, the strong monoidal functor S et −→ C thatsends a set S to the S -fold coproduct of copies of the unit of C takes every multicategory to a C -enrichedmulticategory. A morphism between enriched, symmetric multicategories F : P −→ Q , or multifunctor , consists of a setmap of objects F : obj( P ) −→ obj( Q ) together with a family of Σ n -equivariant C -morphisms { F : P ( d , ..., d n ; d ) −→ Q ( F ( d ) , ..., F ( d n ); F ( d )) } d ,...,d n ,d ∈P which are compatible with the composition structure maps. When P and Q are enriched over simplicial sets,the multifunctor is enriched when the maps on n -operations preserve the enrichment. We denote the categoryof all small symmetric multicategories enriched in C by M ulti ( C ) and denote the morphisms between twoobjects as M ulti ( P , Q ).2.1. Structure of M ulti ( C ) . Multicategories are often called colored operads, or just operads(See, forexample, [BM06, BV73, May, CGMV10], etc.), but we use the term multicategory in this paper because wewant to emphasize the relationship between multicategory theory with classical category theory. Informally,we can say that inside every multicategory lies a category which makes up the linear part (i.e. the 1-operations). We make this explicit by assigning to each multicategory P a category [ P ] with the sameobject set as P and with morphisms given by [ P ] ( p, p (cid:48) ) = P ( p ; p (cid:48) ) for any two objects p, p (cid:48) in P (i.e. just Note that a multicategory enriched over small categories can be considered enriched over simplicial sets by applying the nervefunctor to the n -operations, since the nerve functor preserves categorical products. ook at the operations of P which have only one input). The functor [ − ] takes all higher operations, i.e. P ( p , ..., p n ; p ), to be trivial. Composition and identity operations are induced by P .This relationship with category theory is useful in making sense of ideas which do not have obvious meaningin the multicategory setting. For example, we will often want to discuss the “connected components” of amulticategory, but it is difficult to say that an n -ary operation φ is an “isomorphism” in P . This is wherethe relationship between categories and multicategories can be useful, we can say that φ is an isomorphism in P if [ φ ] is an isomorphism in the category [ P ] .2.1. Definition.
Let P and Q be two multicategories. A multifunctor F : P → Q is essentially surjective if [ F ] is essentially surjective as a functor of categories. We say that F is full if for any sequence p , ..., p n , p the function F : P ( p , ..., p n ; p ) → Q ( F p , ..., F p n ; F p ) is surjective. We say that F is faithful if for anysequence p , ..., p n , p the function F : P ( p , ..., p n ; p ) → Q ( F p , ..., F p n ; F p ) is injective. The multifunctor F is called fully faithful if it is both full and faithful. Definition.
Let F : P → Q be a functor between two symmetric multicategories. We say that F is an equivalence of multicategories if, and only if, F is both fully faithful and essentially surjective. Let C be the category of simplicial sets with the standard model structure. Given a simplicial category A ,we can form a genuine category π ( A ) which has the same set of objects as A and whose set of morphisms π ( A )( x, y ) := [ , A ( x, y )]. This induces a functor π ( − ) : C at ( C ) → C at, with values in the category of smallcategories and, moreover, a functor Ho( C at ( C )) −→ Ho( C at ) . In other words, any F : C −→ D in Ho( C at ( C ))induces a morphism π ( C ) −→ π ( D ) which is well defined up to a non-unique isomorphism. This lack ofuniqueness will not be an issue, as we are only interested in properties of functors which are invariant up toisomorphism.As with the non-enriched case, we can consider the linear part of a simplicial multicategory P , [ P ] , whichis in this case a simplicial category. Applying the functor π to the simplicial category [ P ] gives us the underlying category of the multicategory P . In order to cut back on notation, we denote this category by[ P ] rather than π ([ P ] ).2.3. Theorem. [R]
The category of small C -enriched symmetric multicategories admits a right propercofibrantly generated model category structure in which a multifunctor F : P −→ Q is a weak equivalence if: W1: for any n ≥ and for any signature x , ..., x n ; x in P the map of C -objects F : P ( x , ..., x n ; x ) −→ Q ( F x , ..., F x n ; F x ) is a weak equivalence in the model category structure on C . W2: the induced functor [ F ] is a weak equivalence of categories.A simplicial multifunctor F : P −→ Q is a fibration if: F1: for any n ≥ and for any signature x , ..., x n ; x in P the map of C -objects F : P ( x , ..., x n ; x ) −→ Q ( F x , ..., F x n ; F x ) is a fibration in the model category structure on C . F2: the induced functor [ F ] is a fibration of categories.The cofibrations (respectively, acyclic cofibrations) are the multifunctors which satisfy the left lifting property(LLP) with respect to the acyclic fibrations (respectively, fibrations). The fibrant objects in M ulti ( C ) are those objects which are locally fibrant, i.e. P ( x , ..., x n ; x ) is a Kancomplex for each n ≥ x , ..., x n ; x in P .2.4. Lemma.
There exists a fibrant replacement functor on M ulti ( C ) which fixes objects, i.e. ( P ) f −→ P is the identity on object sets. .2. Cofibrant Replacements.
There exists several explicit cofibrant resolutions of operads in literature,and in this paper we will focus on two, the cotriple resolution and the W -construction of Boardman andVogt.Let P , Q and R be simplicial multicategories, let X be a P - Q -bimodule, and let Y be a R - P -bimodule.We define the bar complex B ( X, P , Y ) to be the simplicial object in the category of R - Q -bimodules with n th -degree B n ( X, P , Y ) = X ◦ P ◦ n ◦ Y with the obvious face and degeneracy maps. Applying the diagonal,we get an R - Q -bimodule together with an augmentation map η : diag ( B ( X, P , Y )) −→ X ◦ P Y .2.5. Proposition.
The bar complex diag ( B ( X, P , Y )) is cofibrant in the category of R - Q -bimodules and theaugmentation map η : diag ( B ( X, P , Y )) −→ X ◦ P Y is a weak equivalence.Proof. See [Rezk96] and [DH]. (cid:3)
The
Hochschild resolution of a simplicial multicategory P is a simplicial object in the category of P - P -bimodules with B n ( P , P , P ) := P ◦ ( n +2) where face maps come from the composition of P and degeneracymaps come from the unit maps of P . To shorten notation we will denote B n ( P , P , P ) by H n P .The diagonal of H ∗ ( P ), denoted diag ( H ∗ ( P )), is a P - P -bimodule with n -simplicies the n -simplicies of H n ( P ). The composition operations of P induce maps P ◦ n −→ P . Composition and identity maps arepreserved by taking diagonals, so we have natural maps η : diag ( H ∗ P ) −→ P and P ◦ P −→ H n P . Themaps P ◦ P −→ H n P come from the image of H P under degeneracy maps. Taken together, all of thedegeneracy maps induce a basepoint P ◦ P −→ diag ( H ∗ P ). It follows from arguments similar to [Fre09,17.2.2. 17.2.3],[DH, 5.2],[Rezk96, 5] that diag ( H ∗ P ) is cofibrant as a pointed P - P -bimodule and that theaugmentation map η : diag ( H ∗ P ) −→ P is a weak equivalence of pointed P - P -bimodules. Note that beingcofibrant as a pointed P - P -bimodule is equivalent to saying that P ◦ P −→ diag ( H ∗ P ) is a cofibration.We will also want to know how this resolution interacts with extension-restriction of scalars (see sec-tion 4.1). Given a multifunctor F : P → Q , a P - Q -bimodule Y determined by F , and X := P , themulticategory P considered as a P - P -bimodule over itself, then the bar complex B ( X, P , Y ) = X ◦ P ◦ n ◦ Y is a simplicial object in the category of P - Q -bimodules together with the augmentation map η : diag( B ( X, P , Y )) −→ F ∗ ( X ) . Corollary.
The bar complex diag B ( X, P , Y ) is cofibrant as a P - Q -bimodule. Moreover, the augmentationmap η : diag B ( X, P , Y ) −→ F ∗ ( X ) is a weak equivalence in the category of P - Q -bimodules. The W -construction. The main idea behind the Boardman-Vogt W -construction is to enrich the freeoperad construction by assigning lengths to edges in trees. The composition F ( P ) (cid:26) W ( P ) ∼ −→ P is identified with the counit of the free-forgetful adjunction between pointed collections and operads (See,for example, [EM06, Theorem 4.2], [BM07, Section 3]). If the collection underlying P is cofibrant and well-pointed, then the counit F ( P ) −→ P can be factored into a cofibration F ( P ) (cid:26) W ( P ) followed by a trivialfibration W ( P ) ∼ (cid:16) P . Since F ( P ) is a cofibrant operad, the W -construction provides a cofibrant resolution for P [BM06, 5.1].We also have the notion of a relative W -construction, which resolves a morphism between multicategories u : P −→ Q . This relative version produces an object W ( Q ) P which is characterized by the property thatalgebras over this operad satisfy the operations from Q up to coherent homotopy, while they satisfy theoperations from P on the nose.2.7. Example.
Stasheff ’s A ∞ -operad can be obtained as the relative Boardman-Vogt resolution W ( I ∗ → A ) where I ∗ is the operad for pointed objects. hile most things work in multicategories the same as they do for the more classical operads, we do haveto keep track of objects when doing the W -construction. More explicitly, given a map α : D −→ C betweensets of objects, we can consider the adjunction α ∗ : M ulti D (cid:28) M ulti C : α ∗ between multicategories with D -objects and multicategories with C -objects. For Q ∈ M ulti D and P ∈ M ulti C , there exist natural maps α ∗ W ( Q ) −→ W ( α ∗ Q ) and W ( α ∗ P ) −→ α ∗ W ( P ) , but in general these maps are not isomorphisms. If α is injective, we know that there is an explicit descriptionof α ∗ ( Q ), as α ∗ ( Q )( d , . . . , d n ; d ) = Q ( c , . . . , c n ; c ) if d i = α ( c i ) , d = α ( c ) ,I if n = 1 , d = d (cid:54)∈ Im ( α ) , , and, using this description, it is easy to show that the map α ∗ W ( Q ) −→ W ( α ∗ Q ) is an isomorphism.Consider a cofibration u : P −→ Q between operads. In [BM03, Appendix] they construct what is calledthe free extension P [ u ] of P by u . This free extension is determined by the universal property that operadmaps out of P [ u ] are in one-to-one correspondence with maps of collections out of Q , whose restriction to P (along u ) is an operad map. In particular, the identity on Q induces a factorization of u into maps P → P [ u ] → Q . These maps, in turn, factor into cofibrations P (cid:26) P [ u ] (cid:26) W ( Q ) P followed by a weakequivalence W ( Q ) P ∼ −→ Q in such a way that the operad W ( Q ) P is a quotient of the operad W ( Q ).While this factorization always exists [BM07, Theorem 4.1], if we consider a category of operads whichis a left proper model category , then we may define the required factorization simply by taking a pushout: P −→ P ∪ W ( P ) W ( Q ) ∼ −→ Q . What’s more, the object W ( Q ) P is constructed as a sequential colimit of trivialcofibrations of collections: W ( Q ) P ∼ (cid:26) W ( Q ) P ∼ (cid:26) W ( Q ) P ∼ (cid:26) · · ·
For each k , W k ( Q ) P is a quotient of W k ( Q ), which is the piece of the operad W ( Q ) restricted to operationswith inputs ≤ k . In other words, W ( Q ) P is a quotient of W ( Q ) by a filtration-preserving map.3. Algebra Structures
It is well known to the experts that the category of M ulti ( C ) is a closed , symmetric monoidal category withrespect to the Boardman-Vogt tensor product. By closed, we mean that there exists an internal hom-object Hom satisfying the adjunction relation M ulti ( P ⊗ BV Q , R ) ∼ = M ulti ( P , Hom ( Q , R )) . This internal hom-object
Hom ( P , Q ) is a multicategory with objects the multifunctors P −→ Q , andwhose operations are type of multi-natural transformation.3.1.
