aa r X i v : . [ m a t h . K T ] M a r M onoidal cofibrant resolutions of dg algebras B oris S hoikhet A bstract . Let k be a field of any characteristic. In this paper, we construct afunctorial cofibrant resolution R ( A ) for the Z ≤ -graded dg algebras A over k ,such that the functor A R ( A ) is colax-monoidal with quasi-isomorphismsas the colax maps. More precisely, there are maps of bifunctors R ( A ⊗ B ) → R ( A ) ⊗ R ( B ) , compatible with the projections to A ⊗ B , and obeying the colax-monoidal axiom.The main application of such resolution (which we consider in our next paper)is the existence of a colax-monoidal dg localization of pre-triangulated dg cate-gories, such that the localization is a genuine dg category, whose image in thehomotopy category of dg categories is isomorphic to the Toën’s dg localization. I ntroduction We formulate our main result (Theorem . below) in Section . , discuss its applications tohigher-monoidal Deligne conjecture in Section . , and overview our proof of Theorem . inSection . . . T he M ain T heorem Let k be a field of any characteristic.It is known that any associative algebra A over k admits a free resolution , that is, a freeassociative dg algebra R ( A ) endowed with a quasi-isomorphism of algebras p A : R ( A ) → A .Here we specify this claim as follows. Having two associative algebras A and B over k , A ⊗ B is an associative algebra again. We want to choose a functorial free resolution R ( A ) for all algebras A over k , such that R ( A ⊗ B ) and R ( A ) ⊗ R ( B ) are related by a functorialquasi-isomorphism, defined over A ⊗ B .More precisely, we want to find a functor A R ( A ) with R ( A ) a free associative dgalgebra, such that there exists a morphism of bifunctors β : R ( A ⊗ B ) → R ( A ) ⊗ R ( B ) uch that for any two algebras A , B the map β ( A , B ) is a quasi-isomorphism , the diagram R ( A ⊗ B ) β / / p A ⊗ B & & ▲▲▲▲▲▲▲▲▲▲ R ( A ) ⊗ R ( B ) p A ⊗ p B w w ♦♦♦♦♦♦♦♦♦♦♦ A ⊗ B ( . )is commuative, and such that for any three algebras A , B , C , the diagram R ( A ⊗ B ⊗ C ) / / (cid:15) (cid:15) R ( A ⊗ B ) ⊗ R ( C ) (cid:15) (cid:15) R ( A ) ⊗ R ( B ⊗ C ) / / R ( A ) ⊗ R ( B ) ⊗ R ( C ) ( . )is commutative. A functor F : M → M between two monoidal categories, with a map ofbifunctors β : F ( A ⊗ B ) → F ( A ) ⊗ F ( B ) for which the diagram ( . ) commutes, is called colax-monoidal (see Definition A. below for the precise definition of a colax-monoidal functor).More generally, we construct such a resolution A R ( A ) on the category of Z ≤ -gradeddifferential graded algebras. We think that such a functor hardly exists on the category of all Z -graded dg algebras.Throughout the paper, we use the technique of closed model categories , introduced byQuillen in [Q].Roughly speaking, the theory of closed model categories is a non-abelian pattern of clas-sical homological algebra. A closed model category is a category C with three classes ofmorphisms, called weak equivalences, fibrations, and cofibrations . The class of weak equivalencesis the most essential. The goal is to define non-abelian analogues of derived functors, definedon the localized by weak equivalences categories. In the abelian setting, the underlying cate-gory is the category of complexes, and the weak equivalences are the quasi-isomorphisms ofcomplexes.In this paper, we assume the reader has some familiarity with the foundations of closedmodel categories. We use [Hir], [GS], and [DS] as the references.The closed model structure on the category of associative dg algebras A lg k over a field k ofany characteristic (and as well, for the category of dg algebras over any operad, over a field ofcharacteristic ) was constructed by Hinich in [Hi]. For this structure, the weak equivalencesare the quasi-isomorphisms of dg algebras, the fibrations are the component-wise surjectivemaps of dg algebras. The cofibrations are uniquely defined from the weak equivalences andthe fibrations, by the left lifting property .Recall the description of the cofibrant objects in the Hinich’s closed model category ofassociative dg algebras. dg associative algebra A is called standard cofibrant if there is an increasing filtration on A , A = colim A i , A ⊂ A ⊂ A ⊂ . . .such that the underlying graded algebra A i is obtained from A i − by adding the free genera-tors graded space V i , A i = A i − ∗ T ( V i ) such that d ( V i ) ⊂ A i − for i ≥
1, and A has zero differential. Here ∗ is the free product of algebras, which isthe same as the categorical coproduct of algebras, and T ( V i ) is the free graded algebra ongenerators V i .The general description (valid as well for any operad in characteristic ) is: any cofibrantdg associative algebra is a retract of a standard cofibrant dg algebra. For the category of associativedg algebras in any characteristic, the class of retracts of standard cofibrant objects coincideswith the class of standard cofibrant objects itself.As follows from this description, when restricted to Z ≤ -graded dg algebras, the conceptsof free dg algebras and of cofibrant dg algebras coincide . For general Z -graded dg algebras,the two concepts are different: a cofibrant Z -graded dg algebra is free, with the differentialof some special “triangular” form.Denote the category of Z ≤ -graded associative algebras by A lg ≤ k .Our main result is:T heorem 0 . . Let k be a field of any characteristic. There is a functor R : A lg ≤ k → A lg ≤ k and amorphism of functors w : R → Id with the following properties: . R ( A ) is cofibrant, and w : R ( A ) → A is a quasi-isomorphism, for any A ∈ A lg ≤ k , . there is a colax-monoidal structure on the functor R , such that all colax-maps β A , B : R ( A ⊗ B ) → R ( A ) ⊗ R ( B ) are quasi-isomorphisms of dg algebras, and such that the diagram R ( A ⊗ B ) β A , B / / & & ▲▲▲▲▲▲▲▲▲▲ R ( A ) ⊗ R ( B ) w w ♦♦♦♦♦♦♦♦♦♦♦ A ⊗ Bis commutative, . the morphism w ( k q ) : R ( k q ) → k q coincides with α : R ( k q ) → k q , where α is a part of thecolax-monoidal structure (see Definition A. ) , and k q = A lg ≤ k is the dg algebra equal to theone-dimensional k-algebra in degree , and vanishing in other degrees. ote once again, that in the context of this Theorem, the words cofibrant and free aresynonymous. In the course of proof we use an “auxiliary” closed model category of simpli-cial associative algebras, where the difference between the free and the cofibrant objects isessential. For simplicial associative algebras there is fairly analogous (and much simpler) the-orem, see Theorem . below. We formulate Theorem . with some closed model categorieslanguage to have a unified formulation of the both theorems. . A pplications Here we explain our motivations for Theorem . . The results mentioned here will be provenin our sequel paper(s).The problem which motivates our interest in Theorem . is the following generalizedDeligne conjecture for n-fold monoidal abelian categories . Let M be an abelian n -fold monoidal[BFSV] category, with some weak compatibility of the exact and of the monoidal structures,and let e be its unit element. The generalized Deligne conjecture claims that RHom q M ( e , e ) isa homotopy ( n + ) -algebra, whose commutative product is quasi-isomorphic to the Yonedaproduct.Originally, the name “Deligne conjecture” was given (due to a letter of P.Deligne wherehe announced it) to the particular case of the above claim for n =
1, when M = B imod ( A ) for an associative algebra A , with − ⊗ A − as the monoidal product. Then the claim is thatthe Hochschild complex RHom q B imod ( A ) ( A , A ) admits a homotopy -algebra structure, whosecommutative product is quasi-isomorphic to the Yoneda product, and whose Lie bracket ofdegree - is quasi-isomorphic to the Gerstenhaber bracket. At the moment, there are severalproofs of this claim, see e.g. [MS], [KS], and [T].In their paper [KT], Kock and Toën suggest a very conceptual proof of (a suitable versionof) the Deligne conjecture for n -fold monoidal categories enriched over simplicial sets . The meth-ods of loc.cit. can be applied for the n -fold monoidal categories enriched over arbitrary carte-sian symmetric monoidal category , that is, enriched over a symmetric monoidal category whosemonoidal product of any two objects is isomorphic to their cartesian product. The categoriesof sets, of simplicial sets, ... give examples of cartesian-monoidal categories. The categories ofabelian groups, k -vector spaces, complexes of k -vector spaces, ... are not cartesian-monoidal,as the cartesian product is the direct sum, and the monoidal product is the tensor product.The reason why the methods of [KT] do not work in non-cartesian-monoidal context isthat the authors use essentially the concept of weak Segal monoids whose standard definitionassumes that the Hom-sets belong to a cartesian-monoidal category.However, there appeared a definition of weak Segal monoids in non-cartesian monoidalcategories, due to Tom Leinster [Le]. Working with the Leinster’s definition, one needs tohave a more subtle monoidal property of the Dwyer-Kan-type localization of categories than he one used in [KT]. Let us formulate this necessary property (see Theorem . below).Let C at dg U denotes the category of U -small dg categories over a field k , let CC at dg U denotesthe category of colored U -small dg categories over k , and let, finally, C at pre − tr U (resp., CC at pre − tr U )denotes the category of U -small pre-triangulated dg categories (resp., the category of colored U -small pre-triangulated dg categories) over k . The either of these three categories is sym-metric monoidal, with ⊗ k as the monoidal product.Tabuada [Tab] constructed a closed model structure on the category of U -small dg cate-gories; this structure is assumed in the two following statements.T heorem 0 . (to be proven later) . Let k be a field of any characteristic. There is a functor R : C at pre − tr U → C dg U and a morphism of functors w : R → Id with the following properties: . R ( C ) is cofibrant, and w : R ( C ) → C is a weak equivalence, for any C ∈ C at pre − tr U , . there is a colax-monoidal structure on the functor R , such that all colax-maps β C , D : R ( C ⊗ D ) → R ( C ) ⊗ R ( D ) are weak equivalences of dg categories, and such that the diagram R ( C ⊗ D ) β C , D / / & & ▼▼▼▼▼▼▼▼▼▼ R ( C ) ⊗ R ( D ) w w ♦♦♦♦♦♦♦♦♦♦♦ C ⊗ Dis commutative.
To prove Theorem . , we firstly extend our Main Theorem . from Z ≤ -graded dg al-gebras to Z ≤ -graded dg categories, this should be proven by the methods of this paper andthe ones of [BM]. In the next step, we extend the result from Z ≤ -graded dg categories topre-triangulated dg categories. It seems that the analogous statement is not true for general Z -graded dg categories (not the pre-triangulated ones).T heorem 0 . (to be proven later) . There is a localization functor L : CC at pre − tr U → C at dg [ U ] from colored pre-triangulated U -small dg categories to dg U -categories, with the following properties:(i) L is colax-monoidal, with weak equivalences of dg categories as the colax maps β ( C , D ) ,(ii) β ( C , D ) is defined over C ⊗ D, for any two pre-triangulated dg categories C , D,(iii) the image of L ( C , S ) under the projection to the homotopy category Ho C at dg U coincides with theToën dg localization [To], Sect. . . oën defines a localization of the dg categories with nice homotopy properties (see [To],Corollary . ) as the homotopy colimit of some push-out-angle diagram. It is possible tocompute this homotopy colimit as the genuine colimit of a cofibrant replacement of the initialdiagram. We use the cofibrant resolution R ( C ) , given in Theorem . , for this replacement.(In fact, the push-out-angle diagrams and the pull-back-angle diagrams in a closed modelcategory admit closed model structures, with component-wise weak equivalences as the weakequivalences, see [DS], Section ).Theorem . is essentially the only new toolkit one needs to have, to extend the Kock-Toën’s proof of simplicial n -fold monoidal Deligne conjecture to its genuine k -linear version.Notice that this (way of) proof of it does not use any transcendental numbers which unavoid-ably appear in any “physical” proof. When our n -fold monoidal category is k -linear, for afield k , the output homotopy ( n + ) -algebra (for a homotopy equivalent definition of thisconcept) is defined over the field k as well. . P lan of the paper We start with proving in Section a direct analogue of Theorem . for simplicial associativealgebras over k , instead of Z ≤ -graded dg associative algebras (see Theorem . below). Itturns out that Theorem . becomes much simpler in the simplicial setting. We solve it by anexplicit construction, which reminiscences the construction of Dwyer-Kan in their first paper[DK ] on simplicial localization. We denote by F ( A ) , for a simplicial algebra A , our solutionto Theorem . .The rough idea of the remaining Sections is to “transfer” the solution of Theorem . forsimplicial algebras to a solution to the Main Theorem . , using the Dold-Kan correspondenceand passing to the categories of monoids. Recall (see Section for detail) that the Dold-Kancorrespondence N : M od ( Z ) ∆ ⇄ C − ( Z ) : Γ is an adjoint equivalence between the category M od ( Z ) ∆ of simplicial abelian groups, andthe category C − ( Z ) of Z ≤ -graded complexes of abelian groups. The both categories aresymmetric monoidal, but neither of the functors N and Γ respects the monoidal structure.In Section , we recall some “non-homotopy” results of Schwede-Shipley [SchS ]. Inparticular, having a lax-monoidal structure ϕ on the right adjoint functor R in an adjunction L : C ⇄ D : R between monoidal categories C and D one easily defines a functor R mon : Mon D → Mon C nd, bit more tricky, one defines a left adjoint L mon to R mon , such that there is an adjunction L mon : Mon C ⇄ Mon D : R mon Applying this construction to L = N , R = Γ , and ϕ the lax-monoidal structure on Γ adjoint tothe Alexander-Whitney colax-monoidal structure on N , one gets an adjoint pair of functors N mon : Mon ( M od ( Z ) ∆ ) ⇄ Mon ( C − ( Z )) : Γ mon ( . )We prove in Section that the formula R ( A ) = N mon ( F ( Γ mon ( A ))) ( . )gives a solution to Theorem . , where F ( B ) is a solution to Theorem . .The proof that ( . ) solves Theorem . is rather sophisticated. We use mainly the follow-ing techniques: the “homotopy” results of [SchS ] (which claims, in particular, that ( . ) is aQuillen equivalence); the bialgebra axiom for the Dold-Kan correspondence (proven in [AM]and independently but later in [Sh ]); and the theory developed in Section and in AppendixB of this paper. In general it is not true that, for an adjoint pair of functors, having a solution F ( B ) of Th. . -like theorem, the formulae ( . ) gives a solution of Th. . -like theorem. In fact,we use many specific features of the Dold-Kan correspondence throughout the proof.In Section we find an explicit expression for the Schwede-Schipley’s functor L mon (de-fined a priori as a co-equalizer ( . )), in the particular case of the Dold-Kan correspondence,with L = N , the normalized chain complex functor. It is relatively easy to do, however, weneed to know as well an explicit expression for the canonical colax-monoidal structure on L mon ,that is the one adjoint to the natural lax-monoidal structure on R mon . The latter task requiressome amount of work, and occupies most part of Section . Some of results of very generalnature we need in Section are formulated in proven separately in Appendix B. Section and Appendix B together form a technical core of the paper.The main results of Section are formulated in Theorem . and Theorem . . As well,we prove here more general Theorem . , which describes the colax-monoidal structure on L mon in greater generality.In Section , the different previous results of the paper become connected and interacted,which leads us to a proof of the Main Theorem . (which comes up under the name Theorem . in the body of the paper).In Appendix A we collect, to ease the reader’s reference, some “diagrammatic” definitions,used in the paper. We recall here the definitions of a lax-monoidal and of a colax-monoidalstructures on a functor, and formulate the bialgebra axiom.In Appendix B we develop some techniques, used in proofs of Section . Roughly speak-ing, we study here a property of a pair of functors L : C ⇄ D : R which weakens the property hat “ L and R are adjoint functors”. We define weak right adjoint pairs and weak left adjoint pairs of functors, followed by very weak left and very weak right adjoint pairs. Our goal is tostudy how these weak concepts interact with the lax-monoidal structures on R and the colax-monoidal structures on L , when the categories C and D are supposed to be monoidal. (Forthe case of a honest adjoint functors, there is a - correspondence between the lax structureson R and the colax structures on L , see Lemma . ). The results proven there are very generalin nature, and we decided to organize them in a single Appendix would be better than tospread them out between the proofs of Section . . N otations Throughout the paper, k denotes a field of any characteristic. An “algebra” always means an“associative algebra with unit”.All differentials have degree + , as is common in the algebraic literature.Let ∆ be the category whose objects are [ ] , [ ] , [ ] , [ ] , and so on, where [ n ] denotes thecompletely ordered sets with n + < < < · · · < n . A morphism f : [ m ] → [ n ] is any map obeying f ( i ) ≤ f ( j ) when i ≤ j . A simplicial object in a category C is a functor ∆ opp → C , and a cosimplicial object in C is a functor ∆ → C . We denote by C ∆ the category ofsimplicial objects in C and by C ∆ opp the category of cosimplicial objects in C . This notation isindeed confusing, but seemingly it is traditional now.All categories we consider in this paper are small for some universe. We do not meet hereany set-theoretical troubles related with the localization of categories, and we always skip theadjective “small” in the formulations of our results (except Section . ).A cknowledgments I am grateful to Sasha Beilinson, Volodya Hinich, Ieke Moerdijk, Stefan Schwede, and VadikVologodsky for discussions on the topics related to this paper. I am greatly indebted to MartinSchlichenmaier for his kindness and support during my -year appointment at the Universityof Luxembourg, which made possible my further development as a mathematician. Thework was done during research stay at the Max-Planck Institut für Mathematik, Bonn. I amthankful to the MPIM for hospitality, for financial support, and for very creative workingatmosphere. T he case of simplicial algebras The category A lg ∆ k of simplicial algebras over field k is monoidal, with degree-wise ⊗ k asthe monoidal structure, and it admits a closed model structure. We recall this closed modelstructure in Section . below. We refer to this closed model structure in the following result.T heorem 1 . (Main Theorem for simplicial algebras) . Let k be a field of any characteristic. Thereis a functor F : A lg ∆ k → A lg ∆ k and a morphism of functors w : F → Id with the following properties: . F ( A ) is cofibrant, and w : F ( A ) → A is a weak equivalence, for any A ∈ A lg ∆ k , . there is a colax-monoidal structure on the functor F , such that all colax-maps β A , B : F ( A ⊗ B ) → F ( A ) ⊗ F ( B ) are weak equivalences of simplicial algebras, and such that the diagram F ( A ⊗ B ) β A , B / / % % ▲▲▲▲▲▲▲▲▲▲ F ( A ) ⊗ F ( B ) x x ♣♣♣♣♣♣♣♣♣♣♣ A ⊗ Bis commutative, . the morphism w ( k q ) : F ( k q ) → k q coincides with α : F ( k q ) → k q , where α is a part of the colax-monoidal structure (see Definition A. ) , and k q = A lg ∆ k is the simplicial algebra equal to theone-dimensional k-algebra k in each degree. . T he construction The idea is very easy. Let A q be a simplicial algebra. There is the forgetful functor A lg ∆ k → V ect ∆ k to simplicial vector spaces, having a left adjoint functor of “free objects”. This is thefunctor A q ( TA ) q , with ( TA ) k = T ( A k ) ( . )where T ( A k ) is the free (tensor) algebra. We consider the cotriple, associated with the pair ofadjoint functors ( L is the left adjoint to R ) L : V ect ∆ k ⇄ A lg ∆ k : R (see [W], Section . ). Explicitly, T = L ◦ R ( . ) his implies that there are maps of functors ǫ : T → id and δ : T → T obeying the cotripleaxioms. These axioms guarantee that the following collection of algebras ( FA ) k , k ≥
0, has anatural structure of a simplicial algebra ( FA ) q : ( FA ) k = T ◦ ( k + ) A k (there is the ( k + ) -st iterated tensor power in the r.h.s.), such that the natural map FA q → A q , T ◦ ( k + ) A k ǫ k + −−→ A k , is a weak equivalence of simplicial algebras.Explicitly, having such a functor T with maps of functors ǫ : T → id and δ : T → T , thesimplicial structure on ( FA ) q is defined as follows.When A k = A for any k , ( A q is a constant simplicial algebra), the formulas for simplicialalgebra structure on ( FA ) q are: d i = T ◦ i ǫ T ◦ ( n − i ) : T ◦ ( n + ) A → T ◦ n As i = T ◦ i δ T ◦ ( n − i ) : T ◦ ( n + ) A → T ◦ ( n + ) A ( . )The cotriple axioms then guarantee the simplicial identities (see [W], Section . . for detail).In general case, when A q is not constant, the simplicial algebra ( FA ) q is defined as thediagonal of the bisimplicial set ( FA ) q = diag (( FA q ) q ) ( . )For two simplicial algebras A q and B q , there is a canonical embedding β A , B : F ( A ⊗ B ) q → ( FA ) q ⊗ ( FB ) q defined on the level of algebras by iterations of the map α : T ( A ⊗ B ) → T ( A ) ⊗ T ( B )( a ⊗ b ) ⊗ · · · ⊗ ( a k ⊗ b k ) α −→ ( a ⊗ · · · ⊗ a k ) ⊗ ( b ⊗ · · · ⊗ b k ) ( . )The component ( β A , B ) k : ( T ◦ ( k + ) A k ) ⊗ ( T ◦ ( k + ) B k ) → T ◦ ( k + ) ( A k ⊗ B k ) is defined as the iter-ated power α ◦ ( k + ) .L emma 1 . . The collection of maps { β ℓ } , ℓ ≥ , defines a map of simplicial algebras β : ( F ( A ⊗ B )) q → ( FA ) q ⊗ ( FB ) q Proof.
Denote the product(s) in A k by ⋆ , the product in T ( V ) by ⊗ , and the product in T ( T ( V )) by N . Then the formulas for ǫ : T → id and δ : T → T are as follows: ǫ ( a ⊗ · · · ⊗ a k ) = a ⋆ · · · ⋆ a k δ ( a ⊗ · · · ⊗ a k ) = a O · · · O a k ( . )The statement of Lemma now follows directly from formulas ( . ), expressing the simplicialfaces and degeneracies maps in ǫ and δ . ♦ ur goal in this Subsection is to prove that F ( A q ) = ( FA ) q solves Theorem . . We needto proveL emma 1 . . . For any simplicial algebra A q , the simplicial algebra FA q is cofibrant in the closedmodel structure on A lg ∆ k , . the map β A , B : ( F ( A ⊗ B )) q → ( FA ) q ⊗ ( FB ) q is a weak equivalence. Before proving the above Lemma, we need to remind some results concerning the closedmodel structure on the category A lg ∆ k , which goes back to Quillen [Q], Section . . . T he closed model category of simplicial algebras Firstly recall the model structure on the category V ect ∆ k of simplicial vector spaces over field k . (i) A map f : X → Y in V ect ∆ k is a weak equivalence if it induces an isomorphismon homotopy groups π q ( X ) → π q ( Y ) ,(ii) a map f : X → Y is a fibration if it induces a surjection π q X → π ( X ) × π ( Y ) π q ( Y ) .(iii) A map f : X → Y in V ect ∆ k is a cofibration if it has a form X n → Y n = X n ⊕ V n for some collection of vector spaces { V , V , V , . . . } , such that each simplicial degeneracy map s i : [ n + ] → [ n ] maps V n to V n + , n ≥
0. ( . )The model category V ect ∆ k described above is cofibrantly generated (see [GS], Section , for abeautiful short survey of cofibrantly generated model categories). Recall that it means, inparticular, that there are given sets I of generating cofibrations, and J of generating acycliccofibrations, subject to the following two properties: . the source of any morphism in I obeys the Quillen’s small object argument to the categoryof all cofibrations; the source of any morphism in J obeys the small object argument tothe category of all acyclic cofibrations; . a morphism is a fibration if and only if it satisfies the left lifting property with respectto any morphism in J ; a morphism is an acyclic fibration if and only if it satisfies theleft lifting property with respect to any morphism in I . oncerning the Quillen’s small object argument , see [GS], Section . , or [Hir], Section . , forthorough treatment. The meaning of these two conditions is that they make possible to provethe last axiom (CM ) of a closed model category axioms, which is in a sense the hardest one(see loc.cit.).See [GS], Examples . , for explicit description of the sets I and J in the category V ect ∆ k .There is a pair of adjoint functors L : V ect ∆ k ⇄ A lg ∆ k : R ( . )As the left-hand side category is a cofibrantly generated model category, the model struc-ture can be “transferred” to the right-hand-side category, and this model category is againcofibrantly generated. This transfer principle is explained in [GS], Theorem . , and [Hir],Theorem . . . As is explained in [GS], Sections , , the assumptions of Theorem . aresatisfied in ( . ).In the situation when assumptions of Theorem . of [GS] are fulfilled, the sets L ( I ) and L ( J ) are generating cofibrations (resp., generating acyclic cofibrations) for the category in theright-hand side.The obtained closed model structure on A lg ∆ k has the following explicit description, see[GS], Section . .(i) a map f : X → Y is a weak equivalence if π ∗ f : π ∗ X → π ∗ Y is an isomor-phism,(ii) a map f : X → Y is a fibration if the induced map X → π X × π Y Y is asurjection.(iii) a map f : X → Y is a cofibration in A lg ∆ k if it is a retract of the following freemap : X n → Y n = X n ⊔ T ( V n ) as algebras, for some collection { V , V , V , . . . } of vector spaces, such that alldegeneracy maps s i : [ n + ] → [ n ] maps V n to V n + , n ≥
0. ( . )See [Q], Section . and [GS], Proposition . for a proof. . P roof of T heorem 1 . Firstly we proveL emma 1 . . For any simplicial algebra A q , the simplicial algebra ( FA ) q is cofibrant, and the projectionp : ( FA ) q → A q is an acyclic fibration. roof. We need to find vector spaces V i such that ( FA ) n = T ( V n ) and such that all degen-eracies maps s i : [ n + ] → [ n ] define maps of algebras T ( V n ) → T ( V n + ) induced by somemaps of generators V n → V n + . We set V n = T ◦ n ( V n ) , it is clear that this choice satisfies theboth conditions. The statement that the map ( FA ) q → A q is both a weak equivalence and afibration, is clear. ♦ Next followsL emma 1 . . For any two simplicial algebras A q , B q , the map β A , B : F ( A ⊗ B ) q → ( FA ) q ⊗ ( FB ) q isa weak equivalence.Proof. It is a straightforward and simple check that the diagram F ( A ⊗ B ) q β A , B / / p A ⊗ B & & ▼▼▼▼▼▼▼▼▼▼ ( FA ) q ⊗ ( FB ) q p A ⊗ p B w w ♦♦♦♦♦♦♦♦♦♦♦ A q ⊗ B q ( . )is commutative. The map p A ⊗ B is a weak equivalence by Lemma . , the product p A ⊗ p B is a weak equivalence by Lemma . again. Then, the commutativity of the diagram ( . )implies, by -out-of- axiom of closed model category, that the third arrow β A , B is also a weakequivalence. ♦ M onoids and the B ialgebra axiom . L ax ↔ colax duality Let C and D be two strict monoidal categories, and let F : C → D be a functor with twoproperties: ) F is an equivalence of the underlying categories, ) F is strict monoidal, that is F ( X ⊗ Y ) = F ( X ) ⊗ F ( Y ) for any two X , Y ∈ C .Then one can choose a quasi-inverse to F functor G : D → C such that ( F , G ) is an adjoint equivalence. As well, we can choose a quasi-inverse functor to F functor G : D → C whichis strict monoidal . However, one can not choose in general a quasi-inverse G enjoying the bothproperties at once. This is an explanation how the lax-monoidal functors and the colax-monoidalfunctors come up into the contemporary mathematics. The reader is referred to Appendix Afor the definitions of the (co)lax-monoidal functors.The set-up of the following standard lemma is very common. emma 2 . . Let C and D be two strict monoidal categories, and letL : C ⇄ D : Rbe a pair of adjoint functors, with L the left adjoint. Then there is a - correspondence between thecolax-monoidal structures on L and the lax-monoidal structures on R, given by ( . ) , ( . ) below.Evermore, this - correspondence is involutive.Proof. Let ǫ : LR → Id D and η : Id C → RL are the adjunction maps.For a colax-monoidal structure c on L , written as c X , Y : L ( X ⊗ Y ) → L ( X ) ⊗ L ( Y ) , define alax-monoidal structure ℓ on the functor A as R ( X ) ⊗ R ( Y ) η −→ RL ( R ( X ) ⊗ R ( Y )) c ∗ −→ R ( LR ( X ) ⊗ LR ( Y )) ( ǫ ⊗ ǫ ) ∗ −−−→ R ( X ⊗ Y ) ( . )Vice versa, suppose a lax-monoidal structure ℓ on R is given. Define a colax-monoidal struc-ture c on L as L ( X ⊗ Y ) ( η ( X ) ⊗ η ( Y )) ∗ −−−−−−−→ L ( RL ( X ) ⊗ RL ( Y )) ℓ ∗ −→ LR ( L ( X ) ⊗ L ( Y )) ǫ ∗ −→ L ( X ) ⊗ L ( Y ) ( . )Recall that for the adjoint pairs of functors the two compositions, defined out of ǫ and η L → L RL = LR L → L ( . )and R → RL R = R LR → R ( . )are the identity maps of functors. This identities imply all claims of Lemma by a directcheck. ♦ When L : C ⇄ D : R is an adjoint equivalence , there are more possibilities for the applicationof the above construction. For example, suppose there are given a colax-monoidal structure c L and a lax-monoidal structure ℓ L on the functor L . Then ( . ) and ( . ) define a lax-monoidalstructure ℓ R and a colax-monoidal structure c R on the functor R . There is a compatibilityrelation between c L and ℓ L , which holds if and only if the same relation holds for c R and ℓ R .This relation, called the bialgebra axiom , is elaborated in Section . . . T he category of monoids Let M be a symmetric monoidal category, Mon M be the category of monoids in M . There isthe forgetful functor f : Mon M → M Under some conditions, the functor f has a left adjoint functor , “the free monoid functor”.Recall the following result, from [ML], Chapter VII. : emma 2 . . Let M be a monoidal category with all finite colimits, such that the functors − ⊗ a anda ⊗ − (for fixed a) commute with finite colimits. Then the functor M → Mon M , X T ( X ) , withT ( X ) = M ∐ X ∐ X ⊗ X ∐ . . . ( . ) is left adjoint to the forgetful functor. When the finite colimits exist, and there is the inner Hom functor (a right adjoint to themonoidal product), the functors − ⊗ a and a ⊗ − commute with colimits by general categor-ical arguments.We say that a monoidal category M is good when the assumptions of Lemma . hold.Let now M , M be two symmetric monoidal categories, and let F : M → M be afunctor. Suppose a lax-monoidal structure ℓ F is given. Then there is a functor F mon = F mon ( ℓ F ) : Mon M → Mon M , depending on ℓ F . For a monoid X in M , the underlyingobject of F mon ( X ) is defined as F ( X ) , and the monoid structure is given as F ( X ) ⊗ F ( X ) ℓ F −→ F ( X ⊗ X ) m X −→ F ( X ) ( . )We have immediately:L emma 2 . . In the above notations, the following two diagram is commutative: M F / / M Mon M F mon / / O O Mon M O O ( . ) Here the vertical upward arrows are the forgetful functors. ♦ . T he left adjoint functor on monoids Suppose now that the functor F admits a left adjoint functor L : M → M . In this case, wewant to construct a functor L mon : Mon M → Mon M , left adjoint to the functor F mon .The following Lemma (and the construction in its proof) is due to [SchS ], Section . :L emma 2 . . Suppose the monoidal categories M and M are good, and suppose that the functor Lleft adjoint to F exists. Then the left adjoint functor L mon : Mon M → Mon M exists, and it makesthe diagram M (cid:15) (cid:15) M (cid:15) (cid:15) L o o Mon M Mon M L mon o o ( . ) ommutative. Here the downward vertical arrows are the free monoid functors.Proof. The second claim is a formal consequence from the existence of L mon , as the freemonoid functors are left adjoint to the forgetful functors. Define the value L mon ( X ) (for amonoid X in M ) as the co-equalizer in the category Mon ( M ) : T M ( L ( T M ( X ))) α / / β / / T M ( LX ) ( . )where T M denotes the free (tensor) monoid in a monoidal category M , see ( . ). The map α in ( . ) comes from the map T M ( X ) → X defined from the monoid structure on X , and themap β in ( . ) is defined from the following map L ( T M ( X )) → T M ( LX ) : L ( X ⊗ X ⊗ · · · ⊗ X | {z } n factors ) c Ln − −−−→ L ( X ) ⊗ L ( X ) ⊗ · · · ⊗ L ( X ) | {z } n factors ( . )where c L is the colax-monoidal structure on L adjoint to the lax-monoidal structure ℓ F on F .Let us prove that the functor L mon , defined by ( . ), is left adjoint to the functor F mon . Westart with the following fact:S ub - lemma 2 . . Let M be a good monoidal category, and let X ∈ Mon M be a monoid in M . Then Xis isomorphic to the co-equalizer of the following diagram in Mon M :T M ( T M ( X )) a / / b / / T M ( X ) ( . ) where the map a is defined on generators as the product map in X, m ∗ : T M ( X ) → X, and the map bis defined as the product map for the monoid T M ( X ) , m ∗ : T M ( T M ( X )) → T M ( X ) . The map b doesnot depend on the monoid structure on X. It is clear. ♦ We continue to prove Lemma . .Let X be a monoid in M . Represent X as co-equalizer ( . ). Denote by D ( X ) thecorresponding diagram. We have:Hom Mon ( M ) ( X , R mon ( Z )) = Hom
Mon ( M ) ( colim D ( X ) , R mon ( Z )) = lim Hom Mon ( M ) ( D ( X ) , R mon ( Z )) = lim E ( X , Z ) = lim E ∨ ( X , Z ) ( . )where E ( X , Z ) and E ∨ ( X , Z ) are the following diagrams:Hom M ( T M ( X ) , R ( Z )) Hom M ( X , R ( Z )) B o o A o o ( . ) nd Hom M ( L ( T M ( X )) , Z ) Hom M ( L ( X ) , Z ) B ∨ o o A ∨ o o ( . )correspondingly.Let us compute the maps A and B explicitly. The map A is induced by the product m ∗ : T M ( X ) → X in the monoid X . The map B is little more tricky. Define B ( µ ) for µ ∈ Hom M ( X , R ( Z )) . It is enough to define B ( µ ) n : Hom M ( X ⊗ n , R ( Z )) , for any n ≥
2. The lattermap is defined as µ ⊗ n ∈ Hom M ( X ⊗ n , R ( Z ) ⊗ n ) , followed by the product map R ( Z ) ⊗ n → R ( Z ) in the monoid R ( Z ) , defined out of monoid Z by ( . ).We continue:lim E ∨ ( X , Z ) = lim E ∨ Mon ( X , Z ) = Hom
Mon ( M ) ( colim( . ), Z ) ( . )with E ∨ Mon the following diagram:Hom
Mon ( M ) ( T M ( L ( T M ( X )) , Z )) Hom
Mon ( M ) ( T M ( L ( X )) , Z ) B mon o o A mon o o ( . )The maps A mon and B mon can be explicitly described from the description of A and B givenabove. We are done. ♦ Now we pass to the situation when the functor F admits, besides the lax-monoidal struc-ture ℓ F , a colax-monoidal structure c F , compatible by the bialgebra axiom (see Section . below). . T he B ialgebra axiom Fix some notations on adjoint functors.Let L : A ⇄ B : R be two functors. They are called adjoint to each other, with L the leftadjoint and R the right adjoint, whenMor B ( LX , Y ) ≃ Mor A ( X , RY ) ( . )where “ ≃ ” here means “isomorphic as bifunctors A opp × B → Sets ”.This gives rise to maps of functors ǫ : LR → Id B and η : Id A → RL such that the composi-tions L L ◦ η −−→ LRL ǫ ◦ L −−→ LR η ◦ R −−→ RLR R ◦ ǫ −−→ R ( . )are identity maps of the functors. he inverse is true: given maps of functors ǫ : LR → Id B and η : Id A → RL , obeying ( . ),gives rise to the isomorphism of bifunctors, that is, to adjoint equivalence (see [ML], SectionIV. , Theorems and ).In particular, the case of adjoint equivalence is the case when ǫ : LR → Id B and η : Id A → RL are isomorphisms of functors . In this case, setting ǫ = η − and η = ǫ − , we obtain anotheradjunction, with L the right adjoint and R the left adjoint .Let φ ∈ Mor B ( LX , Y ) . The following explicit formula for its adjoint ψ ∈ Mor A ( X , RY ) willbe useful: X η −→ RLX R ( φ ) −−→ RY ( . )and analogously for the way back: LX L ( ψ ) −−→ LRY ǫ −→ Y ( . )(see [ML], Section IV. ).Let now C and D be two symmetric monoidal categories, F : C → D a functor. Supposea lax-monoidal structure ℓ F and a colax-monoidal structure c F on F are given. The bialgebraaxiom is some compatibility condition on the pair ( c F , ℓ F ) , see Section A. . Recall the followingsimple fact from [Sh ], Section :L emma 2 . . Let C and D be two strict symmetric monoidal categories, and let F : C ⇄ D : G be anadjoint equivalence of the underlying categories. Given a pair ( c F , ℓ F ) where c F is a colax-monoidalstructure on F, ℓ F is a lax-monoidal structure on F, assign to it a pair ( c G , ℓ G ) of analogous structureson G, by ( . ) and ( . ) . If C and D are symmetric monoidal, and if the pair ( c F , ℓ F ) satisfies thebialgebra axiom (see Section A. ) , the pair ( c G , ℓ G ) satisfies the bialgebra axiom as well.Proof. Suppose ( c F , ℓ F ) are done. Define ( c G , ℓ G ) by ( . ) and ( . ). When we write down thebialgebra axiom diagram (see Section A. ) for ( c G , ℓ G ) we see due to cancelations of ǫ with ǫ − and of η with η − , that the diagram is commutative as soon as the diagram for ( c F , ℓ F ) is. ♦ L emma 2 . . Let C , D be two symmetric monoidal categories, and let F : C → D be a functor. Supposea lax-monoidal structure ℓ F on F is given. Consider the functorF mon = F