Definition. [EM06, Definition 2.2]
For notational convenience, we denote the sequence c , . . . , c k as { c i } ki =1 . Given symmetric multicategories P and Q , we define Hom ( P , Q ) to be a multicategory with objectsthe multifunctors from P to Q . Given a sequence of multifunctors F , . . . , F k : P → Q of multifunctors and atarget multifunctor G : P → Q , we define a k -natural transformation from F , . . . , F k to G to be a function ξ that assigns to each object a of P a k -operation ξ a : ( F a, . . . , F k a ) → Ga of Q , such that for any m -ary peration φ : ( a , . . . , a m ) → b in P , the following diagram commutes: {{ F j a i } kj =1 } mi =1 { ξ ai } (cid:47) (cid:47) ∼ = (cid:15) (cid:15) { Ga i } mi =1 Gφ (cid:15) (cid:15) {{ F j a i } mi =1 } kj =1 { F j φ } (cid:15) (cid:15) { F j b } kj =1 ξ b (cid:47) (cid:47) Gb.
The unlabelled isomorphism is the standard block permutation that shuffles m blocks of k entries each into k blocks of m entries each. The k -natural transformations form the k -ary operations in the multicategoryHom ( P , Q ) . Composition and symmetric actions are induced by the composition and symmetric actions in Q . In particular, if we restrict to the linear operations, the object [
Hom ] , gives M ulti an enrichment over thecategory of small categories. For any two multicategories P and Q , there exists a tensor product multicategory P ⊗ BV Q and a universal bilinear map ( P , Q ) → P ⊗ BV Q . This tensor product makes M ulti into a symmetricmonoidal category. We will only give a brief description of the construction now, but refer the reader to thehighly readable version in [EM06].3.2. Definition. [EM06]
Let P , Q , and R be multicategories. A bilinear map f : ( P , Q ) → R consists of the following data: (1) A function f : obj( P ) × obj( Q ) → obj( R ) , (2) For each m -ary operation φ ∈ P ( a , . . . , a m ; a ) of P and each object b of Q , an m -ary operation f ( φ, b ) ∈ R ( f ( a , b ) , . . . , f ( a m , b ); f ( a, b )) of R , (3) For each n -ary operation ψ ∈ Q ( b , . . . , b n ; b ) of Q and object a of P , an n -operation f ( a, ψ ) ∈ R ( f ( a, b ) , . . . , f ( a, b n ); f ( a, b )) of R such that: (1) For each object a of P , f ( a, − ) is a multifunctor from Q to R , (2) For each object b of Q , f ( − , b ) is a multifunctor from P to R , (3) Given an m -operation φ ∈ P ( a , . . . , a m ; a ) and an n -operation ψ ∈ Q ( b , . . . , b n ; b ) in Q , the followingdiagram commutes: {{ f ( a i , b j ) } mi =1 } nj =1 { f ( φ,b j ) } (cid:47) (cid:47) ∼ = (cid:15) (cid:15) { f ( a, b j ) } nj =1 f ( a,ψ ) (cid:15) (cid:15) {{ f ( a i , b j ) } nj =1 } mi =1 { f ( a i ,ψ ) } (cid:15) (cid:15) { f ( a i , b ) } mi =1 f ( φ,b ) (cid:47) (cid:47) f ( a, b ) . The set of bilinear maps is denoted as Bilin ( P , Q ; R ) . A multifunctor
P × Q → R assigns a k -operations in R to each pair of k -operations from P and Q . On theother hand, a bilinear map assigns an m × n -operation in R , to each pair ( φ, ψ ), where φ is and m -operation P and ψ is an n -operation Q . When restricted to the linear operations, a bilinear map f : ( P , Q ) → R isprecisely a functor [ f ] : [ P ] × [ Q ] → [ R ] of the underlying categories. Objects of Bilin( M, N ; P ) are theobjects of a multicategory naturally isomorphic to both Hom ( P , Hom ( Q , R )) and Hom ( Q , Hom ( P , R )). .1. Construction of ⊗ BV . Let P and Q be fixed multicategories, and construct the coproducts of multi-categories (cid:97) a ∈ obj( P ) Q and (cid:97) b ∈ obj( Q ) P . The coproduct is a universal morphism, i.e. given a multicategory R , then (cid:96) a ∈ obj( P ) Q is the universal sourcefor any multifunctor which maps the objects obj( P ) × obj( Q ) to obj( R ) and is a multifunctor with respectto Q . Similarly, (cid:96) b ∈ obj( Q ) P is universal for maps that send objects of obj( P ) × obj( Q ) to obj( R ) which aremultifunctors in P . If we are given a bilinear map f : ( P , Q ) → R , it follows that we have multifunctors fromboth (cid:96) a ∈ obj( P ) Q and (cid:96) b ∈ obj( Q ) P to R . Therefore the bilinear map f induces a map from the pushout F (obj( P ) × obj( Q )) (cid:47) (cid:47) (cid:15) (cid:15) (cid:97) b ∈ obj( Q ) P (cid:15) (cid:15) (cid:97) a ∈ obj( P ) Q (cid:47) (cid:47) P + Q , to the multifunctor R . The object in the upper lefthand corner is the free symmetric multicategory on theset of objects obj( P ) × obj( Q ). It follows that the pushout P + Q is universal with respect to maps that aremultifunctors in each variable separately.The BV -tensor product is the quotient P + Q after we force the bilinearity relations to commute. Weconstruct this quotient as follows. For each m -ary operation φ ∈ P ( a , . . . , a m ; a ) in P and each n -aryoperation ψ ∈ Q ( b , . . . , b n ; b ) in Q , define two non-symmetric collections X ( φ, ψ ) and Y ( φ, ψ ). The objectsets of both X and Y will be the set( { a , . . . , a m } × { b , . . . , b n } ) ∪ { ( a, b ) } . We give X ( φ, ψ ) precisely two operations, both with source {{ ( a i , b j ) } mi =1 } nj =1 and target ( a, b ). We give Y ( φ, ψ ) exactly one operation with source {{ ( a i , b j ) } mi =1 } nj =1 and target ( a, b ). We then define a map ofcollections X ( φ, ψ ) −→ Y ( φ, ψ ) which sends the two operations of X to the unique operation of Y . Wedefine a second map of collections from X ( φ, ψ ) to the underlying collection of P + Q , denoted U ( P + Q ), bysending each operation of X one way around the diagram {{ ( a i , b j ) } mi =1 } nj =1 { ( φ,b j ) } (cid:47) (cid:47) ∼ = (cid:15) (cid:15) { ( a, b j ) } nj =1( a,ψ ) (cid:15) (cid:15) {{ ( a i , b j ) } nj =1 } mi =1 { ( a i ,ψ ) } (cid:15) (cid:15) { ( a i , b ) } mi =1 ( φ,b ) (cid:47) (cid:47) f ( a, b ) . Then apply the free, non-symmetric multicategory functor to these collections and form the followingpushout: (cid:97) ( φ,ψ ) F X ( φ, ψ ) (cid:47) (cid:47) (cid:15) (cid:15) F U ( P + Q ) (cid:15) (cid:15) (cid:97) ( φ,ψ ) F Y ( φ, ψ ) (cid:47) (cid:47) P ⊗ BV Q here F ( − ) denotes the free non-symmetric multicategory functor(see appendix). This quotient is preciselywhat it means to force the diagrams in the definition of a bilinear map to commute, so P ⊗ BV Q is a universalbilinear target. We then go back and add the symmetric actions in a symstematic way.The unit of the BV -tensor product is the multicategory I with one object and only the identity morphismon that object (see Example 7.3).3.3. Remark.
The symmetric actions are critical to the definition of the Boardman-Vogt tensor product. Itis possible define a version of the tensor product on planar (a.k.a. non-symmetric) multicategories whichforgets the bilinear relations. This is just P + Q , which we called the coproduct of operads . This does stillform a closed monoidal structure, but the internal hom-objects for this structure are not as well behaved asthose presented above. In particular, one cannot define multilinear transformations between planar operads.We can still define transformations where the domain consists of a single multifunctor. For a general symmetric monoidal category C , the construction of the Boardman-Vogt tensor productstill makes sense if either C is Cartesian closed or we restrict to Hopf operads P and Q . Hopf operadsare characterized by the property that their algebra categories Alg P ( C ) and Alg Q ( C ) are again symmetricmonoidal categories. In this case, the BV -tensor product tells us that a ( P ⊗ BV Q )-algebra in C is the samething as a P -algebra in Alg Q ( C ), and is also the same thing as a Q -algebra in Alg P ( C ).3.2. Derived Tensor Products.
The Bordman-Vogt tensor product can be derived in the usual way, i.e.
P ⊗ L BV Q := ( P ) c ⊗ BV Q . Unfortunately, M ulti ( C ) is not a monoidal model category, and ⊗ BV does not preserve weak equivalences ingeneral. It is true, however, that given a weak equivalence P −→ P (cid:48) which fixes objects that P + Q −→ P (cid:48) + Q is a weak equivalence (see [DH], [FV11]).3.3. Adjunction Relations.
It can be helpful to think about these various monoidal structures in termsof generators and relations. Let S be a fix set of operations of a multicategory P . The multicategory < S > generated by S is the smallest sub-multicategory of P that contains all the operations in S . If < S > = P ,we say that P is generated by S . Let P and Q be multicategories, φ an operation of P and b an objectof Q . Then we write φ ⊗ bv b for the operation of P ⊗ BV Q induced from φ and b by the universal bilinearmap ( P , Q ) → P ⊗ BV Q . Similarly, given an object a of P and an operation ψ of Q we write a ⊗ bv ψ forthe operation of P ⊗ BV Q induced by a and ψ . The universal property of the tensor product implies thefollowing proposition.3.4. Proposition. [EM06, ? ] The operations a ⊗ bv ψ and φ ⊗ bv b generate the multicategory P ⊗ BV Q . Using this characterization, one can prove that we obtain the following adjunction M ulti ( P ⊗ BV Q , R ) ∼ = M ulti ( P , Hom ( Q , R ))which enriches to a natural isomorphism of multicategories Hom ( P ⊗ BV Q , R ) ∼ = Hom ( P , Hom ( Q , R )) . Proposition. [EM06]
The k -operations of Hom ( P ⊗ BV Q , R ) are precisely those functions as inLemma 3.6 which are natural with respect to all morphisms of the form a ⊗ bv ψ or φ ⊗ bv b . Lemma. [EM06]
Fix two multicategories P and Q , and suppose that < S > is a generating set ofoperations for P . Then given a sequence of multifunctors F , . . . , F k , G : P → Q and a map ξ which assigns o each object a of P a k -operation ξ a : ( F a, . . . , F k a ) → Ga of Q such that the diagram {{ F j a i } kj =1 } mi =1 { ξ ai } (cid:47) (cid:47) ∼ = (cid:15) (cid:15) { Ga i } mi =1 Gφ (cid:15) (cid:15) {{ F j a i } mi =1 } kj =1 { F j φ } (cid:15) (cid:15) { F j b } kj =1 ξ b (cid:47) (cid:47) Gb commutes for all φ in < S > . Then the diagram commutes for all operations of P , so ξ is a k -naturaltransformation.Proof. This is proved in [EM06], but we include the proof here because it is useful. First, we assume thatwe are given elements φ , . . . , φ n in < S > with φ i ∈ P ( a i , . . . , a im i ; b i ) and ψ in P ( b , . . . , b n ; c ). Thenthe following diagram shows that our given transformation ξ is natural with respect to the composition ψ ◦ ( φ , . . . , φ n ) in P : {{{ F j ( a is ) } m i s =1 } ni =1 } kj =1 {{ F j φ i } ni =1 } kj =1 (cid:47) (cid:47) ∼ = (cid:15) (cid:15) {{ F j b i } ni =1 } kj =1 { F j ψ } kj =1 (cid:47) (cid:47) ∼ = (cid:15) (cid:15) { F j c } kj =1 ξ c (cid:15) (cid:15) {{{ F j ( a is ) } m i s =1 } kj =1 } ni =1 {{ F j φ i } kj =1 } ni =1 (cid:47) (cid:47) ∼ = (cid:15) (cid:15) {{ F j b i } kj =1 } ni =1 { ξ bi } ni =1 (cid:15) (cid:15) {{{ F j ( a is ) } kj =1 } m i s =1 } ni =1 {{ ξ ais } mis =1 } ni =1 (cid:15) (cid:15) {{ G ( a is ) } m i s =1 } ni =1 { Gφ i } ni =1 (cid:47) (cid:47) { Gb i } ni =1 Gψ (cid:47) (cid:47) Gc.