mon ( ℓ F ) : Mon C → Mon D defined in ( . ) . Then the map F mon ( X ) ⊗ F mon ( Y ) → F mon ( X ⊗ Y ) , defined on the underlyingobjects as ℓ F , is a map of monoids, and, therefore, gives a lax-monoidal structure on F mon . Let now c F be a colax-monoidal structure on F. If ( ℓ F , c F ) satisfies the bialgebra axiom, the map F mon ( X ⊗ Y ) → F mon ( X ) ⊗ F mon ( Y ) , defined on the underlying objects as c F , is a map of monoids, and, therefore,gives a colax-monoidal structure on F mon . he both claims are straightforward checks. The second claim was, in fact, our motivationfor introduction of the bialgebra axiom in [Sh ]. T he D old -K an correspondence We use the following notations: C ( Z ) is the category of unbounded complexes of abelian groups, C ( Z ) + (resp., C ( Z ) − ) arethe full subcategories of Z ≥ -graded (resp., Z ≤ -graded) complexes. The category of abeliangroups placed in degree (with zero differential) is denoted by M od ( Z ) , thus, M od ( Z ) = C ( Z ) − ∩ C ( Z ) + . . The Dold-Kan correspondence is the following theorem:T heorem 3 . (Dold-Kan correspondence) . There is an adjoint equivalence of categoriesN : M od ( Z ) ∆ ⇄ C ( Z ) − : Γ where N is the functor of normalized chain complex (which is isomorphic to the Moore complex). We refer to [W], Section . , and [SchS ], Section , which both contain excellent treat-ment of this Theorem.The both categories M od ( Z ) ∆ and C ( Z ) − are symmetric monoidal in natural way. However,neither of functors N and Γ is monoidal.There is a colax-monoidal structure on N , called the Alexander-Whitney map AW : N ( A ⊗ B ) → N A ⊗ NB and a lax-monoidal structure on N , called the shuffle map ∇ : N ( A ) ⊗ N ( B ) → N ( A ⊗ B ) .Recall the explicit formulas for them.The Alexander-Whitney colax-monoidal map AW : N ( A ⊗ B ) → N ( A ) ⊗ N ( B ) is definedas AW ( a k ⊗ b k ) = ∑ i + j = k d i fin a k ⊗ d j b k ( . )where d and d fin are the first and the latest simplicial face maps.The Eilenberg-MacLane shuffle lax-monoidal map ∇ : N ( A ) ⊗ N ( B ) → N ( A ⊗ B ) is de-fined as ∇ ( a k ⊗ b ℓ ) = ∑ ( k , ℓ ) -shuffles ( α , β ) ( − ) ( α , β ) S β a k ⊗ S α b ℓ ( . ) here S α = s α k . . . s α and S β = s β ℓ . . . s β Here s i are simplicial degeneracy maps, α = { α < · · · < α k } , β = { β < · · · < β ℓ } , α , β ⊂ [
0, 1, . . . , k + ℓ − ] , α ∩ β = ∅ .Let us summarize their properties in the following Proposition, see [SchS ], Section ,and references therein, for a proof.P roposition 3 . . The colax-monoidal Alexander-Whitney and the lax-monoidal shuffle structures onthe functor N enjoy the following properties: . the composition N A ⊗ NB ∇ −→ N ( A ⊗ B ) AW −−→ N A ⊗ NBis the identity, . the composition N ( A ⊗ B ) AW −−→ N A ⊗ NB ∇ −→ N ( A ⊗ B ) is naturally chain homotopic to the identity, . the shuffle map ∇ is symmetric, . the Alexander-Whitney map AW is symmetric up to a natural chain homotopy. ♦ Recall a Theorem proven independently in [AM], Sect. . , and (later) in [Sh ], Sect. :T heorem 3 . . The pair ( ∇ , AW ) of the lax-monoidal shuffle structure and the colax-monoidal Alexander-Whitney structure, defined on the normalized chain complex functor N : M od ( Z ) ∆ ⇄ C ( Z ) − , obeysthe bialgebra axiom. ♦ This Theorem, along with Proposition . ( .), is essentially used in the proof of MainTheorem . in Section . . M onoidal properties Let F , G : C → D be two functors between monoidal categories.D efinition 3 . . Suppose the functor F is colax-monoidal, with the colax-monoidal structure c F , and G is lax-monoidal, with the lax-monoidal structure ℓ G . A morphism of functors Φ : F → G is called colax-monoidal if for any X , Y ∈ C , the diagram F ( X ⊗ Y ) c F (cid:15) (cid:15) Φ / / G ( X ⊗ Y ) F ( X ) ⊗ F ( Y ) Φ ⊗ Φ / / G ( X ) ⊗ G ( Y ) ℓ G O O ( . )As well, when F is lax-monoidal with the lax-monoidal structure ℓ F , and G is colax-monoidal with the colax-monoidal structure c G , a morphism Ψ : F → G is called lax-monoidal ,if for any X , Y ∈ C the diagram F ( X ) ⊗ F ( Y ) Ψ ⊗ Ψ / / ℓ F (cid:15) (cid:15) G ( X ) ⊗ G ( Y ) F ( X ⊗ Y ) Ψ / / G ( X ⊗ Y ) c G O O ( . )Each of the functors N and Γ admits both lax-monoidal and colax monoidal structures.Therefore, the compositions N ◦ Γ and Γ ◦ N are both lax- and colax-monoidal.Here are the main monoidal properties concerning the Dold-Kan correspondence, fromwhich (ii) is used essentially in our proof of Main Theorem . below.L emma 3 . . (i) the adjunction map ǫ : N ◦ Γ → Id is lax-monoidal,(ii) the adjunction map ǫ : N ◦ Γ → Id is colax-monoidal.Proof. The claim (i) is Lemma . of [SchS ]. The claim (ii) is proven analogously, wepresent here the proof for completeness. We need to prove the commutativity of the diagram N Γ ( X ⊗ Y ) ϕ / / , , ❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨ N ( Γ ( X ) ⊗ Γ ( Y )) AW / / N Γ ( X ) ⊗ N Γ ( Y ) ǫ ⊗ ǫ (cid:15) (cid:15) X ⊗ Y ( . )Here ϕ is the colax-monoidal structure on Γ adjoint to the shuffle lax-monoidal structure on N , see ( . ). The explicit formula for ϕ is: Γ ( X ⊗ Y ) ǫ − ⊗ ǫ − −−−−→ Γ ( N Γ ( X ) ⊗ N Γ ( Y )) ∇ −→ Γ N ( Γ ( X ) ⊗ Γ ( Y )) η −→ Γ ( X ) ⊗ Γ ( Y ) ( . ) ow the horizontal composition map in ( . ) is: N Γ ( X ⊗ Y ) ǫ − ⊗ ǫ − −−−−→ N Γ ( N Γ ( X ) ⊗ N Γ ( Y )) ∇ −→ N Γ N ( Γ ( X ) ⊗ Γ ( Y )) η − −−→ N ( Γ ( X ) ⊗ Γ ( Y )) AW −−→ N Γ ( X ) ⊗ N Γ ( Y ) ( . )where the map η − is applied to the “boxed” Γ N factor.Now the idea is to use the identity AW ◦ ∇ = Id (Lemma . ( .)) to “cancel” the secondand the fourth arrows in ( . ). We have:L emma 3 . . The following two compositions are equal:N Γ N ( Γ ( X ) ⊗ Γ ( Y )) η − −−→ N ( Γ ( X ) ⊗ Γ ( Y )) AW −−→ N Γ ( X ) ⊗ N Γ ( Y ) ( . ) and N Γ N ( Γ ( X ) ⊗ Γ ( Y )) AW −−→ N Γ ( N Γ ( X ) ⊗ N Γ ( Y )) ǫ −→ N Γ ( X ) ⊗ N Γ ( Y ) ( . ) where in the first (corresp., second) equation the map η − (corresp., ǫ ) is applied to the boxed factors. Clearly Lemma . (ii) follows from Lemma . and Lemma . ( .). Proof.
For an adjoint equivalence L : C ⇄ D : R with the adjunction isomorphisms ǫ : LR → Idand η : Id → RL , the two arrows LR L ǫ −→ L and L RL η − −−→ L coincide. ♦ R emark 3 . . The adjunction map η : Id → Γ ◦ N is both lax-monoidal and colax-monoidal only up to a homotopy , see Remark . in [SchS ]. The reason is that the another ordercomposition ∇ ◦ AW is equal to identity only up to a homotopy. T he functor L mon for L = N Here we compute explicitly the functor L mon (defined in general as a co-equalizer ( . )), forthe functor L = N , the normalized chain complex functor in the Dold-Kan correspondence N : M od ( Z ) ∆ ⇄ C − ( Z ) : Γ The functor L mon here is defined out of the Alexander-Whitney colax-monoidal structure AW on the functor N , see ( . ), ( . ). Theorems . , . proven here are essentially used in Section , in course of the proof of the Main Theorem . for Z ≤ -graded dg algebras. In fact, theyare used at the very end of the proof of Theorem . , in the proof of Proposition . (iii), seeSection . .We use the notations: L = N , R = Γ , where N , Γ are the functors in the Dold-Kancorrespondence. . Consider the Alexander-Whitney colax-monoidal structure AW : L ( X ⊗ Y ) → L ( X ) ⊗ L ( Y ) .It follows from Proposition . ( .) that the map AW is surjective for any X , Y . Thus, the mapof monoids β : T dg ( L ( T ∆ X )) → T dg ( LX ) , defined on the generators L ( T ∆ X ) as the iteratedAlexander-Whitney map (see ( . ), with c L = AW ) is surjective .Denote K ( X ) = Ker (cid:0) β : T dg ( L ( T ∆ X )) → T dg ( LX ) (cid:1) ( . )Now let X be a monoid in M od ( Z ) ∆ . Consider the iterated product m n : X ⊗ X ⊗ · · · ⊗ X | {z } n factors → X ( . )(with m = m ).The map of monoids α : T dg ( L ( T ∆ X )) → T dg ( LX ) is defined on the generators L ( X ⊗ n ) ⊂ L ( T ∆ X ) as ( m n ) ∗ : L ( X ⊗ n ) → L ( X ) .Denote by I ( X ) = α ( K ( X )) ( . )Then I ( X ) ⊂ T dg ( LX ) . Consider now the monoid structure on LX , defined by ( . ) withthe shuffle lax-monoidal structure ∇ on L . This monoid structure defines the iterated productmap m ∇ : T dg ( LX ) → LX ( . )Finally, denote J ( X ) = m ∇ ( I ( X )) ⊂ LX ( . )In fact, both I ( X ) ⊂ T dg ( LX ) and J ( X ) ⊂ LX are monoid-ideals.We give the following explicit form for the functor L mon , for L = N :T heorem 4 . . The functor L mon : Mon M od ( Z ) ∆ → Mon C ( Z ) − , defined by ( . ) , is isomorphic tothe functor ˜ L mon , defined for a monoid X in M od ( Z ) ∆ as ˜ L mon ( X ) = L ( X ) / J ( X ) ( . ) where in the right-hand side there is a quotient-monoid of the monoid L ( X ) (defined out of the lax-monoidal structure ∇ on L) by the monoid-ideal J ( X ) .Proof. Let X be a monoid in M od ( Z ) ∆ . We want to compute the co-equalizer ( . ) as thequotient-dg-algebra of T dg ( L ( X )) by some dg ideal I , L mon ( X ) = T dg ( L ( X )) / I ( . ) he ideal I is spanned, by definition of co-equalizer, by elements I = h α ( t ) − β ( t ) , t ∈ T dg ( L ( T ∆ X )) i ( . )The problem is how to compute the ideal I ⊂ T dg ( LX ) and the quotient T dg ( LX ) explicitly.Consider A ∈ ( LX ) ⊗ n , n ≥
1. The map AW ◦ ( n − ) : L ( X ⊗ n ) → ( LX ) ⊗ n is surjective, byProposition . ( .). Moreower, this Proposition gives a section of the projection AW ◦ ( n − ) ,given as s ( A ) = ∇ ◦ ( n − ) ( A ) ( . )where ∇ is the shuffle lax-monoidal structure on the functor L .Consider t = s ( A ) in ( . ), which gives, for any A , a particular element in the ideal I .Compute this element. One has β ( s ( A )) = A by construction of s ( A ) . Next, α ( s ( A )) is anelement in LX . This element is nothing but the iterated product m ∇ in the monoid L ( X ) ,defined by ( . ) out of the lax-monoidal structure ∇ on L .Thus, for any n ≥ A ∈ ( LX ) ⊗ n , one has A − m ◦ ( n − ) ∇ ( A ) ∈ I ( . )As the map β is surjective, and for any target element we found some its pre-image, onehas: I = h I , α ( K ( X )) i ( . )where I is the vector space spanned by all A − m ◦ ( n − ) ∇ ( A ) for all A and all n ≥ ♦ In our notations, I ( X ) = α ( K ( X )) , see ( . ), ( . ).We have proved that L mon ( X ) = T dg ( LX ) / h I , I ( X ) i The latter quotient-monoid is isomorphic to LX / J ( X ) , see ( . ). We are done. ♦ . . E xample Consider the value of the functor L mon on a free monoid X = T ∆ ( V ) , generated by a simplicialabelian group V . Due to the commutative diagram ( . ), L mon ( T ∆ ( V )) = T dg ( L ( V )) ( . )On the other hand, our result in Theorem . gives an isomorphism L mon = ˜ L mon , withformula for ˜ L mon , given in ( . ). It is interesting to compare ˜ L mon ( T ∆ ( V )) , given in ( . ), withformula ( . ). e get some identity of the form L ( T ∆ ( V )) / J ( T ∆ V ) = T dg ( L ( V )) ( . )It seems the latter identity is rather non-trivial combinatorial fact, if one tries to prove itdirectly. . Here we specify Theorem . , as follows (see Theorem . below).Let L : C ⇄ D : R be an adjunction between monoidal categories, and let ϕ be a lax-monoidal structure on R . Then the functor L mon : Mon ( C ) → Mon ( D ) , defined in ( . ), is leftadjoint to R mon , defined as in ( . ). Then, the functor R mon is equal to R on the underlyingobjects, and the lax-monoidal structure ϕ on R defines a lax-monoidal structure ϕ mon on R mon .Consequently, the left adjoint functor L mon comes with the colax-monoidal structure, adjointto ϕ mon . We refer to this colax-monoidal structure on L mon as canonical .Here we describe explicitly this colax-monoidal structure on the functor ˜ L mon , L mon ≃ ˜ L mon .The shuffle lax-monoidal structure ∇ makes L ( X ) a monoid, for any monoid X , by ( . ).We always assume this monoid structure on LX , and denote the functor L : Mon M od ( Z ) ∆ → Mon C ( Z ) − by L ∇ . The pair ( AW , ∇ ) of lax-monoidal and colax-monoidal structures on thefunctor L obeys the bialgebra axiom , see Theorem . . Then Lemma . says that the Alexander-Whitney map AW : L ∇ ( X ⊗ Y ) → L ∇ ( X ) ⊗ L ∇ ( Y ) , for monoids X and Y , is a map of monoids ,and therefore defines a colax-monoidal structure on the functor L ∇ : Mon ( M od ( Z ) ∆ ) → Mon ( C ( Z ) − ) .We claim that this colax-monoidal structure on L ∇ descents to a colax-monoidal structureon the functor ˜ L mon , and this colax-monoidal structure is the one which ˜ L mon enjoys outof isomorphism of functors L mon ≃ ˜ L mon , given in Theorem . , and out of the canonicalcolax-monoidal structure on L mon .T heorem 4 . . (A) The Alexander-Whitney colax-monoidal structure on the functor L ∇ descents toa colax-monoidal structure on the functor ˜ L mon ≃ L ∇ / J ,(B) Within the isomorphism of functors L mon and ˜ L mon , given in Theorem . , the colax-monoidalstructure on ˜ L mon from part (A) is the one which is corresponded to the canonical colax-monoidalstructure on L mon . We prove this Theorem in Section . below, after formulating and proving a more generalTheorem . in Section . . . T he colax - monoidal structure on L mon , the general case Let C , D be general symmetric monoidal categories, and let L : C ⇄ D : R be a pair of adjoint functors, with R the right adjoint.Suppose a lax-monoidal structure ϕ on R is given, so that the functor R mon : Mon D → Mon C is defined, as in ( . ). The functor R mon comes with a lax-monoidal structure ϕ mon ,equal to ϕ on the underlying objects.Denote by c L the colax-monoidal structure on the functor L , adjoint to the lax-monoidalstructure ϕ on R .Recall formula ( . ) for the functor L mon left adjoint to R mon . We call canonical the colax-monoidal structure on L mon , adjoint to the lax-monoidal structure ϕ mon on R mon . Here wediscuss the following question: how to express the canonical colax-monoidal structure on L mon in terms of the co-equalizer definition ( . )?Consider the co-equalizer diagram for L mon : T D ( L ( T C ( X ))) α / / β / / T D ( LX ) ( . )Denote the diagram ( . ) by D ( X ) , with L mon ( X ) = colim D ( X ) . As well, denote by D ( X ) and D ( X ) the left-hand (corresp., the right-hand) algebras in the diagram D ( X ) .Firstly, we construct, under some assumptions, a map Ψ X , Y : L mon ( X ⊗ Y ) → colim ( D ( X ) ⊗ D ( Y )) ( . )where D ( X ) ⊗ D ( Y ) is the diagram D ( X ) ⊗ D ( Y ) α X ⊗ α Y / / β X ⊗ β Y / / D ( X ) ⊗ D ( Y ) ( . )see Lemma . .Next, there is a canonical map i X , Y : colim ( D ( X ) ⊗ D ( Y )) → colim ( D ( X )) ⊗ colim ( D ( Y )) = L mon ( X ) ⊗ L mon ( Y ) ( . )Indeed, by the universal property of colimit, there are maps i ( X ) : D ( X ) → colim D ( X ) and i ( X ) : D ( X ) → colim D ( X ) , and analogous maps for Y . Then the maps i ( X ) ⊗ i ( Y ) : D ( X ) ⊗ D ( Y ) → colim ( D ( X )) ⊗ colim ( D ( Y )) nd i ( X ) ⊗ i ( Y ) : D ( X ) ⊗ D ( Y ) → colim ( D ( X )) ⊗ colim ( D ( Y )) define the map i X , Y in ( . ).The composition Θ X , Y = i X , Y ◦ Ψ X , Y : L mon ( X ⊗ Y ) → L mon ( X ) ⊗ L mon ( Y ) ( . )which is defined once Ψ X , Y is, will enjoy the colax-monoidal property, and is isomorphic tothe canonical colax-monoidal structure on L mon .Turn back to the map Ψ X , Y , see ( . ).To define a map of monoids Ψ X , Y : L mon ( X ⊗ Y ) → colim ( D ( X ) ⊗ D ( Y )) , it is enough todefine two maps of monoids Ψ ′ X , Y : T D ( L ( T C ( X ⊗ Y ))) → T D ( L ( T C ( X ))) ⊗ T D ( L ( T C ( Y ))) and Ψ ′′ X , Y : T D ( L ( X ⊗ Y )) → T D ( LX ) ⊗ T D ( LY ) such that the diagrams T D ( L ( T C ( X ⊗ Y ))) Ψ ′ X , Y / / α (cid:15) (cid:15) T D ( L ( T C ( X ))) ⊗ T D ( L ( T C ( Y ))) α ⊗ α (cid:15) (cid:15) T D ( L ( X ⊗ Y )) Ψ ′′ X , Y / / T D ( LX ) ⊗ T D ( LY ) ( . )and T D ( L ( T C ( X ⊗ Y ))) Ψ ′ X , Y / / β (cid:15) (cid:15) T D ( L ( T C ( X ))) ⊗ T D ( L ( T C ( Y ))) β ⊗ β (cid:15) (cid:15) T D ( L ( X ⊗ Y )) Ψ ′′ X , Y / / T D ( LX ) ⊗ T D ( LY ) ( . )are commutative.Define the maps Ψ ′ X , Y and Ψ ′′ X , Y as the compositions Ψ ′ X , Y : T D ( L ( T C ( X ⊗ Y ))) θ ∗ / / T D ( L ( T C ( X ) ⊗ T C ( Y ))) ( c L ) ∗ / / T D ( L ( T C ( X )) ⊗ L ( T C ( Y ))) θ ∗ / / T D ( L ( T C ( X ))) ⊗ T D ( L ( T C ( Y ))) ( . )and Ψ ′′ X , Y : T D ( L ( X ⊗ Y )) c L ∗ / / T D ( LX ⊗ LY ) θ ∗ / / T D ( LX ) ⊗ T D ( LY ) ( . )correspondingly.We note: emma 4 . . The both maps Ψ ′ X , Y and Ψ ′′ X , Y are maps of monoids.Proof. As in the both cases the source monoids are free, to have maps of monoids it is enoughto define them somehow on the generators L ( T C ( X ⊗ Y )) (correspondingly, on L ( X ⊗ Y ) ), andthen to extend them by multiplicativity. This is precisely how the definitions ( . ) and ( . )are organized. ♦ Here c L ∗ is the map induced by the colax-monoidal structure c L on the functor L , and themap θ ∗ is induced by the map θ : T ( X ⊗ Y ) → T ( X ) ⊗ T ( Y ) , see ( . ).The both maps c L and θ are colax-monoidal, therefore, the compositions Ψ ′ and Ψ ′′ alsoare, in appropriate sense.Before formulating the following Lemma, we give a definition:D efinition 4 . . Let C and D be symmetric monoidal categories, and let F : C → D be a functor. Acolax-monoidal structure Θ on F is called quasi-symmetric if for any four objects X , Y , Z , W in C thediagram F ( X ⊗ Y ) ⊗ F ( Z ⊗ W ) Θ ⊗ Θ / / F ( X ) ⊗ F ( Y ) ⊗ F ( Z ) ⊗ F ( W ) σ F ( Y ) , F ( Z ) / / F ( X ) ⊗ F ( Z ) ⊗ F ( Y ) ⊗ F ( W ) F ( X ⊗ Y ⊗ Z ⊗ W ) Θ O O σ Y , Z / / F ( X ⊗ Z ⊗ Y ⊗ W ) Θ / / F ( X ⊗ Z ) ⊗ F ( Y ⊗ W ) Θ ⊗ Θ O O ( . ) is commutative. L emma 4 . . For Ψ ′ and Ψ ′′ as defined in ( . ) , ( . ) :(i) the diagram ( . ) always commutes,(ii) the diagram ( . ) commutes if the colax-monoidal structure c L on the functor L is quasi-symmetric, see Definition . .Proof. As all maps in the diagrams ( . ) and ( . ) are maps of monoids, with the source afree monoid, it is enough to prove the commutativity of these diagrams on the generators inthe source(s).Now the claim (i) is reduced to the following diagram, for n ≥ ub - lemma 4 . . Let X , Y be monoids in C , and let n ≥ . Then the following diagram is commutative: L (( X ⊗ Y ) ⊗ · · · ⊗ ( X ⊗ Y )) m X ⊗ Y (cid:15) (cid:15) σ / / L (( X ⊗ · · · ⊗ X ) ⊗ ( Y ⊗ · · · ⊗ Y )) c L / / L ( X ⊗ · · · ⊗ X ) ⊗ L ( Y ⊗ · · · ⊗ Y ) m X ⊗ m Y (cid:15) (cid:15) L ( X ⊗ Y ) c L / / L ( X ) ⊗ L ( Y ) ( . ) where there is n of X’s and n of Y’s in each entry of the diagram. roof. The diagram L (( X ⊗ Y ) ⊗ · · · ⊗ ( X ⊗ Y )) m X ⊗ Y ) ) ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ σ / / L (( X ⊗ · · · ⊗ X ) ⊗ ( Y ⊗ · · · ⊗ Y )) m X ⊗ m Y t t ✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ L ( X ⊗ Y ) ( . )by trivial reasons. Therefore, we need to prove that the diagram L (( X ⊗ · · · ⊗ X ) ⊗ ( Y ⊗ · · · ⊗ Y )) c L / / m X ⊗ m Y (cid:15) (cid:15) L ( X ⊗ · · · ⊗ X ) ⊗ L ( Y ⊗ · · · ⊗ Y ) m X ⊗ m Y (cid:15) (cid:15) L ( X ⊗ Y ) c L / / L ( X ) ⊗ L ( Y ) ( . )commutes.But c L : L ( A ⊗ B ) → L ( A ) ⊗ L ( B ) is a map of bifunctors, what easily implies the commu-tativity of ( . ). ♦ The claim (i) of Lemma is proven. Let us prove (ii).Written on the generators of free source monoid, the diagram ( . ) is reduced to thefollowing diagram: L (( X ⊗ Y ) ⊗ · · · ⊗ ( X ⊗ Y )) β X ⊗ Y (cid:15) (cid:15) σ / / L (( X ⊗ · · · ⊗ X ) ⊗ ( Y ⊗ · · · ⊗ Y )) c L / / L ( X ⊗ · · · ⊗ X ) ⊗ L ( Y ⊗ · · · ⊗ Y ) c ◦ ( n − ) L ⊗ c ◦ ( n − ) L (cid:15) (cid:15) L ( X ⊗ Y ) ⊗ · · · ⊗ L ( X ⊗ Y ) c ⊗ nL / / L ( X ) ⊗ L ( Y ) ⊗ · · · ⊗ L ( X ) ⊗ L ( Y ) σ / / ( L ( X ) ⊗ · · · ⊗ L ( X )) ⊗ ( L ( Y ) ⊗ · · · ⊗ L ( Y )) ( . )Suppose c L is quasi-symmetric. Then the diagram ( . ) is commutative for any X , Y , Z , W in C . This commutativity implies the commutativity of ( . ).Claim (ii) is proven.Lemma is proven. ♦ Suppose that c L is quasi-symmetric. Then we have defined a map Θ X , Y : L mon ( X ⊗ Y ) → L mon ( X ) ⊗ L mon ( Y ) on co-equalizers ( . ).T heorem 4 . . Let C and D be two symmetric monoidal categories, and letL : C ⇄ D : Rbe a pair of adjoint functors, with R the right adjoint. Let ϕ be a lax-monoidal structure on R, such thatthe adjoint colax-monoidal structure c L on L is quasi-symmetric (see Definition . ). Define Ψ X , Y as n ( . ) , ( . ) above. Define Θ X , Y = i X , Y ◦ Ψ X , Y . Then Θ X , Y : L mon ( X ⊗ Y ) → L mon ( X ) ⊗ L mon ( Y ) is a colax-monoidal structure on the functor L mon , isomorphic to the canonical one. We prove Theorem . in Section . below. . P roof of T heorem 4 . The proof is essentially based on the theory of (very) weak adjoint functors, developed inAppendix B. The reader is advised to read the statements of Appendix B, especially TheoremB. before proceeding to read this Section.We keep the notations of Section . . Denote by I the category a • β / / α / / • b ( . )We introduce two diagrams indexed by I , D : I → F un ( Mon C , Mon D ) and R : I → F un ( Mon D , Mon C ) By definition, D is the diagram ( . ), with D ( a )( X ) = D ( X ) = T D ( L ( T X )) , D ( b )( X ) = D ( X ) = T D ( LX ) , and with the maps α , β from ( . ).The diagram R is defined as the constant diagram, R ( a )( Y ) = R ( b )( Y ) = R mon ( Y ) , withthe both maps α , β equal to the constant maps.The idea of proof of Theorem . is as follows. We construct a very weak right adjoint pairstructures on ( D , R mon ) and on ( D , R mon ) , such that they define a very weak right adjoint pairof diagrams D : C ⇄ D : R ( . )(see Definitions B. , B. in Appendix B).Next, we show that the pair ( Ψ ′ , ϕ mon ) of (co)lax-monoidal structures on the functors ( D ( a ) , R mon , as well as the pair ( Ψ ′′ , ϕ mon ) of (co)lax-monoidal structures on the functors ( D ( b ) , R mon ) , are weak compatible , see Definition B. (the colax-monoidal structures Ψ ′ and Ψ ′′ are defined ( . ), ( . )).Then we are in the set-up of Corollary ?? , and this Corollary immediately gives Theorem . . emark 4 . . It is most likely that neither of pair of functors ( D ( a ) , R mon ) and ( D ( b ) , R mon ) has a natural structure of a weak right adjoint pair, see Definition B. . In fact, ǫ ( a ) and ǫ ( b ) are uniquely defined, see proof of Key-Lemma . , ( .). Then, most likely there do not existany natural adjunctions η ( a ) : Id C → R mon D ( a ) and η ( b ) : Id C → R mon D ( b ) , such that thecomposition (B. ) is the identity, for ǫ ( a ) = ǫ and ǫ ( b ) = ǫ (see ( . ) and ( . ) below).Nevertheless, the “corresponding” colax-monoidal structures Ψ ′ and Ψ ′′ exist. This remarkexplains our decision to work with very weak adjoint functors of Appendix B. than with weakadjoint functors of Appendix B. .Our proof of Theorem . is based on the followingK ey - lemma 4 . . In the notations and assumptions of Theorem . : . there are very weak right adjoint structures on the pairs of functors ( D , R mon ) and ( D , R mon ) such that they form a very weak right adjoint pair of diagrams ( . ) , . the colax-monoidal structure Ψ ′ on the functor D and the colax-monoidal structure Ψ ′′ on thefunctor D , form a diagram of colax-monoidal structures, . the pair ( Ψ ′ , ϕ mon ) of (co)lax-monoidal structures on the functors ( D , R mon ) , and the pair ( Ψ ′′ , ϕ mon ) of (co)lax-monoidal structures on the functors ( D , R mon ) , are weakly compatible,see Definition B. , . the colimit colax-monoidal structure colim I ( Ψ ′ , Ψ ′′ ) on the functor L mon (which is well-definedby ( .)) is precisely the colax-monoidal structure Θ on L mon , see ( . ) . We firstly show how Theorem . follows from the Key-lemma above, and then prove theKey-Lemma in the rest of this Subsection. Proof of Theorem . : Equip the both functors R ( a ) = R ( b ) = R mon with the lax-monoidal structure ϕ mon , withthe identity maps between them. Then we get a diagram of lax-monoidal structures . Its limitis a lax-monoidal structure on the functor lim I R = R mon , equal to ϕ mon . We know, by Key-lemma . , that the pairs of (co)lax-monoidal structures ( Ψ ′ , ϕ mon ) , and ( Ψ ′′ , ϕ mon ) are weaklycompatible, see Definition B. . Now, by Theorem B. , the colax-monoidal structure on thefunctor L mon = colim I D adjoint to the limit lax-monoidal diagram ϕ mon , and the colimit Ψ colim of the colax-monoidal structures on D ( a ) and D ( b ) adjoint to ϕ mon , are also weaklycompatible. By Key-lemma . ( .), the colax-monoidal structure Ψ colim on L mon is equal tothe colax-monoidal structure Θ on L mon , see ( . ). We conclude, that the colax-monoidalstructure Θ on L mon is weakly compatible with the lax-monoidal structure ϕ mon on R mon .Now, the functors ( L mon , R mon ) form a (genuine) adjoint pair of functors, with ǫ : L mon R mon → Id D equal to the limit ǫ lim of ǫ ( a ) and ǫ ( b ) , and with some η : Id C → R mon L mon . Therefore, heorem B. implies that Θ is isomorphic to the colax-monoidal structure on L mon , adjointto the lax-monoidal structure ϕ mon on R mon . ♦ Proof of Key-lemma . :Proof of ( .): Recall our notations D ( X ) = T D ( L ( T C X )) and D ( X ) = T D ( LX ) , for X ∈ Mon C . Weneed to construct morphisms of functors ǫ : D ◦ R mon → Id Mon D ( . )and ǫ : D ◦ R mon → Id Mon D ( . )such that the diagrams D R mon ( Y ) ǫ / / α ∗ (cid:15) (cid:15) Y id (cid:15) (cid:15) D R mon ( Y ) ǫ / / Y ( . )and D R mon ( Y ) ǫ / / β ∗ (cid:15) (cid:15) Y id (cid:15) (cid:15) D R mon ( Y ) ǫ / / Y ( . )for any Y ∈ Mon D .Recall the the functor L mon is the co-equalizer ( . ). Therefore, there is a canonical map p : D ( X ) → L mon ( X ) ( . )such that the two compositions α ◦ p and β ◦ p are equal maps D ( Y ) → L mon ( Y ) .Next, there is the adjunction map ǫ mon : L mon ◦ R mon → Id Mon D . Define ǫ as the composi-tion ǫ = p ∗ ◦ ǫ mon : D ◦ R mon → Id Mon D ( . )and define ǫ = α ∗ ǫ = β ∗ ǫ : D ◦ R mon → Id Mon D ( . )Then the diagrams ( . ) and ( . ) commute by the construction. Proof of ( .): The claim means the commutativity of diagrams ( . ) and ( . ). It wasproven in Lemma . , with assumption that the colax-monoidal structure c L on L , adjoint o the lax-monoidal structure ϕ on R , is quasi-symmetric, see Definition . (the latter isassumed in the statement of Key-Lemma . ). Proof of ( .): We need firstly to know an explicit expression for ǫ : D R mon → Id Mon D .S ub - lemma 4 . . In the above notations, the map ǫ : D R mon → Id Mon D is equal to the compositionT D ( L ( R mon ( Y ))) = T D ( LR ( Y )) ǫ ∗ −→ T D ( Y ) m Y −→ Y ( . ) where the last map uses the monoid structure on Y.Proof. By our definition ( . ), the following compositionHom Mon D ( T D L (( R mon Y ) , Z )) t ←− t ←− lim (cid:16) ( Hom
Mon D ( T D ( L ( R mon Y )) , Z ) ⇒ Hom
Mon D ( T D ( T D ( L ( R mon Y ))) , Z ) (cid:17) ≃≃ Hom
Mon D ( L mon R mon ( Y ) , Z ) ( . )is induced by the map D ( X ) = T D ( L ( X )) → L mon ( X ) for X ∈ Mon C (here t is the universallimit map).Thus, we need to find (a unique) κ ∈ Hom
Mon D ( T D ( L ( R mon Y )) , Y ) and κ ∈ Hom
Mon D ( T D ( T D ( L ( R mon Y ))) , Y ) such that α ∗ κ = β ∗ κ = κ , and such that the corresponding element κ ∈ Hom
Mon D ( L mon R mon Y , Y ) is equal to ǫ mon , or, in other words, is corresponded to the identity in Hom Mon C ( R mon Y , R mon Y ) .We follow the proof of Lemma . , and find, step by step, corresponding representativesin (co)limits, starting with the identity map in Hom Mon C ( R mon Y , R mon Y ) .By Sub-lemma . , the monoid R mon Y is isomorphic to R mon ( Y ) ≃ colim (cid:16) T C T C R mon ( Y ) ⇒ T C R mon ( Y ) (cid:17) ( . )where the two arrows are as in Sub-lemma . .We have: Hom Mon C ( R mon Y , R mon Y ) ≃ lim (cid:16) Hom
Mon C ( T C R mon Y , R mon Y ) ⇒ Hom
Mon C ( T C T C R mon Y , R mon Y ) (cid:17) ( . )Consider ω ∈ Hom
Mon C ( T C R mon Y , R mon Y ) defined as the map induced by the product in themonoid R mon Y , and consider ω ∈ Hom
Mon C ( T C T C R mon Y , R mon Y ) defined as the compositionof the map in Hom Mon C ( T C T C R mon Y , T C R mon Y ) induced by the monoid structure in T C R mon Y ,followed by ω . ne easily checks that the pair ( ω , ω ) is compatible in the limit diagram, and is corre-sponded, by the isomorphism ( . ), to the identity map of R mon Y .We proceed as in Lemma . :lim (cid:16) Hom
Mon C ( T C R mon Y , R mon Y ) ⇒ Hom
Mon C ( T C T C R mon Y , R mon Y ) (cid:17) ≃ lim (cid:16) Hom C ( R mon Y , R mon Y ) ⇒ Hom C ( T C R mon Y , R mon Y ) (cid:17) ( . )Consider χ ∈ Hom C ( RY , RY ) defined as the identity map, and consider χ ∈ Hom C ( T C ( RY ) , RY ) defined from the product map. One easily sees that ( χ , χ ) is corresponded to ( ω , ω ) underthe isomorphism ( . ).The next step is: lim (cid:16) Hom C ( R ( Y ) , R ( Y )) ⇒ Hom C ( T C R ( Y ) , R ( Y )) (cid:17) ≃ lim (cid:16) Hom D ( LR ( Y ) , Y ) ⇒ Hom D ( L ( T C ( RY )) , Y ) (cid:17) ( . )One sees that ( σ , σ ) corresponded to ( χ , χ ) under ( . ) are defined as σ = ǫ ( L ( χ )) and σ = ǫ ( L ( χ )) ( . )where ǫ : LR → Id D is the adjunction.Now lim (cid:16) Hom D ( LR ( Y ) , Y ) ⇒ Hom D ( L ( T C ( RY )) , Y ) (cid:17) ≃ lim (cid:16) Hom
Mon D ( T D ( L ( R ( Y )) , Y )) ⇒ Hom
Mon D ( T D ( L ( T C ( RY ))) , Y ) (cid:17) ≃ Hom