Now, for every σ ∈ Σ n , the following diagram shows that ξ is natural with respect to the symmetricactions ψ · σ : {{ F j b σ ( i ) } ni =1 } kj =1 ∼ = (cid:47) (cid:47) { σ } (cid:15) (cid:15) {{ F j b σ ( i ) } kj =1 } ni =1 { ξ bσ ( i ) } ni =1 (cid:47) (cid:47) { σ } (cid:15) (cid:15) { Gb σ ( i ) } ni =1 { σ } (cid:15) (cid:15) {{ F j b i } ni =1 } kj =1 ∼ = (cid:47) (cid:47) { F j ψ } kj =1 (cid:15) (cid:15) {{ F j b i } kj =1 } ni =1 { ξ bi } ni =1 (cid:47) (cid:47) { Gb i } ni =1 Gψ (cid:15) (cid:15) { f j c } kj =1 ξ c (cid:47) (cid:47) Gc.
Since we know that ξ is natural with respect to the generating operations, it now follows that ξ is naturalwith respect to all morphisms in P . Therefore ξ is a k -natural transformation. (cid:3) The bijection on objects M ulti ( Q , Hom ( P , R )) ↔ M ulti ( P ⊗ BV Q , R ) ↔ M ulti ( P , Hom ( Q , R ))can be extended to functors between simplicial multicategories Hom ( Q , Hom ( P , R )) (cid:28) Hom ( P ⊗ BV Q , R ) (cid:28) Hom ( P , Hom ( Q , R )) . n particular, before we take the quotient forcing the bilinear relations, we have a functor of simplicialcategories U : [ Hom ] ( P + Q , R ) −→ [ Hom ] ( P , [ Hom ] ( Q , R )) . Lemma.
There exists a left adjoint to U which we will denote by E .Proof. This can be checked explicitly on generators and relations. (cid:3)
In the next section we show that the adjoint pair (
E, U ) can be easily extend an adjoint pair of functors U : P + Q (cid:38) M ulti ( C ) (cid:28) P M Q : E , where P M Q denotes the category of pointed P - Q -bimodules. In thiscase, we will refer to E as the enveloping functor . The functor E is left adjoint to a functor which preservesfibrations and weak equivalences, and thus E preserves cofibrant objects and weak equivalences betweencofibrant objects. We define the left derived functor of E , denoted L E , as L E ( M ) := E ( M c ), where M c is acofibrant resolution of the object M as a P - Q -bimodule. Note that this left derived functor L E lands in thecategory P + Q (cid:38) M ulti ( C ) rather than the homotopy category.An operad P , like any monoid, can be considered as a P - P -bimodule over itself. An important propertyof L E is that L E ( P ) = P , when the cofibrant resolution of P we take is the Hochschild resolution, H ∗ ( P ).3.8. Example (Endomorphism Modules) . For any two objects
X, Y in a symmetric monoidal category ( M , ⊗ , M ) we can define a collection whose k -operations are given by End
X,Y ( k ) := M ( X ⊗ k , Y ) . This collection can be given a Σ k -action by permuting the source factors. We have natural compositionproducts ◦ i : M ( X ⊗ k , Y ) ⊗ M ( X ⊗ l , X ) −→ M ( X ⊗ k + l − , Y ) which implies that End X,Y is a right module over the endomorphism operad End X . Given that X is a Q -algebra, i.e. that there exists an operad homomorphism α : Q → End X then we can say that End X,Y is aright Q -module by restriction along the structure map.At the same time, we could consider the composition maps M ( Y ⊗ r , Y ) ⊗ M ( X ⊗ n , Y ) ⊗ ... ⊗ M ( X ⊗ n r , Y ) −→ M ( X ⊗ n + ... + n r , Y ) which makes End X,Y into a left module over the operad End Y . If Y is a P -algebra, i.e. there exists an map β : P →
End Y then End X,Y is a left P -module by restriction. Example.
Consider an operad P as a P - P -bimodule over itself. One way to do this is to recall thatthere exists a natural collection End P , P := End P , which has two structure maps which are isomorphisms P →
End P which commute in a universal way. In other words, we could consider the P - P -bimodule structureon End P as belonging to the space Hom ( P , Hom ( P , End P )) . Once we apply the functor E , we are considerEnd P as a multicategory, together with a map from P + P −→
End P . This map is induced by the structuremaps P →
End P , which are isomorphisms, and thus P + P −→
End P = P is just the fold map. Lemma.
The diagonal functor commutes with L E . In particular, E ( diag ( H ∗ P )) is isomorphic todiag ( E ( H ∗ P )) in ( P + P ) (cid:38) M ulti ( C ) .Proof. The proof follows as in [DH, 5.3], once we note that, by construction, the multicategories E (diag( H ∗ P )) and diag( E ( H ∗ P )) have the same set of objects. (cid:3) Proposition.
The multicategory diag E ( H P ) is a cofibrant object in ( P + P ) (cid:38) M ulti ( C ) . Moreover,there exists a weak equivalence of multicategories diag ( EH ∗ P ) −→ P . This is a many objects version of [DH, 5.4]. We prolong the functor E , applying E to H ∗ P degree-wise.Since in each degree H n P is a pointed bimodule (the basepoints come from the image of H ( P ) underdegeneracies), it follows that E ( H n ( P )) is a simplicial object in the category ( P + P ) (cid:38) M ulti ( C ) for each ≥
0. The functor E commutes with composition and units, i.e. the face and degeneracy maps, and thus E ( H ( P )) is a simplicial object in ( P + P ) (cid:38) M ulti ( C ).It follows that the multicategory diag E ( H ∗ P ) is an object under P + P . Since we know that E ( P )is isomorphic to P in the category ( P + P ) (cid:38) M ulti ( C ), this implies that the augmentation map η :diag( E ( H ∗ P )) −→ E ( P ) factors the the fold map P + P −→ P . Proof of Proposition.
The key observation is that E ( P ) is isomorphic to P under P + P . It follows that E ( H n P ) = E ( P ◦ ( n +2) ) is isomorphic to the free multicategory on U ( P ◦ n ), the underlying collection of P ◦ n , together with a map from P + P (coming from the basepoint). (cid:3) Spaces of Algebra Structures.
The category of right Q -modules, M Q , is a symmetric monoidal C -category (see appendix) and, as such, it makes sense to define generalized P -algebras taking values in M Q .More explicitly, a P -algebra structure on a right Q -module M is a multifunctor from P to End Q ( M ) (seealso, [Fre09]). The endomorphism multicategory End Q ( M ) has n -ary operations given by Hom M Q ( M ⊗ n , M )which is the space of right Q -module homomorphisms from the n -fold tensor product of M to M . This formsa simplicial multicategory in the usual way, and has both a natural left End Q ( M )-action and a natural right Q -action which makes End Q ( M ) into an End ( M )- Q -bimodule. The P - Q -bimodule structures are in one-to-onecorrespondence with P -algebra structures, i.e. operad homomorphisms P −→
End ( M ).3.12. Example.
The multicategory Q is naturally a right Q -module over itself, and the left action of Q onitself gives an equivalence End Q ( Q ) ∼ = Q . For a fixed object X in M , the simplicial set [ Hom ] ( P , End ( X )) is the space of P -algebra structures on the object X . This is again due to the fact that the endomorphism operad of an object X ∈ M is theuniversal object in M ulti ( C ) acting on X , i.e. that any action on X by an object P in M ulti ( C ) is therestriction of the End ( X )-action on X along a uniquely determined morphism P →
End ( X ).3.13. Example.
Let C denote the commutative operad and let (cid:100) S et denote the underlying multicategory of S et , namely the n -ary operations of (cid:100) S et are given by S et ( x × ... × x n ; x ) . One can check straight from the definitions that [ Hom ] ( C , (cid:100) S et ) is isomorphic to the category of commutativemonoids. The Moduli Space of Algebra Structures.
Fix a simplicial multicategory P with object setobj( P ) = S . Let M be a symmetric monoidal category which is tensored and cotensored over s S et , and let M S denote the category obtained as the product of copies of the category M indexed over S . If M is a (sim-plicial) monoidal model category, the the product category M S inherits a (simplicial) model structure fromthe model structure on M , where the fibrations, cofibrations, and weak equivalences formed coordinatewise.The simplicial category [ Hom ( P , M )] is a category of algebras over a triple , or monad , T . Explicitly, if wefix an object x of P , and let A be an object in M S , then we have a triple T := ( T ( A )) x = (cid:113) n ≥ ( (cid:113) x ,...,x n ∈ obj( P ) P ( x , . . . , x n ; x ) ⊗ Σ n ( A ( x ) ⊗ ... ⊗ A ( x n )) , let η : A → T ( A ) be the map A ( x ) −→ { id x } ⊗ A x → P ( x ; x ) ⊗ A ( x ) → ( T ( A ( x )) , and let µ : T T ( A ) → T ( A ) just be the map induced by the composition operations of P .3.14. Remark.
If we consider T as a functor T : M S −→ [ Hom ] ( P , M ) , hen one can easily check that T is the left adjoint to the forgetful functor [ Hom ] ( P , M ) −→ M S . In other words, T ( A ) is precisely the free P -algebra on A := { A ( x ) | A ( x ) ∈ M} x ∈ obj( P ) . Denote the free P -algebra on A by F P ( A ) . The following theorem is an easy generalization of [EM06, Theorem 11.2], which is itself a generalizationof [May].3.15.
Theorem. [EM06][May]
Given the triple T above on the category M , a T -algebra structure on an objectof M is equivalent to a simplicial multifunctor from P to M , and the simplicial category of T -algebras isisomorphic to the simplicial category [ Hom ] ( P , M ) . Corollary.
The category [ Hom ] ( P , M Q ) is a symmetric monoidal category over C and has all smalllimits and colimits. We know that the category of right Q -modules M Q admits a cofibrantly generated monoidal model categorystructure over s S et . In this case, a map of right Q -modules is a cofibration of right Q -modules if, and only ifit, is a retract of a relative I -complex where I := { K ◦ Q −→ L ◦ Q | n ≥ } and K → L runs over the generating cofibrations of Coll( C ). A map of right Q -modules is an acycliccofibration if, and only if, it is a retract of a relative J -complex, where J := { K ◦ Q −→ L ◦ Q | n ≥ } and K → L runs over the generating acyclic cofibrations of Coll( C ) (for more on the model structure ofColl( C ), see [BM07]).The generating (acyclic) cofibrations for the product category M S Q are defined similarly. Explicitly, for afixed object x in P we let ι x : M Q −→ M S Q be the left adjoint to the evaluation functor Ev x : M S Q −→ M Q ,i.e. given a fixed right Q -module A and an arbitrary object y in P , the object ( ι x A ) y in M S Q is either A if x = y or trivial otherwise. Now, we can define the sets ι ∗ I := { ι x f | f ∈ I, x ∈ obj( P ) } and ι ∗ J := { ι x f | f ∈ J, x ∈ obj( P ) } . So, a map is a cofibration of M S Q if, and only if, it is a retract of ι ∗ I ; a map of M Q is an acyclic cofibrationof M S Q if, and only if, it is a retract of ι ∗ J .3.17. Theorem (Model Structure) . Let P and Q be two simplicial multicategories. If the category of right Q -modules admits a cofibrantly generated model category structure then the category [ Hom ] ( P , M Q ) admits acofibrantly generated model category structure where a morphism is a weak equivalence (respectively, fibration)if the underlying map of right Q -modules is a weak equivalence (respectively, fibration). We will defer the proof of the theorem to the appendix.3.18.
Definition.
A right Q -module M is pointed if it comes with a unit map Q = ◦ Q −→ M ◦ Q −→ M. A pointed right Q -module M will be called quasi-free if the natural map Q −→ End Q ( M ) is weak equivalence of right Q -modules. We will say that a P - Q -bimodule is pointed (respectively, right quasi-free ) if it is pointed (respectively, quasi-free ) as a right Q -module. he category of pointed P - Q - bimodules is equivalent to the category of P - Q -bimodules M which havea natural homomorphism of bimodules F P ( Q ) −→ M . Moreover, the category of pointed P - Q bimodulesadmits a cofibrantly generated model structure [Fre09, Chapter 14].3.19. Remark.
In Fresse [Fre09, Chapter 14] the model structure on P - Q -bimodules depends on the multicat-egory Q being reduced . We claim this is equivalent to the condition of our bimodules being pointed . Noticethat endomorphism-operads are not reduced, since the zero operations End ( A )( − ; A ( x )) = A ( x ) . However,any object A under defines a reduced endomorphism multicategory (cid:103) End ( A ) . If P is reduced, a P -algebrastructure on A is also equivalent to a base point → A together with an operad map P → (cid:103) End ( A ) . Corollary.
The simplicial category [ Hom ] ( P , M Q ) is simplicially Quillen equivalent to the categoryof pointed P - Q -bimodules P M Q . Morita Theory
We will denote the category of right quasi-free P - Q -bimodules by P M ∗ Q . The model structure in theprevious section clearly induces a model structure on P M ∗ Q . This section is devoted to showing that thederived mapping space Map h ( P , Q ) is weakly homotopy equivalent to the moduli space of quasi-free P - Q -bimodules.4.1. Extension and Restriction of Scalars.
Given a multicategory P with obj( P ) := S and a map ofsets, F : T −→ S , We can construct a multicategory F ∗ ( P ) with object set T , with operations given by F ∗ ( P )( d , · · · , d n ; d ) := P ( F d , ..., F d n ; F d ) . It follows that given a multifunctor ψ : R −→ S , we can show that R operates on any right S -module N through the morphism ψ : R −→ S . This R -action defines a right R -module ψ ∗ N associated to N byrestriction. This structure has a natural left adjoint, which defines a right S -module ψ ∗ M by extension. Thefollowing results easily generalize from operads to general multicategories.4.1. Proposition.
Let ψ : R → S be a multifunctor. The extension-restriction functors ψ ∗ : M R −→ M S : ψ ∗ are functors of symmetric monoidal categories over C and define an adjunction relation in the -category ofsymmetric monoidal categories over C . Given another multifunctor φ : P → Q , we have extension and restriction functors on algebra categories φ ∗ : Alg P ( M ) (cid:29) Alg Q ( M ) : φ ∗ . By definition, the restriction functor φ ∗ : Alg Q ( M ) → Alg P ( M ) reduces to the identity functor φ ∗ ( B ) = B if we forget operad actions.4.2. Lemma.
The extension functor φ ∗ preserves weak equivalences and fibrations. In particular, the extension functor φ ∗ is the right adjoint in a Quillen adjunction. We delay the proof ofthe following proposition to the appendix.4.3. Proposition.
Let φ : P → Q be a multifunctor between two admissible, Σ -cofibrant simplicial multicat-egories, and let M be a left proper, cofibrantly generated monoidal model category over C . Them φ ∗ : Alg P ( M ) (cid:29) Alg Q ( M ) : φ ∗ defines a Quillen adjunction. Furthermore, if φ : P → Q is a weak-equivalence in M ulti ( C ) , then ( φ ∗ , φ ∗ ) defines a Quillen equivalence. In particular, extensions and restrictions on the left commute with extensions and restrictions on the rightup to coherent functor isomorphisms . .4. Proposition.
Let P be any well pointed, Σ -cofibrant multicategory enriched in C . Given ρ ∗ : M (cid:29) N : ρ ∗ a Quillen adjunction of monoidal model categories over C . The functors ρ ∗ : Alg P ( M ) (cid:29) Alg P ( N ) : ρ ∗ induced by ρ ∗ and ρ ∗ define a Quillen adjunction. If ρ ∗ : M (cid:29) N : ρ ∗ forms a Quillen equivalence, then ρ ∗ : Alg P ( M ) (cid:29) Alg P ( N ) : ρ ∗ forms a Quillen equivalence. Lifting Extension-Restriction of Scalars.
Any multifunctor ψ : R → S also induces a map G : P + R −→ P + S and a Quillen adjunction G ∗ : ( P + R ) (cid:38) M ulti ( C ) (cid:28) ( P + S ) (cid:38) M ulti ( C ) : G ∗ . The coproduct of operads preserves weak equivalences (see [DH, 4.3] [ ? ]). Therefore, if ψ is a weak equivalencewhich fixes objects , then by [Rezk02, Prop. 2.5], we know that G ∗ : ( P + R ) (cid:38) M ulti ( C ) (cid:28) ( P + S ) (cid:38) M ulti ( C ) : G ∗ is a Quillen equivalence. This depends heavily on the fact that categories of simplicial multicategories withfixed sets of objects is a left proper model category (combine [Rezk02, 4] with [BM07, 1.5]).4.5. Proposition.
Let P be an admissible, Σ -cofibrant multicatgory. Let ψ : R → S be a multifunctor betweentwo locally cofibrant multicategories which fixes objects. If ψ is a weak equivalence, then L E ( ψ ) : ( P + R ) (cid:38) M ulti ( C ) −→ ( P + S ) (cid:38) M ulti ( C ) is a weak equivalence.Proof. Let ( ψ ∗ , ψ ∗ ) and ( G ∗ , G ∗ ) be the Quillen equivalences above induced by ψ. We want to show thatgiven B in P M R , L E ( B ) is weakly equivalent to L E ( A ) for some A in P M S . If A is cofibrant, then ψ ∗ ( A )is a cofibrant object weakly equivalent to B . Since E is left adjoint to a functor that preserves fibrationsand weak equivalences, we know that E preserves cofibrant objects and weak equivalences between cofibrantobjects, and thus that Eψ ∗ ( A ) is weakly equivalent to L E ( B ) = E ( B c ). The adjoint functor theorem impliesthat G ∗ L E ( B ) = Eψ ∗ ( A ). Since G ∗ is the left adjoint of a Quillen equivalence, G ∗ takes weak equivalencesbetween cofibrant objects to weak equivalences, and we are done. (cid:3) Lemma.
Let P be an admissible, Σ -cofibrant simplicial multicategory and let Q be locally cofibrant. Let f : M → N be a homomorphism between cofibrant pointed P - Q -bimodules. Assume that L E ( M ) is an objectof ( P + Q ) (cid:38) M ulti ( C ) so that the natural map Q → ( P + Q ) → L E ( M ) is a weak equivalence. Then if f isa weak equivalence, the natural map Q → ( P + Q ) → L E ( N ) is also a weak equivalence. Proposition.