Mon D ( L mon ( Y ) , Y ) ( . )To define ( , ) corresponded to ( σ , σ ) by ( . ), we recall that Y is a monoid, and define ( , ) as the maps from free monoids, induced by the maps ( σ , σ ) , and by the monoid structure in Y . We finish to prove the Sub-lemma.By ( . ), ǫ = , and for we have an explicit formula, which gives ( . ) immediately. ♦ We continue to prove the claim ( .): having the above Sub-lemma as a toolkit, it is not hard toprove the commutativity of the corresponding diagrams, by a direct check. We need to prove he commutativity of the diagrams: D ( R mon X ⊗ R mon Y ) ϕ mon / / Ψ ′ (cid:15) (cid:15) D R mon ( X ⊗ Y ) ǫ (cid:15) (cid:15) D R mon X ⊗ D R mon ( Y ) ǫ ⊗ ǫ / / X ⊗ Y ( . )and D ( R mon X ⊗ R mon Y ) ϕ mon / / Ψ ′′ (cid:15) (cid:15) D R mon ( X ⊗ Y ) ǫ (cid:15) (cid:15) D R mon X ⊗ D R mon ( Y ) ǫ ⊗ ǫ / / X ⊗ Y ( . )To prove the commutativity of ( . ), divide it into two diagrams, as follows: T D L ( RX ⊗ RY ) ϕ / / c L (cid:15) (cid:15) T D ( LR ( X ⊗ Y )) ǫ ∗ (cid:15) (cid:15) T D ( LRX ⊗ LRY ) Θ (cid:15) (cid:15) ǫ ⊗ ǫ / / T D ( X ⊗ Y ) m X ⊗ Y (cid:15) (cid:15) T D LR ( X ) ⊗ T D LR ( Y ) ǫ ∗ ⊗ ǫ ∗ (cid:15) (cid:15) T D X ⊗ T D Y m X ⊗ m Y / / X ⊗ Y ( . )where c L is the colax-monoidal structure on L adjoint to the lax-monoidal structure ϕ on R , and the map Θ is ( . ). The commutativity of the diagram ( . ) is equivalent to thecommutativity of the diagram ( . ), by Sub-lemma . . The commutativity of the both sub-diagrams of ( . ) is clear: the upper one commutes as c L and ϕ are adjoint (co)lax-monoidalstructures, and the lower one commutes by tautological reasons.Now prove the commutativity of ( . ). The diagram ( . ) is divided into two diagrams,as follows: D ( R mon X ⊗ R mon Y ) Ψ ′ s s ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ α ∗ = β ∗ (cid:15) (cid:15) ϕ mon / / D R mon ( X ⊗ Y ) α ∗ = β ∗ (cid:15) (cid:15) D R mon ( X ) ⊗ D R mon ( Y ) α ∗ ⊗ α ∗ + + ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ D ( R mon X ⊗ R mon Y ) Ψ ′′ (cid:15) (cid:15) ϕ mon / / D R mon ( X ⊗ Y ) ǫ (cid:15) (cid:15) D R mon ( X ) ⊗ D R mon ( Y ) ǫ ⊗ ǫ / / X ⊗ Y ( . ) he commutativity of the left “triangle-like” diagram is exactly the claim ( .) of the Key-Lemma. The lower-right square diagram is the diagram ( . ), and its commutativity is justproven above. The commutativity of the upper-right square diagram is a general nonsense.We are done. Proof of ( .): The colax-monoidal structure Θ on L mon is the colimit of Ψ ′ and Ψ ′′ bydefinition, see Definition B. .Key-lemma . is proven. ♦ . A proof of T heorem 4 . Here we prove Theorem . .We want to apply Theorem . . We need to show that the assumptions of this Theoremare fulfilled, in the case L = N , the normalized chain complex functor in the Dold-Kancorrespondence, and c L = AW , the Alexander-Whitney colax-monoidal structure on L .L emma 4 . . The Alexander-Whitney colax-monoidal structure AW : L ( X ⊗ Y ) → L ( X ) ⊗ L ( Y ) isquasi-symmetric, see Definition . .Proof. We need to prove that diagram ( . ) commutes, for F = L = N , and Θ = AW , and forany X , Y , Z , W ∈ M od ( Z ) ∆ .To this end, recall the definition of the Alexander-Whitney map, see ( . ): AW ( a k ⊗ b k ) = ∑ i + j = k d i fin a k ⊗ d j b k As well, the identity d d fin = d fin d ( . )is valid for any simplicial set.Then the both pathes in diagram ( . ), applied to x k ⊗ y k ⊗ z k ⊗ w k (all of degree k ) gives ∑ a + b + c + d = k ∑ b + b = bc + c = c d a fin ( x k ) ⊗ d c d c fin ( z k ) ⊗ d b d b fin ( y k ) ⊗ d d ( w k ) ( . ) ♦ Now we apply Theorem . .Firstly we compute, in the particular case of L = N , the colimit colim ( D ( X ) ⊗ D ( Y )) , see( . ). As in the computation of colim D ( X ) in Section . , we writecolim ( D ( X ) ⊗ D ( Y )) = (( T dg ( LX )) ⊗ T dg ( LY )) / I X , Y ( . ) or some ideal I X , Y ⊂ ( T dg ( LX )) ⊗ ( T dg ( LY )) . The same surjectivity argument reduces theanswer to the computation of ( m ∇ , X ⊗ m ∇ , Y ) ◦ ( m X ⊗ m Y )( Ker ( β X ⊗ β Y )) .Recall the following trivial observation: Let V , W , V ′ , W ′ be vector spaces over a field k ,and let f : V → W , g : V ′ → W ′ be k -linear maps. ThenKer ( f ⊗ g ) = ( Ker f ) ⊗ W + V ⊗ ( Ker g ) ( . )Due to ( . ),Ker ( β X ⊗ β Y ) = ( Ker β X ) ⊗ T dg ( L ( T ∆ Y )) + T dg ( L ( T ∆ X )) ⊗ ( Ker β Y ) ( . )Consequently,colim ( D ( X ) ⊗ D ( Y )) = ( colim D ( X )) ⊗ ( colim D ( Y )) = L mon ( X ) ⊗ L mon ( Y ) ( . )and the map i X , Y (see ( . )) is an isomorphism in the case we consider.We have the following commutative diagram, with Θ X , Y given in Theorem . , and thevertical isomorphisms just described,colim D ( X ⊗ Y ) Θ X , Y / / ≃ (cid:15) (cid:15) colim ( D ( X ) ⊗ D ( Y )) ≃ (cid:15) (cid:15) L ( X ⊗ Y ) / J ( X ⊗ Y ) X / / ( L ( X ) / J ( X )) ⊗ ( L ( Y ) / J ( Y )) ( . )with undefined X . It follows from Theorem . that one necessarily has X = c L = AW We are done. ♦ R emark 4 . . Indirectly, we have proven that the Alexander-Whitney map AW : L ( X ⊗ Y ) → L ( X ) ⊗ L ( Y ) where X , Y are monoids, descents to the quotients L ∇ ( X ⊗ Y ) / J ( X ⊗ Y ) → ( L ∇ ( X ) / J ( X )) ⊗ ( L ∇ ( Y ) / J ( Y )) In other words, we proved that AW ( J ( X ⊗ Y )) ⊂ h J ( X ) ⊗
1, 1 ⊗ J ( Y ) i ( . )We do not see any “direct” way to prove ( . ). M ain theorem for Z ≤ - graded dg algebras . Here we prove our main result:T heorem 5 . . Let k be a field of any characteristic. There is a functor R : A lg ≤ k → A lg ≤ k and amorphism of functors w : R → Id with the following properties: . R ( A ) is cofibrant, and w : R ( A ) → A is a quasi-isomorphism, for any A ∈ A lg ≤ k , . there is a colax-monoidal structure on the functor R , such that all colax-maps β A , B : R ( A ⊗ B ) → R ( A ) ⊗ R ( B ) are quasi-isomorphisms of dg algebras, and such that the diagram R ( A ⊗ B ) β A , B / / & & ▲▲▲▲▲▲▲▲▲▲ R ( A ) ⊗ R ( B ) w w ♦♦♦♦♦♦♦♦♦♦♦ A ⊗ Bis commutative, . the morphism w ( k q ) : R ( k q ) → k q coincides with α : R ( k q ) → k q , where α is a part of thecolax-monoidal structure (see Definition A. ) , and k q = A lg ≤ k is the dg algebra equal to theone-dimensional k-algebra in degree , and vanishing in other degrees. The idea is to use the solution of the analogous problem for simplicial algebras (given inTheorem . in Section ), and “transfer” it to dg algebras using the Dold-Kan correspon-dence.More precisely, consider the Dold-Kan correspondence N : M od ( Z ) ∆ ⇄ C ( Z ) − : Γ The functors N and Γ form an adjoint equivalence of categories; therefore, we have somefreedom which of these two functors to consider as the left (right) adjoint. We consider Γ asthe right adjoint. From now on, we use the notations: Γ = R , N = L .The functor L comes with the colax-monoidal (Alexander-Whitney) structure AW andwith the lax-monoidal (shuffle) structure ∇ . They induce a lax-monoidal structure ℓ R and acolax-monoidal structure c R on the functor R by the adjunction, as is explained in ( . ) and( . ).Consider the functor R mon : A lg ≤ k → A lg ∆ k , induced by the functor R and from its lax-monoidal structure ℓ R , on the categories of monoids (see Section . ). It admits a left adjointfunctor L mon , defined in Section . . ecall that for a simplicial algebra A we denote by F ( A ) a solution of Theorem . .Define R ( A ) = L mon ( F ( R mon ( A ))) ( . )where A ∈ A lg ≤ k .There is a projection p F : F ( R mon ( A )) → R mon ( A ) . Define the projection p A : R ( A ) → A ( . )as the composition of the projection p F with the adjunction map L mon ◦ R mon → Id.We claim that this functor R gives a solution to Theorem . . We need to prove thefollowing statements:P roposition 5 . . (i) the functor R : A lg ≤ k → A lg ≤ k has a natural colax-monoidal structure β ,(ii) R ( A ) is cofibrant, and the projection p A : R ( A ) → A is a weak equivalence, for any A ∈ A lg ≤ k ,(iii) the diagram R ( A ⊗ B ) β A , B / / p A ⊗ B & & ▲▲▲▲▲▲▲▲▲▲ R ( A ) ⊗ R ( B ) p A ⊗ p B w w ♦♦♦♦♦♦♦♦♦♦♦ A ⊗ B ( . ) commutes; consequently, it follows from (ii) and from the -out-of- axiom that β A , B is a weakequivalence for any A , B ∈ A lg ≤ k . The three items of this Proposition rely on three different theories: they are the bialgebraaxiom for (i), the Schwede-Shipley theory of weak monoidal Quillen pairs for (ii), and themonoidal property of the Dold-Kan correspondence (Lemma . ) for (iii). We give the detailedproof in the rest of this Section. . P roof of P roposition 5 . , ( i ) The functor R = L mon ◦ F ◦ R mon is a composition of three functors. The functor L mon comeswith its colax-monoidal structure (adjoint to the lax-monoidal structure on R mon ), and thefunctor F has the colax-monoidal structure ( . ). It remains to define a colax-monoidal struc-ture on R mon (a priori R mon has only a lax-monoidal structure).Recall that the functor L = N has the Alexander-Whitney colax-monoidal structure AW and a lax-monoidal structure ∇ , compatible by the bialgebra axiom (see Theorem . ). Then itfollows from Lemma . that the adjoint lax-monoidal and colax-monoidal structures on R bey the bialgebra axiom as well. In general, the lax-monoidal structure on R induces a lax-monoidal structure on R mon : Mon D → Mon C (the same as the lax-monoidal structure on R for the underlying objects), but that is not true for the colax-monoidal structure . When the bothstructures are compatible by the bialgebra axiom, Lemma . says the colax-monoidal struc-ture, defined on the underlying objects as the one on R , defines a colax-monoidal structureon R mon .Thus, R is a composition of three functors, each of which comes with natural colax-monoidal structure. Therefore, R is colax-monoidal. We always assume this structure whenrefer to a colax-monoidal structure on R . ♦ . Q uillen pairs and weak monoidal Q uillen pairs . . Q uillen pairs and Q uillen equivalences To prove the statement (ii) of Proposition, we need firstly to recall some definitions on Quileenpairs of functors between two closed model categories, and to recall some results of Schwede-Schipley [SchS ] on weak monoidal Quillen pairs.In classical homological algebra one can derive left exact or right exact functor. Whenwe work with closed model categories, we try to extend the classical homological algebra tonon-abelian (and non-additive) context. A typical examples are the category of topologicalspaces and the category of dg associative algebras. How we can define the notions of a left(right) exact functor (i.e., of those functors we can derive) in such generality? The answer isgiven (by Quillen) in the concept of a Quillen pair of functors .To motivate the definition below, recall the following simple fact:L emma 5 . . Suppose A , B are two abelian categories, and letL : A ⇄ B : Rbe a pair of adjoint functors, with L left adjoint. Then L is right exact and R is left exact.
Prove as an exercise, or see the proof in [W], Theorem . . . ♦ Morally, we can not say in non-abelian categories what is the right (left) exactness, but weknow what adjoint functors are. These are (among other assumptions) the functors we canderive. Therefore, they come in pairs.D efinition 5 . . Let C , D be two closed model categories, and let L : C ⇄ D : R is a pair of adjoint functors, with L the left adjoint. The pair ( L , R ) is called a Quillen pair offunctors if ) L preserves cofibrations and trivial cofibrations,( ) R preserves fibrations and trivial fibrations.It is proven (see, e.g., [Hir], Prop. . . ) that, under these conditions, L takes weak equiv-alences between cofibrant objects to weak equivalences, and R takes weak equivalences be-tween fibrant object to weak equivalences.It is proven that a Quillen pair of functors defines an adjoint pair of functors between thehomotopy categories, L : Ho C ⇄ Ho D : R The next step is to find conditions on ( L , R ) under which the pair ( L , R ) is an adjointequivalence. This is the case when ( L , R ) is a Quillen equivalence .D efinition 5 . . A Quillen pair L : C ⇄ D : R is called a Quiilen equivalence if for any cofibrant object X in C and for every fibrant object Y in D , a map f : X → RY is a weak equivalence if and only if the adjoint map f ♯ : LX → Y is aweak equivalence.It is proven (see e.g. [Hir], Theorem . . ) that if ( L , R ) is a Quillen equivalence, thefunctors L : Ho C ⇄ Ho D : R form an adjoint equivalence of categories.All these results are due to D.Quillen [Q]. . . W eak monoidal Q uillen pairs Here we recall a result on weak monoidal Quillen pairs which is essential for our proof ofProposition . , (ii) below. Our intention here is not to give a throughout treatment (as itwould be just a copy of published papers), but rather to recall very briefly the definitions andresults, for convenient reference in the next Subsection.The categories we consider here are at once closed model and monoidal. There is somereasonable compatibility between these two structures on a category C , which guarantee, inparticular, that Ho C is a monoidal category. The concept is called a monoidal model category .We do not reproduce this definition here as we do not use it practically, all our categories inthis paper fulfill this definition. The interested reader is referred to [SchS ].The following definition is due to Schwede-Shipley [SchS ] (Definition . ). efinition 5 . . Let C , D be monoidal model categories , and let L : C ⇄ D : R be a Quillen pair of the underlying closed model categories. Suppose there is a lax-monoidalstructure ℓ on the functor R , denote by ϕ the adjoint colax-monoidal structure on L .The triple ( L , R , ℓ ) is called a weak monoidal Quillen pair if(i) for all cofibrant objects X , Y in C , the colax-monoidal map ϕ X , Y : L ( X ⊗ Y ) → L ( X ) ⊗ L ( Y ) is a weak equivalence,(ii) for some cofibrant replacement q : I c → I of the unit object in C , the composition L ( I c C ) L ( q ) −−→ L ( I C ) µ −→ I D is a weak equivalence (where µ is a part of colax-monoidal structure, see Definition A. ).A triple ( L , R , ℓ ) is called a weak monoidal Quillen equivalence , if is a weak monoidal Quillenpair, such that the underlying Quillen pair ( L , R ) is a Quillen equivalence.We use essentially the following result from [SchS ] (Theorem . ( )).T heorem 5 . . Let ( L , R , ℓ ) be a weak monoidal Quillen equivalence, and letL : C ⇄ D : Rbe the underlying Quillen pair. Suppose that the unit objects in C and D are cofibrant, and suppose thatthe forgetful functors Mon C → C and Mon D → D create model structures in Mon C and Mon D (see the explanation just below). ThenL mon : Mon C ⇄ Mon D : R mon ( . ) is a Quillen equivalence. R emark 5 . . . See [GS], Section , or [Hir], Chapter , for detailed explanation of themeaning of “the forgetful functors generate closed model structures”. This concept refers tothe transfer of closed model structures for cofibrantly generated model categories . See loc.cit. forall these concepts, as well as for a proof that our categories C = M od ( Z ) ∆ and D = C ( Z ) − satisfy this assumptions. . The Quillen equivalence ( . ) is not a weak monoidal Quillen equivalence. In fact, thenatural monoidal structure on Mon M for a monoidal model category M , is not a monoidalmodel category in general. For instance, the monoidal bifunctor does not commute with thecoproducts as a functor of one argument, for fixed another one. . P roof of P roposition 5 . , ( ii ) We need to prove that, for any dg algebra A ∈ A lg ≤ k , the dg algebra R ( A ) is cofibrant, andthe projection p A : R ( A ) → A ( . )is a quasi-isomorphism of dg algebras.Consider the Dold-Kan correspondence. In [SchS ], Section . , there is given a criteriumfor when a triple ( L , R , ℓ ) is a weak Quillen equivalence. It is proven as well, that this criteriumworks in the following two cases. The first case is the case of the Dold-Kan correspondence,with Γ = R the right adjoint, with the lax-monoidal structure being the adjoint to the colax-monoidal structure AW on N = L . The second case is the case when N = R is the rightadjoint, with the lax-monoidal shuffle structure ∇ . In our applications, we need only the firstpossibility. Let us summarize.L emma 5 . . Consider the Dold-Kan correspondenceN : M od ( Z ) ∆ ⇄ C ( Z ) − : Γ Use the notations L = N, R = Γ , and let ℓ be the lax-monoidal structure on R, adjoint to thecolax-monoidal structure AW on L. Then ( L , R , ℓ ) is a weak Quillen equivalence. ♦ Now we apply Theorem . .C orollary 5 . . In the above notations, the adjoint pair of functorsL mon : Mon ( M od ( Z ) ∆ ) ⇄ Mon ( C ( Z ) − ) : R mon ( . ) is a weak Quillen equivalence.Proof. It follows immediately from Lemma . and from Theorem . . ♦ Now we pass to a proof of the claims of Proposition . , (ii). Proof.