Let F : P −→ Q be a multifunctor, and let P be admissible Σ -cofibrant and Q be locallycofibrant. Then F equips P with the structure of a right quasi-free P - Q -bimodule by restriction on the right.Moreover, the induced operad L E ( F ) is an object of ( P + Q ) (cid:38) M ulti ( C ) so that the map Q → ( P + Q ) → L E ( F ) is a weak equivalence.Proof. The first part follows from the isomorphism
End P ( P ) ∼ = P . Let F ∗ : Alg P ( M Q ) −→ Alg P ( M P )be the restriction functor of proposition 4.4, and let F ∗ denote the left adjoint. It follows from standardarguments ( [BM07, Before Theorem 4.1]) that F ∗ commutes with free algebras. It follows that F ∗ takes ourbasepoint F P ( P ) → P to the required basepoint F P ( Q ) → Q .Let G : P + P −→ P + Q be the map id + F and let ( G ∗ , G ∗ ) denote the induced Quillen adjunction onmulticategories under P + Q . Let η : diag( H ∗ P ) −→ P be the cofibrant resolution of P in the category of P - P -bimodules we described in 3.11. For the remainder of this proof, denote this cofibrant bimodule by P c . otice that both P c and P are cofibrant objects at the level of right P -modules. This implies that inducedmap j : F ∗ ( P c ) → F ∗ ( P ) = Q is a weak equivalence at the level of right Q -modules. Now, by Lemma 4.6, weknow that we may assume that F ∗ ( P c ) and Q are also fibrant objects (taking the fibrant replacement whichfixes objects). It then follows from Theorem 6.8 that the P -algebra structure maps are compatible with thisweak equivalence in a homotopy coherent way. It now follows that EF ∗ ( P c ) = G ∗ E ( P c ).Now, using the Hochschild resolution as a cofibrant replacement for P , we have shown that L E ( P ) isprecisely P under the fold map P + P −→ L E ( P ) = P .It remains to check two things:(1) that the object E ( P c ) is a cylinder object for E ( P ) and(2) that the composite(4.8) Q in (cid:47) (cid:47) P + Q i (cid:47) (cid:47) E ( F ∗ P c ) E ( η ) (cid:47) (cid:47) L E ( F ∗ Q )is a weak equivalence.To prove (1), we consider what E ( P c ) looks like. H ∗ ( P ) is a simplicial object in P - P -bimodules, so E ( H ∗ ( P )) is applied degree-wise. This means that E ( H ∗ ( P )) is a simplicial object under P + P . If we nowapply the diagonal, we can consider the augmentation map E ( η ) : diag ( E ( H ∗ ( P )) −→ E ( P ) = P , which,being a morphism in ( P + P ) (cid:38) M ulti ( C ), is a factorization of the fold map P + P i (cid:47) (cid:47) diagE ( H ∗ ( P )) E ( η ) (cid:47) (cid:47) E ( P ) = P . We know that diag E ( P c )) is isomorphic to E ( P c ) as simplicial multicategories under P + P . Moreover,we know that diag E ( P c ) is a cofibrant multicategory under P + P and that the augmentation diag E ( P c ) −→ E ( P ) = P is a weak equivalence. Since diag E ( P c ) is cofibrant, it follows that the map i : P + P −→ diag E ( P c )is a cofibration. Putting this all together, we have(4.9) P + P i (cid:47) (cid:47) E ( P c ) E ( η ) (cid:47) (cid:47) E ( P ) = P with E ( η ) a weak equivalence and i a cofibration.Now to show (2), notice that (1) implies that the composite map(4.10) P in (cid:47) (cid:47) P + P i (cid:47) (cid:47) E ( P c )is a weak equivalence of multicategories. This follows from the 2-out-of-3 property and the observation that(4.11) P in (cid:47) (cid:47) P + P i (cid:47) (cid:47) E ( P c ) E ( η ) (cid:47) (cid:47) E ( P ) = P is the identity map.We restrict along F to get the the map β : Q → E ( F ∗ P c ) which is the composite(4.12) Q in (cid:47) (cid:47) P + Q i (cid:47) (cid:47) E ( F ∗ P c ) . Consider the diagram(4.13) Q β (cid:47) (cid:47) EF ∗ ( P c ) E ( η ) (cid:47) (cid:47) U (cid:15) (cid:15) EF ∗ ( P ) U (cid:15) (cid:15) F ∗ ( P c ) j (cid:47) (cid:47) F ∗ ( P ) . It follows that β is a weak equivalence, as we are looking at the identity map Q −→ F ∗ ( P ) = Q factoredby β and a weak equivalence. (cid:3) utting together Lemma 4.6, Proposition 4.5 and Proposition ?? we have now shown that right quasi-free P - Q -bimodules get lifted to zig-zags [ P −→ M ∼ ←− Q ].4.14. Theorem. If M is right quasi-free P - Q -bimodule then the natural map Q → P + Q → L E ( M ) is a weakequivalence in ( P + Q ) (cid:38) M ulti ( C ) .Proof. Let M be a right quasi-free P - Q -bimodule. We may assume that M is fibrant and cofibrant byLemma 4.6 and that the morphism F P ( Q ) −→ M is a cofibration. The model structure on P M ∗ Q impliesthat we still have a cofibration after forgetting the P -algebra structure U P F P ( Q ) −→ U P ( M ) . Consider the square of right Q -modules:(4.15) End Q ( f ) (cid:47) (cid:47) (cid:15) (cid:15) End Q ( Q ) (cid:15) (cid:15) End Q ( M ) (cid:47) (cid:47) End Q ( Q , M ) . By Theorem 6.8, we know that we have weak equivalences
End Q ( f ) −→ End Q ( M ) and End Q ( f ) −→ End Q ( Q ) = Q . Now, if we remember that M is actually a P - Q -bimodule, and thus we have a P -algebrastructure map α : P −→
End Q ( M ). As in the proof of Theorem 6.8, we can lift the P -algebra structure on End Q ( M ) to a P -algebra structure on End Q ( f ) in a homotopy coherent way. Moreover, we can compose toget the following right quasi-free P (cid:48) - Q -bimodule β : P −→
End Q ( f ) −→ End Q ( Q ) = Q . By the proposition ?? , we know that Q → P + Q → L E ( β ) is a weak equivalence by proposition 4.5 we knowthat this implies that Q → P + Q → L E ( α ) is a weak equivalence. (cid:3) Equivalences of Moduli Spaces.
The main theorem of this paper is the following.4.16.
Theorem.
The derived mapping space
Map h ( P , Q ) is weakly homotopy equivalent to the moduli spaceof right quasi-free P - Q -bimodules, P M ∗ Q . The adjoint pair (
E, U ) is a Quillen pair, in which U preserves all weak equivalences. Before we can provethe main theorem we assume that α : P −→
End Q ( M ) is a fixed P - Q -bimodule structure, and consider thehomotopy fiber of U : P + Q (cid:38) M ulti ( C ) −→ P M ∗ Q at the basepoint M .Let B be the subcategory of M ulti ( C ) consisting of simplicial multicategories with fixed object setsobj( P ) × obj( Q ). In this case, B is a left proper model category (see, [Rezk02]). For this next proposition,we consider the restriction of the adjoint pair ( E, U ) to B .4.17. Proposition.
Suppose that ( E, U ) is the Quillen pair above, and that the right adjoint U, preserves allweak equivalences. Then if either: (1) M in P M ∗ Q is cofibrant, or (2) B is a left proper model category and E ( M c ) −→ E ( M ) is a weak equivalence in B for all M in P M ∗ Q the homotopy fiber of wU over M in B is equivalent to the nerve of the under category M (cid:38) wU .Proof. The assumption that our right adjoint U : B −→ P M Q preserves weak equivalences, implies that theunder category M (cid:38) wU is isomorphic to the union of the components of the moduli space w ( EM (cid:38) B )containing maps α : EM −→ X whose adjoint α (cid:48) : M −→ U X is a weak equivalence in P M ∗ Q . Objectsof the category M (cid:38) wU look like [ M ∼ −→ U ( X )] and morphisms [ M ∼ −→ U ( X )] → [ M ∼ −→ U ( X )] areweak equivalences U ( f ) : U ( X ) ∼ −→ U ( X ) in P M ∗ Q such that the obvious diagram commutes. Objects of w ( EM (cid:38) B ) look like [ EM ∼ −→ X ] and morphisms [ EM ∼ −→ X ] → [ EM ∼ −→ X ] are weak equivalences f : X ∼ −→ X in B such that the obvious diagram commutes. f we assume condition (1), it follows from [Rezk02, After 2.5] that given a weak equivalence M → N between cofibrant objects in P M ∗ Q that the induced map M (cid:38) wU −→ N (cid:38) wU is a weak equivalence. Ifwe instead assume condition (2), then we know that the natural map M (cid:38) wU −→ M c (cid:38) wU is a weakhomotopy equivalence by Rezk’s characterization of left properness [Rezk02, 2.5].Now, consider the subcategory of cofibrant pointed P - Q -bimodules ( P M ∗ Q ) c ⊂ P M ∗ Q . The objects ofthe category w ( P M Q ) c (cid:38) wU look like [ M ∼ −→ U ( X )], where M is a cofibrant pointed P - Q -bimodule.Morphisms [ M ∼ −→ U ( X )] −→ [ M ∼ −→ U ( X )] are pairs ( φ, f ) where φ : M ∼ −→ M is a weak equivalencebetween cofibrant pointed P - Q -bimodules and U ( f ) : U ( X ) ∼ −→ U ( X ) is a weak equivalence (i.e. f : X → X is a weak equivalence in B ).The category w ( P M ∗ Q ) c (cid:38) wU is a path space construction. In particular, we have a commutativediagram:(4.18) w ( P M Q ) c (cid:38) wU pr (cid:47) (cid:47) pr (cid:15) (cid:15) w B wU (cid:15) (cid:15) w ( P M Q ) c in (cid:47) (cid:47) w ( P M Q ) . The map pr takes the object [ M ∼ −→ U ( X )] to M and pr takes [ M ∼ −→ U ( X )] to X in B . The map in is the induced inclusion of moduli spaces. Both the maps in and pr induce weak homotopy equivalenceson moduli spaces.For a fixed cofibrant pointed P - Q -bimodule M we can construct a functor M (cid:38) wU −→ M (cid:38) pr , whichis natural in M and induces a weak homotopy equivalence on nerves. We want to show that the homotopyfiber of pr over a cofibrant M is weakly homotopy equivalent to the nerve of M (cid:38) pr . Now, we havealready argued that every weak equivalence between (the cofibrant objects) M ∼ −→ M induces a weakequivalence of the categories M (cid:38) pr ∼ −→ M (cid:38) pr . Now, we apply Quillen’s Theorem B to show thatthe homotopy fiber of pr over a cofibrant M is weakly homotopy equivalent to the nerve of M (cid:38) pr . Theproposition now follows from the diagram 4.18. (cid:3) Under the same hypotheses as the previous proposition, we can consider what happens to derived mappingspaces under the adjunction (
E, U ).4.19.
Theorem.
Suppose that ( E, U ) is as above, and that the right adjoint U, preserves all weak equivalences.Then there is a natural weak homotopy equivalence Map h ( M, U ( X f )) ∼ −→ Map h ( E ( M c ) , X ) . Proof.
Assume that M is cofibrant in P M ∗ Q and that X is fibrant in B . We can construct a category Z whichhas objects [ M ∼ ←− E ( M ) → X ∼ ←− X ] and morphisms are pairs ( E ( φ ) , f ) where φ : E ( M ) −→ E ( M )and f : X −→ X are morphisms which make the obvious diagram commute. We will show that the nerveof Z is weakly homotopy equivalent to Map h ( EM, X ).Let ( P M Q ) c be the subcategory of cofibrant pointed P - Q -bimodules; let M ∼ ←− M be a weak equivalencebetween cofibrant pointed bimodules. A functor F : Z −→ w ( P M Q ) c (cid:38) M is given by[ M ∼ ←− E ( M ) → X ∼ ←− X ] −→ [ M ∼ ←− M ] . Fix the object Y := [ M ∼ ←− M ]. Then consider the over category Y (cid:38) F . Objects of Y (cid:38) F can be re-written as [ M ∼ ←− M → X ∼ ←− X ]. The nerve of Y (cid:38) F is weakly homotopy equivalentto Map h ( E ( M ) , X ). By the previous proposition, Map h ( E ( M ) , X ) is weakly homotopy equivalent tothe homotopy fiber of the functor F . Since w ( P M ∗ Q ) c (cid:38) M has a terminal object, the moduli space w ( P M ∗ Q ) c (cid:38) M is contractible, and thus Map h ( E ( M ) , X ) is weakly homotopy equivalent to the nerve of Z . n a similar manner, we can consider a category Z (cid:48) with objects that look like[ M ∼ ←− M → U ( X ) ∼ ←− X ] . A dual argument to above shows that Z (cid:48) is weakly homotopy equivalent to Map h ( M, U ( X )). It is thenclear that there is a weak homotopy equivalence Z −→ Z (cid:48) . (cid:3) Proof of the Main Theorem.
The subcategory of objects of P + Q (cid:38) M ulti ( C ) which satisfy the propertythat the natural map Q → P + Q → R is a weak equivalence is a category of zig-zags [ P −→ R ∼ ←− Q ]. What’smore, if we consider the restrictions of the functor pair ( E, U ) to right quasi-free objects, then the functorsfix objects. The theorem now follows from proposition 4.17 and theorem 4.19. (cid:3) The Cosimplicial Model
As we mentioned in Section 1, given that we assume Q is fibrant, we can also take a cosimplicial resolutionof P to obtain a model for Map h ( P , Q ). The fact that we have a fibrant replacement functor that fixes objectsis key to what follows. The point of this section is to explicitly describe the components for this model ofMap h ( P , Q ) , from which we can then describe internal hom-objects.5.1. Cosimplicial Operads.
The category M ulti ( C ) is fibered over varying sets of objects ([BM07, 1.6]).A cosimplicial multicategory is a cosimplicial object in this fibered category. More explicitly, a multicategory P • is given by a cosimplicial set of objects C • , such that for each n we have an operad P n , with object set C n . The maps P n (cid:28) P m are induced by arrows C n (cid:28) C m that come from maps [ n ] −→ [ m ] in ∆.The geometric realization of a cosimplicial operad P • over C • , is given by a functor s S et −→ M ulti ( C )which sends X to the multicategory | X | P • := X ⊗ ∆ P • with objects X ⊗ ∆ C • . An algebra over | X | P • consists of an algebra A x over the C n -colored operad P n , where we allow x to vary over the n -simplices of X .As one might expect, this cosimplicial object P • is completely determined by its 2-skeleton (see [BM07,6]). This means that given a simplicial set X , the inclusion sk ( X ) −→ X of the 2-skeleton of X induces aweak equivalence | sk ( X ) | P • ∼ = −→ | X | P • of multicategories.5.1. Example.
Let A = A be the operad whose algebras are associative unitary monoids. Let A = BiM od be the operad with -objects whose algebras are triples ( A , M, A ) , where A , A are A -algebras, and M isan A - A -bimodule. The operads A and A form part of a cosimplicial operad A • over the cosimplicial setgiven by C n = { a i | ≤ i ≤ n } ∪ { b ij | ≤ i < j ≤ n } , the objects for n + 1 monoids A i and n +12 bimodules M ij . Applying the Boardman-Vogt resolution, one getsa cosimplicial operad W ( A • ) . The W ( A ) -algebras are A ∞ -algebras, the W ( A ) -algebras are ∞ -bimodulesover such A ∞ -algebras, and so on. In a similar manner, we can select any P and create a cosimplicial P • which parameterizes strings ofmorphisms between P -algebras described as follows. Let P be P , and P n be the multicategory whosealgebras are n -simplicies A → A → · · · → A n of P -algebra homomorphisms.In the case where P has one object, the multicategory P has been extensively studied by Markl [ ? ].The operad P has objects { , } and algebras triples ( A , A , f ) where A and A are P -algebras, and : A → A is a map of P -algebras. The operations can be given explicitly by(5.2) P ( x , . . . , x n ; x ) = (cid:40) P ( n ) if max( x , . . . , x n ) ≤ x ;0 otherwise . Notice that a P -algebra consists of exactly two objects A and A . the structure of a P -algebra, and theunit morphism 1 : I → P (1) corresponds to a map α : I → P (0; 1) , which corresponds to a homomorphismof P -algebras, f : A → A . One can write out an explicit description of P n in a similar way.The multicategory P n is defined to be the pushout(5.3) P n = P (cid:116) P P (cid:116) P P ... P (cid:116) P P , over θ i : P −→ P n where θ i : [1] −→ [ n ] ranges over the inclusions { , } to { i, i + 1 } , 0 ≤ i ≤ n −
1. Thereare extension and restriction functors which are induced by the face and degeneracy maps(See, [BM07, 6]).For a cosimplicial operad P • , the categories of algebras Alg P n ( M ) , n ≥
0, together form a (very large) simplicial category, which we denote by Alg P • ( M ).5.4. Observation.
The simplicial category
Alg P • ( M ) is the nerve of the category Alg P ( M ) . We can apply the W -construction to the cosimplicial operad P • levelwise, and this provides a functorialcofibrant replacement W ( P • ) . In level 0, W ( P ) is just W ( P ) , i.e. a (functorial) cofibrant replacement for P . It was observed by Berger-Moerdijk [6][BM07] that the category Alg W ( P • ) ( M ) should be equivalent tothe nerve of the category Alg P ( M ) up to weak equivalence. We use the results of the previous section tomake this precise.Explicitly, we lift already know that P M Q is weakly homotopy equivalent to Map h ( P , Q ) and, given acosimplicial resolution of P we should be able to construct a weak homotopy equivalence from P M Q to azig-zag of the form [ P ∼ (cid:17) P n −→ Q ] when Q is fibrant (see Section 1). We will need the following theorem.5.5. Theorem. [BM07, Theorem 6.4]
Let M be a left proper, cofibrantly generated, monoidal model categoryover C and let P be a Σ -cofibrant C -enriched multicategory. If all of the cofibrant operads W ( P n ) all areadmissible, then for n ≥ , the map θ induces a Quillen equivalence (5.6) Alg W ( P ) ( M ) × Alg W ( P ( M ) × · · · × Alg W ( P ( M ) Alg W ( P ) ( M ) ∼ −→ Alg W ( P n ) ( M ) . Given a morphism f : M → N between P - Q -bimodules we know that we can construct a P - P -bimodule End ( M, N ) as in Theorem 6.8. As in the previous section, we know that we can consider these bimodules asmulticategories under P + P . Our first proposition says that, under certain hypotheses, the endomorphismbimodule End ( M, N ) is a P -algebra in the category P + P (cid:38) M ulti ( C ).5.7. Proposition.
Assume that f : M → N is a weak equivalence between cofibrant and fibrant pointed P - Q -bimodules. Then there exists a P -algebra structure on End ( M, N ) in the homotopy category of P + P (cid:38)M ulti ( C ) .Proof. By our assumption that M and N are cofibrant and fibrant right quasi-free P - Q -bimodules we knowthat L E ( M ) = E ( M ) and L E ( N ) = N . Moreover, we know that E preserves weak equivalences betweencofibrant objects, so that E ( f ) is a weak equivalence.Now, by Theorem 6.8 we know that exists a pullback square (of right Q -modules):(5.8) End ( f ) (cid:47) (cid:47) (cid:15) (cid:15) End ( M ) (cid:15) (cid:15) End ( N ) (cid:47) (cid:47) End ( M, N ) . he object End ( f ) is characterized by the fact that a morphism P →
End ( f ) is equivalent to giving a P -algebra structure on End ( N ), a P -algebra structure on End ( M )in such a way that f is a morphism of P -algebras. In other words, there exists a map P → End ( f ) and thus a P - P -bimodule via the P - P -bimodulestructure on P .The collection End ( M, N ) also has a P - P -bimodule, via the maps P →
End ( M ) → End ( M, N ) and
P →
End ( N ) → End ( M, N ). It is also the case that, by our assumptions that M and N are cofibrant andfibrant that End ( M, N ) is cofibrant, and so we can lift
End ( M, N ) by E to an object in P + P (cid:38) M ulti ( C ).Now, we can consider either composite P → End ( f ) → End ( M ) → End ( M, N )or P → End ( f ) → End ( N ) → End ( M, N )as a morphism between P to End ( M, N ) in ( P + P ) (cid:38) M ulti ( C ) and, in particular, we have the followingdiagram (as objects in P + P (cid:38) M ulti ( C )). P + P α + β (cid:47) (cid:47) (cid:15) (cid:15) End ( M, N ) P . Now, since
End ( M, N ) is fibrant and P is cofibrant in this picture, we can repeat the proof of Theorem 6.8to show that the P - P -bimodule structure on P maps to a P - P -bimodule structure on End ( M, N ) in ahomotopy coherent way. In particular, this tells us that E ( f ) is represented by: W ( P ) + W ( P ) α + β (cid:47) (cid:47) (cid:15) (cid:15) End ( A, B ) W ( P ) E ( f ) (cid:55) (cid:55) (cid:110)(cid:110)(cid:110)(cid:110)(cid:110)(cid:110)(cid:110)(cid:110)(cid:110)(cid:110)(cid:110)(cid:110) . (cid:3) Theorem.
Assume that P is Σ ∗ -cofibrant and Q is fibrant. Then Alg W P • ( M Q ) is weakly homotopyequivalent to Map h ( P , Q ) .Proof. By Theorem ?? we know that W ( P ) M Q ) is weakly homotopy equivalent to Map h ( P , Q ). We alsoknow that there is a natural morphism of simplicial sets [ Hom ] ( W P , Q ) −→ W P • M ∗ Q which sends a string A → ... → A n +1 to the corresponding W ( P n )-algebra in W P • M ∗ Q . The remainder of the proof follows fromProposition 5.7 and Theorem [BM07, 6.4]. (cid:3) The theorem basically provides us with a specific framing ( [Hov99, Chapter 5]). As a consequence weknow that when K is a finite simplicial set, there exist maps in Ho( s S et ) P R K −→ Map h ( K, Alg W P • (Mod Q )) P R K −→ M ap ( K, Map h ( P , Q ))are in fact isomorphisms. Moreover, it implies that for any finite K in Ho( s S et ) and any Σ ∗ -cofibrantmulticategory P , we get an isomorphism[ K, Alg W P • ( M Q )] −→ [ K, Map h ( P , Q )] . Corollary.
The monoidal category Ho ( M ulti ( C )) is closed. Furthermore, for any two simplicial mul-ticategories P and Q there is a natural isomorphism in Ho ( M ulti ( C )) R Hom ( P , Q ) ∼ P M Q . . Algebras over Multicategories
For a given (symmetric) multicategory P , a P -algebra A is an object in the product category C obj( P ) together with a left P -action, i.e. a collection of C -morphisms α x ,...,x n ; x : P ( x , ..., x n ; x ) ⊗ A ( x ) ⊗ ... ⊗ A ( x n ) → A ( x ) , satisfying axioms for associativity, units and equivariance. A P -algebra homomorphism f : A → B is afamily of C -morphisms { f : A ( x i ) → B ( x ) } x i ∈P which fit into the following commutative diagram: P ( x , ..., x n ; x ) ⊗ A ( x ) ⊗ ... ⊗ A ( x n ) −−−−→ A ( x ) id ⊗ f x ⊗ ... ⊗ f xn (cid:121) f x (cid:121) P ( x , ..., x n ; x ) ⊗ B ( x ) ⊗ ... ⊗ B ( x n ) −−−−→ B ( x ) . We denote the resulting category by Alg P ( C ).Equivalently, a P -algebra structure on an object A ∈ C obj( P ) , is a multifunctor P →
End ( A ) which fixesobjects. The classifying object, End ( A ) , is defined by End ( A )( x , ..., x n ; x ) := Hom C ( A ( x ) ⊗ ... ⊗ A ( x n ) , A ( x ))with composition (respectively Σ n -actions) induced by substitution (respectively permutation) on the sourcefactors. This object is called the endomorphism multicategory of A ∈ C obj( P ) .6.1. Example. If P is an ordinary category, then the category of P -algebras is the ordinary functor category [ P , S et ] . If P is a strict monoidal category then an algebra of the underlying multicategory is a lax monoidalfunctor from P to ( S et, × , . Example.
For each multicategory P there exists an algebra A defined by taking A ( x ) to be the set P ( − ; x ) of arrows in P from the empty sequence into x . When P is the multicategory of modules over somecommutative ring R , this P -algebra is just the forgetful functor from R -modules to S et . Example.