Prove that R ( A ) is cofibrant dg algebra for any A ∈ A lg ≤ k .Indeed, R ( A ) = L mon ◦ F ◦ R mon ( A ) We know that F ( − ) is cofibrant (Lemma . ), and that ( L mon , R mon ) is a Quillen equivalence,in particular, a Quillen pair (Corollary . ). By Definition of a Quillen pair, L mon mapscofibrations to cofibrations. As L mon maps the initial object to the initial object, it mapsthe cofibrant objections to cofibrant objects. Therefore, R ( A ) is cofibrant, as F ( R mon ( A )) iscofibrant. rove that, for any A ∈ A lg ≤ k , the projection p A : R ( A ) → A is a weak equivalence.The projection p A : L mon ◦ F ◦ R mon ( A ) → A ( . )is adjoint to p ♯ A : F ◦ R mon ( A ) → R mon ( A ) ( . )According to Corollary . , the pair ( L mon , R mon ) is a Quillen equivalence. We are going toapply the defining property of Quillen equivalences, see Definition . . Moreover, F ◦ R mon ( A ) is cofibrant by Lemma . (as F ( − ) is cofibrant by this Lemma), and R mon ( A ) is fibrant bythe description in Section . . Moreover, the map p ♯ A is a weak equivalence, again by Lemma . . Therefore, p A is a weak equivalence as well, by Definition . . ♦ . P roof of P roposition 5 . , ( iii ) The claim of Proposition . , (iii) follows from the two Lemmas below. The first one isstraightforward, while the second one heavily relies on results of Section .L emma 5 . . For any X , Y, the following diagram is commutative:L mon ( F ( R mon ( X ⊗ Y ))) / / (cid:15) (cid:15) L mon ( F ( R mon ( X ))) ⊗ L mon ( F ( R mon ( Y ))) (cid:15) (cid:15) L mon R mon ( X ⊗ Y ) / / L mon R mon ( X ) ⊗ L mon R mon ( Y ) ( . )L emma 5 . . For any X , Y, the following diagram is commutative:L mon R mon ( X ⊗ Y ) / / p X ⊗ Y ( ( PPPPPPPPPPPP L mon R mon ( X ) ⊗ L mon R mon ( Y ) p X ⊗ p Y u u ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ X ⊗ Y ( . ) Proof of Lemma ( . ) : We have the commutative diagram: L mon ( F ( R mon ( X ⊗ Y ))) c R / / (cid:15) (cid:15) L mon ( F ( R mon ( X ) ⊗ R mon ( Y ))) (cid:15) (cid:15) L mon ( R mon ( X ⊗ Y )) c R / / L mon ( R mon ( X ) ⊗ R mon ( Y )) ( . ) y tautological reasons. Now consider the diagram F ( R mon ( X ) ⊗ R mon ( Y )) / / * * ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ F ( R mon ( X )) ⊗ F ( R mon ( Y )) t t ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ R mon ( X ) ⊗ R mon ( Y ) ( . )from Lemma . . The composition of the diagram ( . ) with the application of L mon to thediagram ( . ) gives the diagram in Lemma . . ♦ Proof of Lemma ( . ) : Theorems . , . describe the functor L mon explicitly, with its colax-monoidal structure. According to these results, L mon ( X ) ≃ ˜ L mon ( X ) = L ∇ ( X ) / J ( X ) and the colax-monoidal structure, adjoint to the lax-monoidal structure on R mon , is inducedby the Alexander-Whitney colax-monoidal structure AW : L ( X ⊗ Y ) → L ( X ) ⊗ L ( Y ) by passing to the quotient-monoids, g AW : L ∇ ( X ⊗ Y ) / J ( X ⊗ Y ) → ( L ∇ ( X ) / J ( X )) ⊗ ( L ∇ ( Y ) / J ( Y )) The diagram ( . ) may be now rewritten as˜ L mon ( R ( X ⊗ Y )) / / p X ⊗ Y ' ' PPPPPPPPPPPP ˜ L mon ( R ( X )) ⊗ ˜ L mon ( R ( Y )) p X ⊗ p Y u u ❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦ X ⊗ Y ( . )where the horizontal map comes from the colax-monoidal structure on R (adjoint to the lax-monoidal structure ∇ on L ), followed by the colax-monoidal structure g AW on ˜ L mon . Theprojections comes from the adjunction maps.Now the commutativity of diagram ( . ) follows immediately from the statement ofLemma . (ii), by passing to quotient-monoids. We are done. ♦ Theorem . is proven. ♦ ppendix A D iagrams A. C olax - monoidal structure on a functor D efinition A. (Colax-monoidal functor). Let M and M be two strict associative monoidalcategories. A functor F : M → M is called colax-monoidal if there is a map of bifunctors β X , Y : F ( X ⊗ Y ) → F ( X ) ⊗ F ( Y ) and a morphism α : F ( M ) → M such that:( ): for any three objects X , Y , Z ∈ Ob ( M ) , the diagram F ( X ⊗ Y ) ⊗ F ( Z ) β X , Y ⊗ id F ( Z ) * * ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ F ( X ⊗ Y ⊗ Z ) β X ⊗ Y , Z ❧❧❧❧❧❧❧❧❧❧❧❧❧❧ β X , Y ⊗ Z ) ) ❘❘❘❘❘❘❘❘❘❘❘❘❘❘ F ( X ) ⊗ F ( Y ) ⊗ F ( Z ) F ( X ) ⊗ F ( Y ⊗ Z ) id F ( X ) ⊗ β Y , Z ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ (A. )is commutative. The functors β X , Y are called the colax-monoidal maps .( ): for any X ∈ Ob M the following two diagrams are commutative F ( M ⊗ X ) (cid:15) (cid:15) β X / / F ( M ) ⊗ F ( X ) α ⊗ id (cid:15) (cid:15) F ( X ) M ⊗ F ( X ) o o F ( X ⊗ M ) (cid:15) (cid:15) β X ,1 / / F ( X ) ⊗ F ( M ) id ⊗ α (cid:15) (cid:15) F ( X ) F ( X ) ⊗ M o o (A. ) A. L ax - monoidal structure on a functor D efinition A. (Lax-monoidal functor). Let M and M be two strict associative monoidalcategories. A functor G : M → M is called lax-monoidal if there is a map of bifunctors γ X , Y : G ( X ) ⊗ G ( Y ) → G ( X ⊗ Y ) and a morphism κ : 1 M → G ( M ) such that:( ): for any three objects X , Y , Z ∈ Ob ( M ) , the diagram G ( X ⊗ Y ) ⊗ G ( Z ) γ X ⊗ Y , Z u u ❦❦❦❦❦❦❦❦❦❦❦❦❦❦ G ( X ⊗ Y ⊗ Z ) G ( X ) ⊗ G ( Y ) ⊗ G ( Z ) γ X , Y ⊗ id G ( Z ) j j ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ id G ( X ) ⊗ γ Y , Z t t ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ G ( X ) ⊗ G ( Y ⊗ Z ) γ X , Y ⊗ Z i i ❙❙❙❙❙❙❙❙❙❙❙❙❙❙ (A. ) s commutative. The functors γ X , Y are called the lax-monoidal maps .( ): for any X ∈ Ob M the following two diagrams are commutative F ( M ⊗ X ) (cid:15) (cid:15) F ( M ) ⊗ F ( X ) γ X o o F ( X ) M ⊗ F ( X ) κ ⊗ id O O o o F ( X ⊗ M ) (cid:15) (cid:15) F ( X ) ⊗ F ( M ) γ X ,1 o o F ( X ) F ( X ) ⊗ M id ⊗ κ O O o o (A. ) A. B ialgebra axiom This axiom, expressing a compatibility between the lax-monoidal and colax-monoidal struc-tures on a functor between symmetric monoidal categories, seems to be new.D efinition A. (Bialgebra axiom). Suppose there are given both colax-monoidal and lax-monoidal structures on a functor F : C → D , where C and D are strict symmetric monoidalcategories. Denote these structures by c F ( X , Y ) : F ( X ⊗ Y ) → F ( X ) ⊗ F ( Y ) , and l F : F ( X ) ⊗ F ( Y ) → F ( X ⊗ Y ) . We say that the pair ( l F , c F ) satisfies the bialgebra axiom, if for any forobjects X , Y , Z , W ∈ Ob C , the following two morphisms F ( X ⊗ Y ) ⊗ F ( Z ⊗ W ) → F ( X ⊗ Z ) ⊗ F ( Y ⊗ W ) coincide: F ( X ⊗ Y ) ⊗ F ( Z ⊗ W ) l F −→ F ( X ⊗ Y ⊗ Z ⊗ W ) F ( id ⊗ σ ⊗ id ) −−−−−−→ F ( X ⊗ Z ⊗ Y ⊗ W ) c F −→ F ( X ⊗ Z ) ⊗ F ( Y ⊗ W ) (A. )and F ( X ⊗ Y ) ⊗ F ( Z ⊗ W ) c F ⊗ c F −−−→ F ( X ) ⊗ F ( Y ) ⊗ F ( Z ) ⊗ F ( W ) id ⊗ σ ⊗ id −−−−−→ F ( X ) ⊗ F ( Z ) ⊗ F ( Y ) ⊗ F ( W ) l F ⊗ l F −−−→ F ( X ⊗ Z ) ⊗ F ( Y ⊗ W ) (A. )where σ denotes the symmetry morphisms in C and in D .Thus, the commutative diagram, expressing the bialgebra axiom, is F ( X ⊗ Y ) ⊗ F ( Z ⊗ W ) (A. ) ( ( (A. ) F ( X ⊗ Z ) ⊗ F ( Y ⊗ W ) (A. ) ppendix B W eak adjoint functors
Here we develop some techniques used in Section of the paper. The main topic here isroughly the following. Recall from ( . ), ( . ) that a pair of adjoint functors L : C ⇄ D : R provides 1 ↔ L and the lax-monoidal structures on R . In Section we need to know whether and to which extend thisduality remains true, when we work with weak adjoint functors , see Definition B. below. Thenwe study the situation (also came up in Section ) when L and R are honest adjoint functorsbut they are represented as (co)limits of pairs of functors L i : C ⇄ D : R i , for i running troughsome category I , such that for each i the functors ( L i , R i ) are weak adjoint. The main resultsof this Appendix are Theorems B. and B. . B. Suppose L : C ⇄ D : R is a pair of adjoint functors, with L the left adjoint. Recall that this means that for any X in C and Y in D , there is an isomorphism of setsHom D ( LX , Y ) ≃ Hom C ( X , RY ) (B. )such that, for all X and Y , the two corresponding bifunctors C ◦ × D → S ets are isomorphic.This gives rise to a morphisms of functors ǫ LR → Id D and η : Id D → RL , such that thecompositions L η ∗ −→ LRL ǫ ∗ −→ L (B. )and R η ∗ −→ RLR ǫ ∗ −→ R (B. )are the identity morphisms of the functor L (correspondingly, of R ), see [ML IV. , Theorem ]. It is well-known that the inverse is also true:If there are two functors L : C → D and R : D → C , with morphisms of functors ǫ : LR → Id D and η : Id C → RL , such that the compositions (B. ) and (B. ) are identity maps of functors,the functors L and R are adjoint ,see [ML IV. , Theorem ].In this Appendix B, we refer to the classical adjoint functors described above as to genuine adjoint functors, and consider various definitions of weak adjoint functors. We define weakright adjoint functors and weak left adjoint functors in Definition B. just below, and define veryweak adjoint functors in Section B. .We give the following definition: efinition B. . Let C and D be two categories, and let L : C → D , R : D → C be two functors.Suppose there are given two morphisms of functors ǫ : LR → Id D and η : Id C → RL , such thatthe composition (B. ) is the identity morphism of the functor R , but the composition (B. )may fail to be the identity morphism of L . Then the data ( L , R , ǫ , η ) is called a weak rightadjunction between L and R . Analogously, when (B. ) is the identity morphism of L , but (B. )may fail to be the identity morphism of R , the data ( L , R , ǫ , η ) is called a weak left adjunction between L and R .The following Lemma gives an equivalent way to describe weak adjoint pairs. We will useit in the proof of Lemma B. below.L emma B. . Let L : C ⇄ D : Rbe a pair of functors, and suppose the two following morphisms of bifunctors C opp × D → S ets Θ : Hom D ( LX , Y ) → Hom C ( X , RY ) Υ : Hom C ( X , RY ) → Hom D ( LX , Y ) (B. ) are given. They produce the maps ǫ : LR → Id D and η : Id C → RL in a standard way, and vice versa.Then:(i) the two diagrams
Hom C ( X , Y ) L ∗ (cid:15) (cid:15) η ◦− / / Hom C ( X , RLY ) Hom D ( LX , LY ) Θ ❧❧❧❧❧❧❧❧❧❧❧❧❧❧ Hom D ( Z , W ) −◦ ǫ / / R ∗ (cid:15) (cid:15) Hom D ( LRZ , W ) Hom C ( RZ , RW ) Υ ❦❦❦❦❦❦❦❦❦❦❦❦❦❦ (B. ) commute,(ii) the pair ( L , R ) is weak right adjoint iff the composition Hom C ( X , RY ) Υ −→ Hom D ( LX , Y ) Θ −→ Hom C ( X , RY ) (B. ) is the identity map;(iii) the pair ( L , R ) is weak left adjoint iff the composition Hom D ( LX , Y ) Θ −→ Hom C ( X , RY ) Υ −→ Hom D ( LX , Y ) (B. ) is the identity map. roof of (i). Let us prove the commutativity of the left-hand-side diagram in (B. ). Let X , Y ∈ C , and let f ∈ Hom C ( X , Y ) .Consider the diagramHom C ( X , X ) L ∗ / / f ◦− (cid:15) (cid:15) Hom D ( LX , LX ) Θ / / L ( f ) ◦− (cid:15) (cid:15) Hom C ( X , RLX ) RL ( f ) ◦− (cid:15) (cid:15) Hom C ( X , Y ) L ∗ / / Hom D ( LX , LY ) Θ / / Hom C ( X , RLY ) (B. )We compute both paths on the identity morphism id X ∈ Hom C ( X , X ) , which gives the left-hand-side diagram in (B. ). The proof of commutativity of the right-hand-side diagram isanalogous. Proof of (ii).