There exists a multicategory Op C , whose category of algebras is the category of operads in C .The set of objects in this case is the natural numbers N .The elements of Op ( n , . . . , n k ; n ) are equivalence classes of triples ( T, σ, τ ) where T is a planar rooted treewith n input edges and k vertices, σ is a bijection { , . . . , k } → V ( T ) (i.e. the set of vertices of T ) with theproperty that the vertex σ ( i ) has valence n i (i.e. n i input edges), and τ is a bijection { , . . . , n } → in ( T ) , theset of input edges of T . Two such triples ( T, σ, τ ) , ( T (cid:48) , σ (cid:48) , τ (cid:48) ) represent the same element of Op ( n , . . . , n k ; n ) if there is a (planar) isomorphism ϕ : T → T (cid:48) with ϕ ◦ τ = τ (cid:48) and ϕ ◦ σ = σ (cid:48) .Any α ∈ Σ k induces a map α ∗ : Op ( n , . . . , n k ; n ) −→ Op ( n α (1) , . . . , n α ( k ) ; n ) sending (the equivalenceclass of ) ( T, σ, τ ) to ( T, σα, τ ) . The identity element n ∈ Op ( n ; n ) is represented by the tree t n (the corollawith n leaves) whose inputs are numbered , . . . , n from left to right with respect to the planar structure.The composition product is defined as follows: given ( T, σ, τ ) as above, and k other such ( T , σ , τ ) , . . . , ( T k , σ k , τ k ) , with n , . . . , n k inputs and p , . . . , p k vertices respectively, one obtains a newplanar rooted tree T (cid:48) by replacing the vertex σ ( i ) in T by the tree T i , identifying the n i input edges of σ ( i ) in T with the n i input edges of T i via the bijection τ i (the l -th input edge of σ ( i ) in the planar order is matchedwith the input edge τ i ( l ) of T i ).The vertices of the new tree T (cid:48) are numbered in the following order: first the vertices of T σ (1) in the ordergiven by σ , then the vertices in T σ (2) in the order given by σ , etc. In other words, the map { , . . . , p + · · · + p k } → V ( T (cid:48) ) is given by ( σ × · · · × σ k ) ◦ σ ( p , . . . , p k ) where σ ( p , . . . , p k ) permutes the blocks of size p i .The new tree T (cid:48) still has n input edges, which are ordered as given by τ and the identifications given by the i . Notice that Op ( n ; n ) = Σ n if n = n , and Op ( n i , n ) = φ otherwise. More precisely, Op ( n, n ) consists ofpairs ( t n , τ ) where t n is the tree above and τ is a numbering of its inputs. The composition product of Op in particular gives a map Op ( n, n ) × Op ( n, n ) −→ Op ( n, n ) which sends (( t n , τ ) , ( t n , ρ )) to ( t n , ρτ ) , so that Op ( n ; n ) is identified with the opposite group of Σ n .The Op -algebras are exactly the operads in sets. Applying the strong symmetric monoidal functor S et → C gives Op C whose algebras are exactly the operads in C . Let M be a C -model category and P a multicategory enriched in C . Then there is an adjoint pair F P : M obj( P ) (cid:47) (cid:47) Alg P ( M ) : U P , (cid:111) (cid:111) where F P is the free P -algebra functor defined by(6.4) F P ( A )( x ) = (cid:97) n ≥ (cid:97) x ,...,x n ∈ obj( P ) P ( x , . . . , x n ; x ) ⊗ Σ n A ( x ) ⊗ · · · ⊗ A ( x n ) for every A = ( A ( x )) x ∈ obj( P ) in M obj( P ) , and U P is the forgetful functor. If a simplicial multicategory P isΣ-cofibrant, i.e. for each x , ..., x n ; x P ( x , ..., x n ; x ) is a cofibrant object in M then the model structure on M obj( P ) is transferred to Alg P ( M ) along the free-forgetful adjunction (see [BM07]).6.1. Endomorphism Modules for Algebras.
Let us denote by M S the product category of copies of M indexed by the set obj( P ) = S . For each A = { A ( x ) } x ∈ S we define a simplicial multicategory End ( A ) ∈M ulti ( C ) S (6.5) End ( A )( x , . . . , x n ; x ) = Hom C ( A ( x ) ⊗ · · · ⊗ A ( x n ) , A ( x )) , which forms a classifying object for P -algebra structures on A . The monoidal product is ordinary compositionin M and the Σ n -actions are given by permuting the source factors. The object End ( A ) is called the endomorphism multicategory or endomorphism S -colored operad of A ∈ C obj( P ) .We can define a similar object which provides a classifying object for the P -algebra homomorphisms f : A −→ B in C S , i.e., an S -indexed family of maps { f x : A ( x ) −→ B ( x ) } x ∈ S in C , there is an endomorphismobject End ( f ), defined as the pullback of the following diagram of collections:(6.6) End ( f ) (cid:47) (cid:47) (cid:15) (cid:15) End ( A ) (cid:15) (cid:15) End ( B ) (cid:47) (cid:47) End ( A, B ) . The collection
End ( A, B ) is called an endomorphism module between P -algebras A and B and is defined(6.7) End ( A, B )( x , . . . , x n ; x ) = Mor C ( A ( x ) ⊗ · · · ⊗ A ( x n ) , B ( x )) . There are natural maps
End ( A ) −→ End ( A, B ) and
End ( B ) −→ End ( A, B ) which come from composingwith f on either side. The collection
End ( f ) inherits a multicategory structure from End ( A ) and End ( B )(cf. [BM03, Theorem 3.5]). It also turns out that End ( A, B ) forms a left
End ( B )-module and a right End ( A )module. Moreover, these actions are compatible, giving us our first example of an operadic bimodule. Wewill discuss the endomorphism modules more in the next section. We choose the definition of
End ( f ) so that it will provide a classifying object for P -algebra homomorphisms f : A −→ B. More specifically, a multifunctor
P −→
End ( f ) is equivalent to providing a P -algebra structureon A and a P -algebra structure on B in such a way that f is P -algebra map between them.Versions of the following theorem appear in [Rezk96],[BM03],[BM07],and [BV73]. We take A ( x ) ⊗ · · · ⊗ A ( x n ) is taken to be the unit if n = 0. Set theoretically,
End f ( n ) = { ( φ, ψ ) ∈ End A ( n ) × End B ( n ) | fφ = ψf ⊗ n } . If P is a non-symmetric multicategory, then endomorphism objects are defined in the same way, by forgetting the symmetricgroup action on End ( A ). .8. Theorem.
Let f : A → B be a map between objects in the diagram category C S , and assume that C be asymmetric monoidal model category which satisfies all of the additional conditions necessary for M ulti ( C ) S to support a model category structure. Further, suppose that P is a Σ -cofibrant object in M ulti ( C ) S , i.e.that the underlying collection of P is a cofibrant object in Coll( C ) S . (1) Assume that B is fibrant as an object in C S , and that f ⊗ n is a trivial cofibration for each n ≥ ,then any P -algebra structure on A extends along f to a P -algebra structure on B . (2) If A is cofibrant as an object in C S , and f is a trivial fibration, then any P -algebra structure on B can be lifted along f to a P -algebra structure on A . (3) If both A and B are bifibrant objects in C S , and f is a weak equivalence, then any P -algebra structureon A (respectively B ) induces a P -algebra structure on B (respectively A ) in such a way that f preserves the P -algebra structures up to homotopy.Proof. We define the collections
End ( A, B ) and
End ( f ) as we did above. The key idea in the proof is thata morphism f is compatible with the P -algebra structure maps P →
End A and P →
End B if and only ifthese are induced by an operad map P →
End ( f ).We are first assuming that f has a fibrant target. The model category structure on C S has weak equiv-alences and fibrations defined objectwise. Further, C S has a symmetric monoidal tensor product inducedby the symmetric monoidal tensor product from the category C , and this structure is compatible with themodel category structure, i.e. C S supports the structure of a monoidal model category over C .Now, given our assumption that B is fibrant, and that C S is a C -model category, we can apply thepushout-product axiom, to show that the horizontal maps of the diagram(6.9) End ( f ) (cid:47) (cid:47) (cid:15) (cid:15) End ( A ) (cid:15) (cid:15) End ( B ) (cid:47) (cid:47) End ( A, B ) , are trivial fibrations. The additional assumption that P is a cofibrant operad, implies that the P -algebrastructure map P →
End A has a lift P →
End f → End B giving the required P -algebra structure on B .The hypothesis of (2) are dual, implying that End f → End B is a trivial fibration, and that the the P -algebra structure map P →
End B lifts to P →
End f → End A . Now assume that f is a weak equivalence between cofibrant-fibrant objects and we assume that A is a P -algebra, i.e. there exists a morphism P →
End A . We can factor f into a trivial cofibration f : A → Z followed by a trivial fibration f : Z → B . Since f is a trivial fibration with a cofibrant target, we mayassume that f admits a trivial cofibration as section. Now consider the following pullback diagram:(6.10) End f φ (cid:47) (cid:47) (cid:15) (cid:15) End Z ( f ) ∗ (cid:15) (cid:15) End B ( f ) ∗ (cid:47) (cid:47) End
Z,B . Since B is fibrant, we can again apply the pushout product axiom to conclude that each of the collectionsin the diagram is locally fibrant (equivalently, each of the collections is fibrant in the model structure onColl( C ) S ). The vertical maps are trivial fibrations, and the horizontal maps are weak equivalences. As weassumed that P is Σ-cofibrant, we have that the upper horizontal map φ induces a bijection[ P , φ ] : [ P , End f ] ∼ = [ P , End Z ] . ow, the map f is a trivial cofibration with a cofibrant source, and so satisfies the conditions of (2).Therefore we can extend the P -algebra structure map P →
End A to a P -algebra structure map φ : P →
End Z . Since End ( Z ) is fibrant, and P is Σ-cofibrant, we can lift the map φ : P →
End Z to a map ψ : P →
End f such that φ and the composite φψ are homotopic, and this map is unique up to homotopy.The composite map P →
End f → End B gives B the structure of a P -algebra.We can make the dual argument so show that a P -algebra structure on B induces a P -algebra structureon A in such a way that f preserves the P -algebra structures up to homotopy. (cid:3) Corollary.
Now consider f : M i → L p → N where i is a cofibration and p is a fibration in M . Thenthe induced map g : End i,p → End f is a fibration. Further, g is a weak equivalence if either i or p is a weakequivalence.Proof. Define
End i,p := End M × End
M,L
End L × End
L,N
End N . Then, by the earlier claim we know that h : End L → End
M,L × End
L,N
End N is a fibration, which is trivial if either i or p is a weak equivalence. Thenwe notice that g is defined as pullback over h :(6.12) g g (cid:47) (cid:47) f (cid:15) (cid:15) End M End
M,L (cid:15) (cid:15)
End N End
L,N (cid:47) (cid:47) h. (cid:3) Appendix: The Homotopy Theory of Operadic Bimodules A left P -module is an object M in Coll( C ) together with a left P -action P ◦ M −→ M . A right Q -module is an object N in Coll( C ) together with a right action N ◦ Q −→ N .7.1. Definition.
For any two multicategories, P and Q , a P - Q -bimodule is consists of an object M in Coll( C ) obj( P ) × obj( Q ) which has a left P action and a compatible right Q action: • for each a , . . . , a n ∈ Q and each b ∈ P , an M -object M ( a , . . . , a n ; b ) • for each a ji ∈ Q and b i , b ∈ P , a left P -action ( M -morphism) P ( b , . . . , b n ; b ) ⊗ M ( a , . . . , a k ; b ) ⊗ · · ·⊗ M ( a n , . . . , a k n n ; b n ) → M ( a , . . . , a k n n ; b ) , ( φ, ξ , . . . , ξ n ) (cid:55)→ φ · ( ξ , . . . , ξ n ) • for each a ji , a i ∈ Q and each b ∈ P , a right Q -action ( M -morphism) M ( a , . . . , a n ; b ) ⊗ Q ( a , . . . , a k ; a ) ⊗ · · ·⊗ Q ( a n , . . . , a k n n ; a n ) → M ( a , . . . , a k n n ; b ) , ( ξ, θ , . . . , θ n ) (cid:55)→ ξ · ( θ , . . . , θ n ) , which satisfy the evident axioms for compatibility with the composition products and identities of both Q and P , in addition to: ( φ · ( ξ , . . . , ξ n )) · ( θ , . . . , θ k n n ) = φ · ( ξ · ( θ , . . . , θ k ) , . . . , ξ n · ( θ n , . . . , θ k n n )) whenever these expressions make sense. The morphisms between P - Q -bimodules are maps of collections which are compatible with both the left P -action and the right Q -action. We denoted the resulting category by P M Q . In the special case where P and Q have only unary arrows, we recover the usual definition of bimodule between enriched categories sometimes called a pro-functor ). If P and Q are both operads, then we recover Rezk’s definition of a( P , Q )-biobject [Rezk96].7.2. Example.
Every multicategory is itself a P – P -bimodule. Example.
Let P = I be the trivial operad, so that Alg I ( C ) = C . An I - I -bimodule M is just a symmetricsequence in C . The category of right Q -modules is a closed, symmetric monoidal category over C . This fact requires somechecking (see [Fre09]), but if we accept that the category of collections in C is a closed, symmetric monoidalcategory over C with respect to the pointwise tensor product, then it remains to check that this structurelifts to the category of objects with right Q -action. Since the circle product commutes with colimits on the left , the tensor product of two right Q -modules has a natural right Q -module structure, where M ◦ Q ⊗ N ◦ Q = ( M ⊗ N ) ◦ Q . Lemma.
Let Q be a multicategory enriched in C . The category of right Q -modules is a cocomplete, closed,symmetric monoidal category over C . Theorem.
Let Q be a multicategory which is locally cofibrant, i.e. for every n ≥ and every sequence x , ..., x n ; x the C -object Q ( x , ..., x n ; x ) is a cofibrant object in C . Then the category of right Q -modules admitsa cofibrantly generated monoidal model category over C . Moreover, if C is right (respectively, left) properthen so is the model category structure on M Q . Remark.
Throughout this paper we have been using the model structure from [BM03] on simplicialoperads (or a generalization there of ), but in the thesis [Rezk96] , Rezk has a different, though Quillenequivalent, model structure on the category of simplicial operads. The reader could choose to use Rezk’smodel structure (properly generalized), and then it would not be necessary to assume that your multicategoryis locally cofibrant.
The theorem follows from somewhat standard arguments that the forgetful functor from right Q -modulesto collections − ◦ Q : Coll( C ) (cid:29) M Q : U preserves and detects weak equivalences and fibrations. Since there are at least two proofs known to theauthor of this theorem in the one-object case (See, [Rezk96, Fre09]) which easily generalize to the manyobjects case , we will not include a complete proof here. The important thing for us is that we can describethe generating (acyclic) cofibrations as tensor products of generating (acyclic) cofibrations of the base cat-egory C . The distribution relation between the composition product and the external tensor product givesidentifications ( i ⊗ K ) ◦ R = i ⊗ ( K ◦ R ) . Now, if the tensor product − ⊗ D : C → C maps acyclic cofibrationsto weak-equivalences for all D ∈ C , then it follows immediately that the model structure lifts from Coll( C )to M Q . Otherwise, we can use our assumption that Q is locally cofibrant to show that the objects K ◦ Q arelocally cofibrant. It follows that the tensor products − ⊗ ( K ◦ Q )) preserve acyclic cofibrations.7.7. Lemma.
The generating (acyclic) cofibrations of the model category structure on M Q are given by i ◦ Q : K ◦ Q −→ L ◦ Q , where i : K → L is a generating (acyclic) cofibration of Coll( C ) . Proposition.
The model category of right Q -modules is a monoidal model category over C .Proof. The claim follows from the fact that the category of C -collections forms a symmetric monoidal modelcategory over C (generalize [Fre09, 14.1] to fixed objects case), and the description of the generating (acyclic)cofibrations for right Q -modules. (cid:3) Note that here “many objects” is still referring to fixed object sets since the category of right Q -modules is really collectionswith | obj( Q ) | objects which are equipped with a right Q -action. .9. Proposition. [Fre09, 14.1] If C is a right (respectively, left) proper model category and Q is locallycofibrant, then M Q is right (respectively, left) proper as well. We can now prove the following theorem.7.10.
Theorem (Model Structure) . Let P and Q be two simplicial multicategories. If the category of right Q -modules admits a cofibrantly generated model category structure then the category [ Hom ] ( P , M Q ) admitsa cofibrantly generated model category structure where a homomorphism f : A → B is a weak equivalence(respectively, fibration) if the underlying map of right Q -modules, U f : U ( A ) → U ( B ) , is a weak equivalence(respectively, fibration). Recall that given any cocomplete category M and any class of maps I in M then the subcategoryof relative I -cell complexes is the subcategory which can be constructed via transfinite compositions andpushouts of the maps in I .7.11. Lemma (Classifying Fibrations) . A map in [ Hom ] ( P , M Q ) is a fibration if, and only if, it has theright lifting property with respect to retracts of relative F P ( ι ∗ J ) -cell complexes.Proof. Fibrations of generalized P -algebras were defined via the free-forgetful adjunction. In particular, f : A → B is a fibration of P -algebras if, and only if, U f : A → B is a fibration of right Q -modules. Themap U f is a fibration if, and only if,
U f has the right lifting property (RLP) with respect to a retract ofsomething in ι ∗ J. The lemma then follows from adjunction. (cid:3)
Lemma (Classifying Trivial Fibrations) . A map in [ Hom ] ( P , M Q ) is a trivial fibration if and only ifit has the right lifting property with respect to retracts of relative F P ( ι ∗ I ) -cell complexes. Lemma.
A relative F P ( ι ∗ J ) complex is a weak equivalence of right Q -modules.Proof. The proof closely follows the proof of Lemma 6.2 [ ? ] and will depend on analyzing pushouts in[ Hom ] ( P , M Q ). Let A be an object in the category [ Hom ] ( P , M Q ) and let M −→ N be map of objectsin M Q and recall that ι x : M Q −→ M S Q is the left adjoint to the “evaluation at object x ” functor. We willwant to study the pushout of the diagram F P ( ι x M ) −−−−→ F P ( ι x N ) (cid:121) A in [ Hom ] ( P , M Q ). The pushout will be given as a colimit, taken in right Q -modules, of a sequence A = X → X → ... → X n → .... We will build up this filtration in layers. First, let’s understand A (cid:96) F P ( M ) by building up the k -aryrelations. For each k ≥ x , ..., x k in P we will construct an object G x ,...,x n ( A )as the coequalizer of the following diagram (cid:97) n ≥ (cid:0) (cid:97) y ,...,y n P ( y , ..., y n , x , ..., x k ; − ) ⊗ Σ n F P ( A ( y )) ⊗ ... ⊗ F P ( A ( y n ) (cid:1) ⇒ (cid:97) n ≥ (cid:0) (cid:97) y ,...,y n P ( y , ..., y n , x , ..., x k ; − ) ⊗ Σ n A ( y ) ⊗ ... ⊗ A ( y n ) (cid:1) −→ G x ,...,x k ( A ) . The top map comes from the structure of the multicategory P and the bottom map comes from the actionof the triple F P on A . The underlying object of the coproduct A (cid:96) F P ( M ) is (cid:97) k (cid:0) (cid:97) x ,...,x n G x ,...,x k ( A ) ⊗ Σ n M ( y ) ⊗ ... ⊗ M ( y n ) . Now, let f : M → N be a map in M Q . We will construct a right Q -module C k,i ( f ) where k ≥ ≤ i ≤ k as follows. Let C k, = M ⊗ k and for 0 < i < k we define C k,i as a pushout [Σ k ] ⊗ Σ k − i × Σ i M ⊗ k − i ⊗ C k,i − −−−−→ [Σ k ] ⊗ Σ k − i × Σ i M ⊗ k − i ⊗ N ⊗ i (cid:121) (cid:121) C k,i − −−−−→ C k,i . We can combine these constructions to get a filtration on the pushout of F P ( ι x M ) −−−−→ F P ( ι x N ) (cid:121) A. Let X = A , fix a k and let fix an object x in P , i.e. x = ... = x k = x. We will define X k as the pushout in M S Q G x ,...,x k ( A ) ⊗ Σ k C k,k − −−−−→ G x ,...,x k ( A ) ⊗ Σ k ( ι x N ) ⊗ i (cid:121) (cid:121) X k − −−−−→ X k . The left hand vertical map comes from the map ι x M → A in M S Q . The pushout of the diagram is then X = colimX k .Now, we may assume that f : M → N is of the form K ◦ Q → L ◦ Q where K → L runs over generatingacyclic cofibrations of Coll( C ). Our goal is to show that the map A → A (cid:96) F P ( ι x M ) F P ( ι x N ) is a weakequivalence of right Q -modules. But, since it is clear that for each k ≥ X k − → X k is a weak equivalence,we are done. (cid:3) Proof of Theorem.
The simplicial category [
Hom ] ( P , M Q ) is cocomplete by 3.16 and it is clear that theclass of weak equivalences and fibrations are closed under retracts and that the class of weak equivalencessatisfies the “2-out-of-3” property. Cofibrations are defined via a lifting property, and so it is easy to checkthat they are closed under retracts.Given an arbitrary map f in [ Hom ] ( P , M Q ) we can apply the small object argument to produce afactorization f = p ◦ i where i is in F P ( ι ∗ I ) and p has the right lifting property with respect to i . Theour lemma 7.12 implies that p is an acyclic fibration. In a similar manner, we factor f = q ◦ j , where j isin F P ( ι ∗ J ) and q has the right lifting with respect to j . The lemma 7.11 implies that q is a fibration ofmulticategories.Finally, we check that given the square A f −−−−→ C (cid:121) i (cid:121) p B g −−−−→ D with i a cofibration and p a fibration. If p is also a weak equivalence, then we find a lift by the classificationof cofibrations. If i is a weak equivalence, then we factor i = q ◦ j : A (cid:44) → (cid:101) B (cid:16) B where q is a fibration and j is an acyclic cofibration. Since we have shown that every acyclic cofibration is a weak equivalence, we knowthat j is a weak equivalence. The “2-out-of-3” property for weak equivalences now implies that q is an acylicfibration.Since j is an acyclic cofibration, and p is a fibration, we know that p has the RLP with respect to j . Inother words, we have a lift h : (cid:101) B −→ C so that j ◦ h = f and q ◦ g = p ◦ h .Now, since i is an acyclic cofibration and q is a trivial fibration, there exists a retract s of q with s ◦ i = j .The composite h ◦ s provides the desired lift. (cid:3) Corollary.
The category [ Hom ] ( P , M Q ) is a simplicial model category. roof. The tensor of an F P -algebra A and a simplicial set K is given by a reflexive coequalizer F P (( F P A ) ⊗ K ) ⇒ F P ( A ⊗ K ) −→ A ⊗ X. We may assume that i is a map in either F P ( ι ∗ I ) or F P ( ι ∗ J ). Since the monad F P , viewed as a functor,is left adjoint to the forgetful functor we can reduce this issue to proving that the product category M S Q satisfies SM
7, which then reduces to showing that M Q satisfies SM
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University of Western Ontario, Department of Mathematics, Middlesex College, London, Ontario, Canada
E-mail address : [email protected]@uwo.ca