Prove implication (B. ) ⇒ (B. ).Consider the map Θ : Hom D ( LX , Y ) → Hom C ( X , RY ) . Let f ∈ Hom D ( LX , Y ) . We havethe following diagram: Hom D ( LX , LX ) f ∗ (cid:15) (cid:15) Θ / / Hom C ( X , RLX ) f ∗ (cid:15) (cid:15) Hom C ( X , RY ) Υ / / Hom D ( LX , Y ) Θ / / Hom C ( X , RY ) (B. )The square commutes as Θ is a map of bifunctors, and the composition of the two maps inthe bottom line is identity, by the assumption.Now substitute X = RY , and let f : LRY → Y be the map ǫ . We haveHom D ( LRY , LRY ) ǫ ∗ (cid:15) (cid:15) Θ / / Hom C ( RY , RLR ( Y )) X (cid:15) (cid:15) Hom C ( RY , RY ) L ∗ ❦❦❦❦❦❦❦❦❦❦❦❦❦❦ Υ / / Hom D ( LRY , Y ) Θ / / Hom C ( RY , RY ) (B. )We know that the right-hand-side square sub-diagram commutes. We start with the identitymorphism id RY ∈ Hom C ( RY , RY ) . Then the left-hand side triangle commutes (we need toknow this claim only with the identity of RY in the left corner, in this case it is tautological).Thus, the overall diagram commutes. Then the map X is equal to the identity of RY , if westart with the identity of RY , as the composition Θ ◦ Υ = Id by assumption. ow define Θ and Υ , out of η and ǫ , byHom C ( X , RY ) Υ / / L ∗ (cid:15) (cid:15) Hom D ( LX , Y ) Hom D ( LX , LRY ) ǫ ◦− ❧❧❧❧❧❧❧❧❧❧❧❧❧❧ Hom D ( LX , Y ) Θ / / R ∗ (cid:15) (cid:15) Hom C ( X , RY ) Hom C ( RLX , RY ) −◦ η ❧❧❧❧❧❧❧❧❧❧❧❧❧❧ (B. )For the implication (B. ) ⇒ (B. ), consider the diagramHom C ( X , RY ) Υ / / L ∗ (cid:15) (cid:15) Hom D ( LX , Y ) Θ / / R ∗ (cid:15) (cid:15) Hom C ( X , RY ) Hom D ( LX , LRY ) R ∗ (cid:15) (cid:15) ǫ ◦− ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ Hom C ( RLX , RY ) −◦ η ❧❧❧❧❧❧❧❧❧❧❧❧❧❧ Hom C ( RLX , R LR Y ) ǫ ( (cid:3) ) ◦− ❥❥❥❥❥❥❥❥❥❥❥❥❥❥ (B. )The diagram is commutative. Thus, we need to compute the compositionHom C ( X , RY ) R ∗ L ∗ −−→ Hom C ( RLX , RLRY ) −◦ η −−→ Hom C ( X , R LR Y ) ǫ ◦− −−→ Hom C ( X , RY ) (B. )(we switched ǫ and η which does dot affect the answer).We have:S ub - lemma B. . Let L : C ⇄ D : Rbe a pair of functors, and let Ψ : Id C → RL is a map of functors. Let Z , W ∈ C . Then the composition Hom C ( Z , W ) ( RL ) ∗ −−−→ Hom C ( RLZ , RLW ) −◦ Ψ ( Z ) −−−−→ Hom C ( Z , RLW ) (B. ) is equal to the composition Hom C ( Z , W ) Ψ ( W ) ◦− −−−−→ Hom C ( Z , RLW ) (B. )For, consider the diagram, for a morphism f : Z → W :Hom C ( Z , Z ) ( RL ) ∗ / / f ∗ ◦− (cid:15) (cid:15) Hom C ( RLZ , RLZ ) −◦ Ψ ( Z ) / / f ∗ ◦− (cid:15) (cid:15) Hom C ( Z , RLZ ) f ∗ ◦− (cid:15) (cid:15) Hom C ( Z , W ) ( RL ) ∗ / / Hom C ( RLZ , RLW ) −◦ Ψ ( Z ) / / Hom C ( Z , RLW ) (B. ) with the identity in the upper-left corner) and Z Ψ ( Z ) / / f (cid:15) (cid:15) RLZ f ∗ (cid:15) (cid:15) W Ψ ( W ) / / RLW (B. ) ♦ By the above Sub-lemma, we rewrite composition (B. ) asHom C ( X , RY ) η ( RY ) ◦− −−−−−→ Hom C ( X , R LR Y ) ǫ ( (cid:3) ) −−→ Hom C ( X , RY ) (B. )Now (B. ) is the identity map by assumption (B. ). Proof of (iii) is analogous to the proof of (ii), and is left to the reader. ♦ We turn to the discussion of interplay between the lax- and the colax-monoidal structures,in the weak adjoint setting. We haveL emma B. . There are the following statements:(i) Let L : C ⇄ D : R be a weak right adjunction, for some ǫ and η . Let ℓ be a lax-monoidal structureon R. Then the compositionL ( X ⊗ Y ) ( η ⊗ η ) ∗ −−−−→ L ( RLX ⊗ RLY ) ℓ ∗ −→ LR ( LX ⊗ LY ) ǫ ∗ −→ LX ⊗ LY (B. ) is a colax-monoidal structure on L;(ii) Analogously, if ( L , R , ǫ , η ) is a left weak adjunction, and c is a colax-monoidal structure on L,the compositionRX ⊗ RY η ∗ −→ RL ( RX ⊗ RY ) c ∗ −→ R ( LR ( X ) ⊗ LR ( Y )) ( ǫ ⊗ ǫ ) ∗ −−−→ R ( X ⊗ Y ) (B. ) is a lax-monoidal structure on L.Proof. We prove (i), the proof of (ii) is analogous.We need to prove the commutativity of the diagram L ( X ⊗ Y ⊗ Z ) c X ⊗ Y , Z / / c X , Y ⊗ Z (cid:15) (cid:15) L ( X ⊗ Y ) ⊗ L ( Z ) c X , Y ⊗ id Z (cid:15) (cid:15) L ( X ) ⊗ L ( Y ⊗ Z ) id X ⊗ c Y , Z / / L ( X ) ⊗ L ( Y ) ⊗ L ( Z ) (B. )where c is the colax-monoidal structure on L , defined out of the lax-monoidal structure ℓ on R by (B. ). ub - lemma B. . There are the following statements:A. The compositionL ( X ⊗ Y ⊗ Z ) c X ⊗ Y , Z −−−→ L ( X ⊗ Y ) ⊗ L ( Z ) c X , Y ⊗ id Z −−−−−→ L ( X ) ⊗ L ( Y ) ⊗ L ( Z ) (B. ) is equal to the compositionL ( X ⊗ Y ⊗ Z ) η ∗⊗ −−→ L ( RLX ⊗ RLY ⊗ RLZ ) ℓ LX , LY ⊗ id RLZ −−−−−−−→ L ( R ( LX ⊗ L ( Y )) ⊗ RLZ ) ℓ LX ⊗ LY , LZ −−−−−→ LR ( LX ⊗ LY ⊗ LZ ) ǫ −→ LX ⊗ LY ⊗ LZ (B. ) B. the compositionL ( X ⊗ Y ⊗ Z ) c X , Y ⊗ Z −−−→ L ( X ) ⊗ L ( Y ⊗ Z ) id LX ⊗ c Y , Z −−−−−→ L ( X ) ⊗ L ( Y ) ⊗ L ( Z ) (B. ) is equal to the compositionL ( X ⊗ Y ⊗ Z ) η ⊗ ∗ −−→ L ( RLX ⊗ RLY ⊗ RLZ ) id RLX ⊗ ℓ LY , LZ −−−−−−−→ L ( RLX ⊗ R ( LY ⊗ LZ )) ℓ LX , LY ⊗ LZ −−−−−→ LR ( LX ⊗ LY ⊗ LZ ) ǫ −→ LX ⊗ LY ⊗ LZ (B. )The claim (i) of Lemma follows easily from the Sub-lemma above. Indeed, (B. )=(B. ),as ℓ is a lax-monoidal functor, then Sub-lemma implies (B. )=(B. ). The latter is exactly thecommutativity of the diagram (B. ).It remains to prove the Sub-lemma. Proof of Sub-lemma:
We prove the claim (A.), the proof of claim (B.) is analogous. In the totaldiagram below the upper-right composition is equal to (B. ), and the lower-left compositionis equal to (B. ). L ( X ⊗ Y ⊗ Z ) / / (cid:15) (cid:15) L ( RL ( X ⊗ Y ) ⊗ RLZ ) (cid:15) (cid:15) L ( RLX ⊗ RLY ⊗ RLZ ) / / ℓ ∗ (cid:15) (cid:15) L ( RL ( RLX ⊗ RLY ) ⊗ RLZ ) ℓ ∗ (cid:15) (cid:15) L ( R ( LX ⊗ LY ) ⊗ RLZ ) η ∗ / / id (cid:15) (cid:15) L ( RLR ( LX ⊗ LY ) ⊗ RLZ ) ǫ (cid:15) (cid:15) L ( R ( LX ⊗ LY ) ⊗ RLZ ) id / / ǫ ◦ ℓ ∗ (cid:15) (cid:15) L ( R ( LX ⊗ LY ) ⊗ RLZ ) ǫ ◦ ℓ ∗ (cid:15) (cid:15) LX ⊗ LY ⊗ LZ id / / LX ⊗ LY ⊗ LZ (B. ) herefore, it is sufficient to prove that the overall diagram (B. ) commutes. The first, thesecond, and the fourth from above sub-diagrams commute by general categorical nonsense,while the third from the above diagram commutes by (B. ). We are done. ♦♦ B. I ntrinsic characterization of the lax - colax duality D efinition B. . Let C and D be monoidal categories, and let L : C ⇄ D : R be a genuine adjunction. ) We say that a colax-monoidal structure c on L and a lax-monoidal structure ℓ on R are compatible if the following diagram commutes: L ( RX ⊗ RY ) ℓ ∗ / / c ∗ (cid:15) (cid:15) LR ( X ⊗ Y ) ǫ ∗ (cid:15) (cid:15) LR ( X ) ⊗ LR ( Y ) ǫ ∗ ⊗ ǫ ∗ / / X ⊗ Y (B. ) ) Suppose a lax-monoidal structure ℓ on R is given. We call the colax-monoidal structure c on L , defined by (B. ), the canonical colax-monoidal structure, induced by ℓ . As well,suppose a colax-monoidal structure c on L is given. We call a lax-monoidal structure ℓ on R , defined by (B. ), the canonical lax-monoidal structure, induced by c .L emma B. . Let C and D be monoidal categories, and letL : C ⇄ D : Rbe a genuine adjunction.(i) given a lax-monoidal structure ℓ on R, it is compatible with the canonical colax-monoidal struc-ture on L, induced by ℓ . Analogously, given a colax-monoidal structure c on L, it is compatiblewith the canonical lax-monoidal structure on R, induced by c;(ii) given a compatible pair ( c , ℓ ) of (co)lax-monoidal structures, each of them is uniquely defined byanother. By (i), it implies that the only compatible pairs of (co)lax-monoidal structures are thecanonically induced ones. t is simple and fairly standard, but the result follows from our more subtle Lemmas B. ,B. as well. ♦ Our goal in this Subsection is to study what happens with Lemma B. in weak adjointcontext. These results are used below in Appendix B, in proofs of Theorem B. and ofTheorem B. .We have:L emma B. . Let L : C ⇄ D : Rbe a weak right adjunction.( ) Suppose a lax-monoidal structure ℓ on R is given. Then the canonical colax-monoidal structure con L, induced by ℓ as in (B. ) , is compatible with ℓ (in the sense that diagram (B. ) commutes);( ) suppose a colax-monoidal structure c on L and a lax-monoidal structure ℓ on R are given, suchthat the diagram (B. ) commutes. Then ℓ can be uniquely reconstructed by c. R emark B. . We would emphasize a rather unexpected claim in (ii): not a colax-monoidalstructure c on L making (B. ) commutative, one particular possibility for it is given in (i), isuniquely defined by ℓ , but, vice versa, ℓ is uniquely reconstructed by c .Before proving Lemma B. , let us formula a pattern of it for left weak adjunctions.Consider the diagram X ⊗ Y η ⊗ η / / η (cid:15) (cid:15) RL ( X ) ⊗ RL ( Y ) ℓ ∗ (cid:15) (cid:15) RL ( X ⊗ Y ) c ∗ / / R ( L ( X ) ⊗ L ( Y )) (B. )Here the maps are defined from a lax-monoidal structure ℓ on R , a colax-monoidal structure c on L , and from the adjunction η : Id C → RL .R emark B. . For a pair of genuine adjoint functors, the diagram (B. ) commutes iff (B. )does. For weak adjunctions, it is not anymore the case, and we have two different diagrams.The “dual” to Lemma B. isL emma B. . Let L : C ⇄ D : Rbe a weak left adjunction. ) Suppose a colax-monoidal structure c on L is given. Then the canonical lax-monoidal structure ℓ on R, induced by c as in (B. ) , is compatible with ℓ in the sense that diagram (B. ) commutes;( ) suppose a colax-monoidal structure c on L and a lax-monoidal structure ℓ on R are given, suchthat the diagram (B. ) commutes. Then c can be uniquely reconstructed by ℓ .Proof of Lemma B. : (i):Suppose ℓ is given, define c by (B. ). Then the lower-left path of diagram (B. ) is: L ( RX ⊗ RY ) η ∗ ⊗ η ∗ −−−→ L ( RLRX ⊗ RLRY ) ℓ ∗ −→ LR ( LRX ⊗ LRY ) ǫ ( ǫ ⊗ ) −−−→ X ⊗ Y (B. )The composition L ( RLRX ⊗ RLRY ) ℓ ∗ −→ LR ( LRX ⊗ LRY ) → X ⊗ Y is the same, by the functoriality, that the composition L ( R LR X ⊗ R LR Y ) ǫ ∗ ⊗ ǫ ∗ −−−→ L ( RX ⊗ RY ) ℓ ∗ −→ LR ( X ⊗ Y ) ǫ −→ X ⊗ Y (in the first arrow, η ’s act on the boxed factors).The latter composition is equal to the upper-right path of diagram (B. ), because thecomposition R η −→ RLR ǫ −→ R is equal to the identity map of R , by assumption.(ii):Build up a top level to the R ( (B. ) ) , as follows: RX ⊗ RY ℓ / / η (cid:15) (cid:15) R ( X ⊗ Y ) η (cid:15) (cid:15) RL ( RX ⊗ RY ) ℓ ∗ / / c ∗ (cid:15) (cid:15) RLR ( X ⊗ Y ) ǫ ∗ (cid:15) (cid:15) R ( LR ( X ) ⊗ LR ( Y )) ǫ ∗ ⊗ ǫ ∗ / / R ( X ⊗ Y ) (B. )It follows from our assumption of right weak adjunction that the upper-right path of theoverall diagram (B. ) gives the lax-monoidal structure ℓ : RX ⊗ RY → R ( X ⊗ Y ) . On theother hand, the lower-left path expresses it in c , ǫ , and η . ♦ The proof of Lemma B. is analogous and is left to the reader. . (C o ) limits of weak adjoint pairs We start with the definition of a morphism between weak adjoint pairs.D efinition B. . Let C and D be two categories, and suppose L : C ⇄ D : R (B. )and L ′ : C ⇄ D : R ′ (B. )are two weak right adjunctions (resp., weak left adjunctions), see Definition B. . A morphism Ψ from (B. ) to (B. ) is given by the following data:(i) two maps of functors Ψ L : L ′ → L (in backward direction) and Ψ R : R → R ′ , whichinduce maps of bifunctors Ψ L ∗ : Hom D ( LX , Y ) → Hom D ( L ′ X , Y ) and Ψ R ∗ : Hom C ( X , RY ) → Hom C ( X , R ′ Y ) (ii) the diagram Hom D ( LX , Y ) Θ / / Ψ L ∗ (cid:15) (cid:15) Hom C ( X , RY ) Ψ R ∗ (cid:15) (cid:15) Hom D ( L ′ X , Y ) Θ ′ / / Hom C ( X , R ′ Y ) (B. )and the diagram Hom C ( X , RY ) Υ / / Ψ R ∗ (cid:15) (cid:15) Hom D ( LX , Y ) Ψ L ∗ (cid:15) (cid:15) Hom C ( X , R ′ Y ) Υ ′ / / Hom D ( L ′ X , Y ) (B. )are commutative, where ( Θ , Υ ) and ( Θ ′ , Υ ′ ) are corresponded to the weak right adjunc-tions (B. ) (correspondingly, (B. )) by (B. ).We now turn to the discussion of (co)limits of weak adjoint functors and of (co)lax-monoidal structures.Let I and C be categories; we refer to a functors from I to C as diagrams . Recall that a limit lim D of a diagram D : I → C is defined, if it exists, by one of the two equivalent ways:Hom C ( X , lim D ) = lim I Hom ( X , D ( i )) = Hom I → C ( ∆ X , D ) (B. ) here ∆ ( X ) is the functor ∆ ( X ) : I → C defined on objects as ∆ ( X )( i ) = X , and defined asthe identity morphism of X on all morphisms in I .Analogously, a colimit colim D of the diagram D : I → C is defined, if it exists, byHom C ( colim D , Y ) = lim I Hom ( D ( i ) , Y ) = Hom I → C ( D , ∆ Y ) (B. )For any two categories P and Q , there is a canonical equivalence F un ( P , Q ) ≃ F un ( P opp , Q opp ) opp (B. )Thus, a functor D : I → C , can be regarded as well as a functor D opp : I opp → C opp . Thenlim I D = ( colim I opp D opp ) opp (B. )Let α : J → I be a functor. Then with any functor D : I → C one assigns the composition D ◦ α : J → C .Recall for the reference belowL emma B. . In the above notations, the functor α : J → I defines canonical morphisms α ∗ : lim D → lim ( D ◦ α ) (B. ) and α ∗ : colim ( D ◦ α ) → colim D (B. ) functorial in diagram D : I → C .Proof. For the case of limits, there are two adjunctionsHom C ( X , lim D ) ≃ Hom I → C ( ∆ I X , D ) (B. )and Hom C ( X , lim ( D ◦ α )) ≃ Hom J → C ( ∆ J X , D ◦ α ) (B. )The map α : J → I defines a map of the right-hand-sidesHom I → C ( ∆ I X , D ) → Hom J → C ( ∆ J X , D ◦ α ) (B. )which defines a map of the left-hand sides (as a map of bifunctors):Hom C ( X , lim D ) → Hom C ( X , lim ( D ◦ α )) (B. )Then (B. ) defines a functor (unique up to a unique isomorphism)lim D → lim ( D ◦ α ) by the Yoneda lemma.The case of colimits is analogous (and dual). ♦ ee [ML, Ch.V] and [KaS, Ch. ] for more detail on (co)limits.Recall a well-known fact on the (co)limits in the categories of functors.P roposition B. . Let C and D be two categories. Consider the category F un ( C , D ) . Let I be anindexing category, and let D : I → F un ( C , D ) be a diagram.(i) When D is complete, the limit lim D exists, and can be computed point-wise: ( lim D )( X ) = lim ( D ( X )) (B. ) Analogously, when D is cocomplete, the colimit colim D exists, and can be computed point-wise ( colim D )( X ) = colim ( D ( X )) (B. ) (ii) Suppose D is cocomplete and C is complete, and let D L : I opp → F un ( C , D ) and D R : I → F un ( D , C ) be two diagrams. Suppose that for each i ∈ I, the pair of functors D L ( i ) : C ⇄ D : D R ( i ) is genuine adjoint. Then the functors colim D L : C ⇄ D : lim D R exist, and are genuine adjoint. Recall a proof, for completeness.Denote by lim and colim the functors, defined by (B. ) and (B. ), correspondingly. Onedirectly shows that Hom F un ( C , D ) ( F , lim D ) = lim i ∈ I Hom F un ( C , D ) ( F , D ( i )) (B. )and Hom F un ( C , D ) ( colim D , G ) = lim i ∈ I Hom F un ( C , D ) ( D ( i ) , G ) (B. )Now the claim (i) follows from (B. ), (B. ), and the Yoneda lemma.The claim (ii) is a direct consequence from the claim (i). ♦ R emark B. . The (co)limits in the categories of functors may exist as well when the targetcategory is not (co)complete. See [Kel], Section . .Henceforth, we suppose that all necessary (co)limits in the target categories exist, and the(co)limits in the categories of functors can be computed point-wise. efinition B. . Let I be an indexing category, and let D L : I → F un ( C , D ) opp and D R : I → F un ( D , C ) be two diagrams. Suppose that for each fixed i ∈ I the maps D L ( i ) : C ⇄ D : D R ( i ) (B. )is a weak right adjunction, for some ǫ ( i ) : D L ( i ) ◦ D R ( i ) → Id D , η ( i ) : Id C → D R ( i ) ◦ D L ( i ) (B. )and for each morphism f : i → j in I , the maps of functors ( D L ( j ) f ∗ −→ D L ( i ) , D R ( i ) f ∗ −→ D R ( j )) (B. )form a morphism between weak right adjunctions , see Definition B. . Then the diagrams D L and D R are called weak right adjoint diagrams .We have:L emma B. . Suppose C , D are two categories, with C complete and D cocomplete. Let D L : C ⇄ D : D R be weak right adjoint diagrams, see Definition B. . Then the functors colim D L : C ⇄ D : lim D R (B. ) has a natural structure of a weak right adjunction.Proof. We use the description of a pair of (right) weak adjoint functors in terms of maps Θ and Υ (see (B. )) satisfying (B. ).We can rephrase the statement of Lemma, saying that there are diagrams of morphismsof bifunctors Θ D ( i ) : Hom D ( D L ( i )( X ) , Y ) → Hom C ( X , D R ( i )( Y )) (B. )and Υ D ( i ) : Hom C ( X , D R ( i )( Y )) → Hom D ( D L ( i )( X ) , Y ) (B. )such that Θ D ( i ) ◦ Υ D ( i ) = Id for any i ∈ I (B. )Now we can definelim Θ D : = lim I Θ D ( i ) : lim Hom D ( D L ( i )( X ) , Y ) → lim Hom C ( X , D R ( i )( Y )) (B. ) nd lim Υ D : = lim I Υ D ( i ) : lim Hom C ( X , D R ( i )( Y )) → lim Hom D ( D L ( i )( X ) , Y ) (B. )By Proposition B. , the above limit morphisms are in factlim Θ D : Hom D ( colim D L ( X ) , Y ) → Hom C ( X , lim D R ( Y )) (B. )and lim Υ D : Hom C ( X , lim D R ( Y )) → Hom D ( colim D L ( X ) , Y ) (B. )Clear the morphisms lim Θ D and lim Υ D are maps of bifunctors. Therefore, they define, byLemma B. , some canonical maps of functorslim ǫ : colim D L ◦ lim D R → Id D and lim η : lim D R ◦ colim D L → Id C It remains to prove that ( lim ǫ , lim η ) define a weak right adjunction on the functorscolim D L : C ⇄ D : lim D R By Lemma B. , it is enough to prove thatlim Θ D ◦ lim Υ D = Id (B. )As for each i ∈ I we have a weak right adjunction, one has Θ D ( i ) ◦ Υ D ( i ) = Id (B. )Then (B. ) follows from (B. ) by passing to limits. ♦ Let D L : C ⇄ D : D R be weak right adjoint diagrams, indexed by a category I , see Defi-nition B. . Suppose that each functor D R ( i ) is endowed with a lax-monoidal structure ℓ ( i ) ,such that the diagram D R ( i )( X ) ⊗ D R ( i )( Y ) ℓ ( i ) / / f ∗ (cid:15) (cid:15) D R ( i )( X ⊗ Y ) f ∗ (cid:15) (cid:15) D R ( j )( X ) ⊗ D R ( j )( Y ) ℓ ( j ) / / D R ( j )( X ⊗ Y ) (B. )is commutative for any morphism f : i → j in I , and for any X , Y in D . In this case we saythat we are given a diagram of lax-monoidal structures . nalogously, using D L ( i ) , i ∈ I , and a colax-monoidal structure c ( i ) on each D L ( i ) , wedefine a diagram of colax-monoidal structures .Suppose we have weak right adjoint diagrams D L : C ⇄ D : D R , and a diagram of lax-monoidal structures on D R . Then, for each i ∈ I , we define a colax-monoidal structure c ℓ ( i ) on D L ( i ) , adjoint to the lax-monoidal structure ℓ ( i ) on D R ( i ) .L emma B. . In the above set-up, the colax-monoidal structures c ℓ ( i ) , i ∈ I, on D L ( i ) , form a diagramof colax-monoidal structures. It is clear. ♦ Suppose there is a diagram D : I → F un ( D , C ) , and a diagram of lax-monoidal structures ℓ ( i ) on D ( i ) .We define the limit lax-monoidal structure lim i ∈ I ℓ ( i ) on the functor lim i ∈ I D ( i ) , as follows.For any i ∈ I , and any X , Y ∈ C , there are universal maps ( lim D )( X ) → D ( i )( X ) and ( lim D )( Y ) → D ( j )( Y ) which define ( lim D )( X ) ⊗ ( lim D )( Y ) → D ( i )( X ) ⊗ D ( j )( Y ) (B. )which is compatible with the morphisms in I × I . Then the universal property gives Ξ ( X , Y ) : ( lim D )( X ) ⊗ ( lim D ( Y )) → lim I × I ( D ( i )( X ) ⊗ D ( j )( Y )) (B. )Next, there is a map Ξ ( X , Y ) : lim I × I ( D ( i )( X ) ⊗ D ( j )( Y )) → lim I ( D ( i )( X ) ⊗ D ( i )( Y )) (B. )The latter map is α ∗ (see (B. )) for the diaginal functor α : I → I × I .Finally, we apply the lax-monoidal structure ℓ ( i ) for each D ( i ) , and take the limit: Ξ ( X , Y ) : lim I ( D ( i )( X ) ⊗ D ( i )( Y )) ℓ ( i ) −−→ lim I D ( i )( X ⊗ Y ) = lim D ( X ⊗ Y ) (B. )Define the limit lax-monoidal structure lim ℓ on lim D as the compositionlim ℓ = Ξ ◦ Ξ ◦ Ξ (B. )Analogously one defines the colimit of colax-monoidal structures.One of our two most principal claims in this Appendix (along with Theorem B. ) is thefollowing: heorem B. . Suppose C , D are categories, with C complete and D cocomplete. Let I be an indexingcategory. Let D L : I → F un ( C , D ) opp , D R : I → F un ( D , C ) are diagrams, such that D L : C ⇄ D : D R is a weak right adjunction of diagrams, see Definition B. , with D R ( i ) the constant functor (notdepending on i ∈ I ) , which we denote by D R . Suppose there is given a lax-monoidal structure ℓ on D R . Consider the natural weak right adjunction colim D L : C ⇄ D : lim D R = D R (B. ) constructed in Lemma B. . Suppose, furthermore, that (B. ) is a genuine adjunction. Then the colimit colax-monoidal structure colim i ∈ I c ℓ ( i ) on the functor colim D L , where eachc ℓ ( i ) is the canonical colax-monoidal structure induced by ℓ , is isomorphic to the canonical colax-monoidal structure on colim D L , induced by the limit lax-monoidal structure lim i ∈ I ℓ = ℓ on thefunctor lim D R = D R .Proof. The proof is based on the following lemma.L emma B. . Let C , D be monoidal categories, with C complete and D cocomplete, and let D L : I → F un ( C , D ) opp and D R : I → F un ( D , C ) be diagrams, with D R ( i ) = D R the constant functor. Sup-pose that for each i the functors D L ( i ) : C ⇄ D : D R ( i ) = D R are weak right adjoint, such that one has a diagram of weak right adjunctions (see Definition B. ).Suppose, furthermore, that there are given a lax-monoidal structure ℓ on D R , and a diagram of colax-monoidal structures c ( i ) on D L ( i ) , such that for each i ∈ I the diagram (B. ) commutes for ( c ( i ) , ℓ ) .Then the pair ( colim c ( i ) , ℓ ) of the colimit colax-monoidal structure colim c ( i ) on the functor colim D L ( i ) and of the lax-monoidal structure lim ℓ = ℓ on the functor lim D R = D R , the diagram (B. ) commutes as well.Proof. First of all, we use (B. ) and replace in (B. ) all entries of colim I D L by lim I opp D opp .According to that, we replace the notation for the colimit colax-monoidal structure colim I c ℓ ( i ) by lim I opp c ℓ ( i ) . First of all, we use (B. ) and replace in (B. ) all entries of colim I D L bylim I opp D opp . According to that, we replace the notation for the colimit colax-monoidal struc-ture colim I c ℓ ( i ) by lim I opp c ℓ ( i ) . ithin our proof, we divide the diagram (B. ) into commutative diagrams, (B. ) and(B. ) below, which implies that the overall diagram (B. ) commutes as well. ( lim I opp D L )( D R ( X ) ⊗ D R ( Y )) ℓ ∗ (cid:15) (cid:15) ≃ , by (B. ) / / lim I opp (cid:16) D L ( i )( D R ( X ) ⊗ D R ( Y )) (cid:17) ( ℓ ) ∗ (cid:15) (cid:15) ( lim I opp D L )( D R ( X )) ⊗ ( lim I opp D L )( D R ( Y )) / / lim I opp × I opp (cid:16) D L ( i ) D R ( X ) ⊗ D L ( j ) D R ( Y ) (cid:17) ∆ / / lim I opp (cid:16) D L ( i ) D R ( X ) ⊗ D L ( i ) D R ( Y ) (cid:17) (B. ) lim I opp (cid:16) D L ( i )( D R ( X ) ⊗ D R ( Y )) (cid:17) ℓ ∗ / / ( c ℓ ( i )) ∗ (cid:15) (cid:15) lim I opp (cid:16) D L ( i )( D R ( X ⊗ Y ))) (cid:17) / / X ⊗ Y id (cid:15) (cid:15) lim I opp (cid:16) D L ( i ) D R ( X ) ⊗ D L ( i ) D R ( Y ) (cid:17) / / X ⊗ Y (B. )The commutativity of (B. ) is an exercise on the definition (B. ) of the limit lax structure (inits colax version), and is left to the reader.The commutativity of (B. ) follows from the assumption that for each i the diagram(B. ) commutes, as (B. ) is the the limit of the commutative diagrams (B. ), and thereforeis commutative by its own.Lemma is proven. ♦ We continue to prove Theorem B. .By assumption, the pair of functorscolim I D L : C ⇄ D : D R (B. )is genuine adjoint.Furthermore, we have two colax-monoidal structures on colim D L , namely the canonicalone c , adjoint to the lax-monoidal structure lim ℓ = ℓ on lim D R = D R , and the second one c is colim I c ℓ ( i ) , the colimit of the canonical lax structures for each i ∈ I . The both pairs ( c , ℓ ) and ( c , ℓ ) make the diagram (B. ) commutative, for adjoint pair (B. ). Indeed, the pair ( c , ℓ ) makes (B. ) commutative by Lemma B. , while the commutativity for ( c , ℓ ) followsfrom Lemma B. .Apply Lemma B. once again (for genuine adjoint pair ( colim D L , D R ) ), we conclude thatthe colax-monoidal structures c and c on the functor colim D L are isomorphic. We are done. ♦ . V ery weak adjoint pairs For our main application to Theorem . , the assumptions of Theorem B. are too strong.Indeed, in the set-up of Theorem . one has not a weak right adjoint pair structure onthe pair ( D L ( i ) , D R ) (where i ∈ I ). In fact, the only what we have are maps of functors ǫ ( i ) : D L ( i ) ◦ D R → Id D , but the maps η ( i ) : Id C → D R ◦ D ( i ) may not exist. See Remarkrefsosna above.It turns out, fortunately, that we did not use the existence of the map η ( i ) anywhere inthe proof Theorem B. . The goal of this (last) Subsection is to formulate and to prove thecorresponding Theorem (see Theorem B. below) in its proper generality.D efinition B. . Let L : C → D and R : D → C be functors. We call ( L , R ) a very weak rightadjunction if there is a morphism of functors ǫ : LR → Id D The latter is equivalent to the existence of a map of bifunctors Υ : Hom D ( X , RY ) → Hom C ( LX , Y ) .D efinition B. . Let L , R be a very weak right adjunction, and let a lax-monoidal structure ϕ on R and a colax-monoidal structure c on L are given. We say that ( c , ϕ ) are weak rightcompatible , if the diagram (B. ) commutes for any X , Y in D .It is crucial to notice that the diagram (B. ) is written only within ǫ : LR → Id C , not η : Id C → RL .A difference between the weak right adjunctions and very weak right adjunctions is thatin the very weak case one can not define the canonical colax-monoidal structure on L out of alax-monoidal structure on R , as (B. ) is not a colax-monoidal structure anymore.As a substitution, we have the “intrinsic” characterization of a pair of a colax-monoidalstructure on L and a lax-monoidal structure on R by (B. ), which is the only one remains touse.We now pass to (co)limits, our goal is to formulate and to prove a result, analogous toTheorem B. , in our very weak generality. Firstly we suitably adjust Definition B. .D efinition B. . Let I be an indexing category, and let D L : I → F un ( C , D ) opp and D R : I → F un ( D , C ) be two diagrams. Suppose that for each fixed i ∈ I the maps D L ( i ) : C ⇄ D : D R ( i ) (B. )form a very weak right adjunction , for some ǫ ( i ) : D L ( i ) ◦ D R ( i ) → Id D (B. ) r, equivalently, for some Υ ( i ) : Hom C ( X , D R ( i ) Y ) → Hom D ( D L ( i ) X , Y ) (B. )such that for each morphism f : i → j in I , the diagramHom C ( X , D R ( i ) Y ) Υ ( i ) / / f ∗ (cid:15) (cid:15) Hom D ( D L ( i ) X , Y ) f ∗ (cid:15) (cid:15) Hom C ( X , D R ( j ) Y ) Υ ( j ) / / Hom D ( D L ( j ) X , Y ) (B. )commutes, for any X in C and Y in D . . Then the diagrams D L and D R are called very weakright adjoint diagrams .Suppose very weak right adjoint diagrams D L ( i ) : C ⇄ D : D R ( i ) indexed by a category I , are given (see Definition B. ).Consider the (co)limit pair of functorscolim D L : C ⇄ D : lim R We haveL emma B. . Let D L : C ⇄ D : D R be very weak right adjoint diagrams, see Definition B. . Thenthe (co)limit functors colim D L : C ⇄ D : lim R form a very weak right adjoint pair of functors. The proof simply repeats the proof in weak adjoint case (see B. and its proof).We define a diagram of lax-monoidal structures and a diagram of colax-monoidal structures , andtheir (co)limits, precisely as in Section B. above.T heorem B. . Suppose C , D are categories, with C complete and D cocomplete. Let I be an indexingcategory. Let D L : I → F un ( C , D ) opp , D R : I → F un ( D , C ) are diagrams, such that D L : C ⇄ D : D R is a very weak right adjunction of diagrams, see Definition B. , with D R ( i ) the constant functor (not depending on i ∈ I), which we simply denote by D R . Suppose there are given a lax-monoidal tructure ℓ on D R , and colax-monoidal structures c ( i ) on D L ( i ) , such that ( ) for any i ∈ I, thediagram (B. ) for ( c ( i ) , ℓ ) on the functors ( D L ( i ) , D R commutes, and ( ) the colax-monoidal struc-tures c ( i ) form a diagram of colax-monoidal structures. Suppose that the natural weak rightadjunction colim D L : C ⇄ D : lim D R = D R (B. ) is a genuine adjunction. Then the colimit colax-monoidal structure colim i ∈ I c ( i ) on the functor colim D L , is isomorphicto the canonical colax-monoidal structure on colim D L , induced by the limit lax-monoidal structure lim i ∈ I ℓ = ℓ on the functor lim D R = D R .Proof. The proof repeats, with minor variations, the proof of Theorem B. . The main point isthat Lemma B. still holds in the very weak context, with literally identical proof. Then weuse Lemma B. , but not for the canonical colax-monoidal structures c ℓ ( i ) (which no longermake sense in the very weak context), but for the given in the data colax-monoidal structures c ( i ) on D L ( i ) , and this this the only difference. ♦ B ibliography [AM] M.Aguiar, S.Mahajan, Monoidal functors, species and Hopf algebras , CRM Monograph Series, , AmericanMathematical Society, Providence, RI, [BFSV] C.Baltenu, Z.Fiedorowicz, R.Schwänzl, R.Vogt, Iterated Monoidal Categories, Advances in Math. ( ), - [BK] A.K.Bousfield, D.M.Kan, Homotopy Limits, Completions, and Localizations , Springer Lecture Notes in Math. , [BM] C.Berger, I.Moerdijk, On the homotopy theory of enriched categories, arxive preprint math . [ChS] W.Chachólski, J.Scherer, Homotopy theory of diagrams , Memoirs of the AMS, , ( )[Dr] V.Drinfeld, DG Quotients of DG Categories, J.Algebra ( )( ), - [Dug] D.Dugger, Spectral enrichments of model categories, Homology, Homotopy Appl , ( ), no. , - [DugS] D.Dugger, B.Shipley, Enriched model categories and an application to additive endomorphism spectra, Theory and Appl. Cat. ( ), - [DK ] W.G.Dwyer, D.M.Kan, Simplicial localizations of categories, Journal of Pure and Appl. Algebra , ( ), - [DK ] W.G.Dwyer, D.M.Kan, Calculating simplicial localizations, Journal of Pure and Appl. Algebra , ( ), - [DK ] W.G.Dwyer, D.M.Kan, Function complexes in homotopical algebra, Topology , ( ), - DS] W.G.Dwyer, J.Spalinski, Homotopy theories and model categories, in:
Handbook of Algebraic Topology , NorthHolland Publ., [GJ] P.Goerss, J.Jardine,
Simplicial Homotopy Theory , Birkhäuser, ∼ pgoerss[GZ] P.Gabriel, M.Zisman, Calculus of fractions and homotopy theory , Springer-Verlag, [Hi] V.Hinich, Homological algebra of homotopy algebras,
Communications in Algebra , ( ), , - [Hir] P.Hirschhorn, Model Categories and Their Localizations , AMS Math. Surveys and Monographs, Vol. , [Ho] M.Hovey, Model Categories , AMS Math. Surveys and Monographs, vol. , [KaS] M.Kashiwara, P.Schapira, Categories and Sheaves , Springer-Verlag [Kel] G.M. Kelly,
Basic concepts of enriched category theory , Cambridge University Press, [Ke ] B.Keller, Deriving dg categories, Ann. sci. Éc. Norm. Sup. , 4 e serie, ( ), - [Ke ] B.Keller, On differential graded categories, Proceedings of the ICM, Madrid, , - [Ke ] B.Keller, Hochschild cohomology and derived Picard groups, J. Pure Appl. Alg. , ( ), - [KS] M.Kontsevich, Y.Soibelman. Deformations of algebras over operads and the Deligne conjecture, In: Con-férence Moshé Flato , Vol. I (Dijon), vol. of Math. Phys. Stud., pp. ˝U , Kluwer Acad. Publ.,Dordrecht, [KT] J.Kock, B.Toën, Simplicial localization of monoidal structures, and a non-linear version of Deligne’s con-jecture, Compos. Math. ( ), , - [Le] T.Leinster, Up-to-Homotopy Monoids, preprint math.QA/ [ML] S.MacLane, Categories for the working mathematician , nd ed., Graduate Text in Mathematics , Springer-Verlag, [MS] J.McClure, J.Smith, A solution of DeligneŠs Hochschild cohomology conjecture, Contemp. Math. , ( ),pp. ˝U [Q] D.G.Quillen, Homotopical Algebra , Springer LMN vol. , [Se] G.Segal, Categories and cohomology theories, Topology , ( ), - [Sch ] S.Schwede, An exact sequence interpretation of the Lie bracket in Hochschild cohomology, J. Reine Angew.Math. , ( ), - [Sch ] S.Schwede, Morita theory in abelian, derived and stable model categories, in Structured ring spectra , – ,London Mathematical Society Lecture Notes SchS ] S.Schwede, B.Shipley, Algebras and modules in monoidal model categories, Proc. London Math. Soc. , ( ), - [SchS ] S.Schwede, B.Shipley, Equivalence of monoidal model categories, Alg. Geom. Topology , ( ), - [SGA ] A.Grothendieck et al., Theorie des topos et cohomologie etale des schemas, SGA .I - , SpringerLNM [Sh ] B.Shoikhet, Hopf algebras, tetramodules, and n -fold monoidal categories, preprint arxiv math . [Sh ] B.Shoikhet, A bialgebra axiom and the Dold-Kan correspondence, preprint arxiv math . [T] D.Tamarkin, Another proof of M.Kontsevich formality theorem, preprint arxiv math. [Tab] G.Tabuada, Une structure decatégorie de modèles de Quillen sur la catégorie des dg-catégories, C.R.Acad.Sci.Paris Sér. I Math. , ( ), , - [To] B.Toën, The homotopy theory of dg-categories and derived Morita theory, Invent. Math. , ( )( ), - [TV] B.Toën, G.Vezzosi, Segal topoi and stacks over Segal categories, preprint arxiv math . [W] Ch.A.Weibel, An introduction to homological algebra , Cambridge studies in advances Mathematocs , Cam-bridge Univ. Press, M ax -P lanck I nstitut für M athematik , V ivatsgasse 7 , B onn ,GERMANY E-mail address : [email protected] Web : http://borya.euhttp://borya.eu