HOMOTOPY THEORY OF MODULES OVER OPERADS AND NON- OPERADS IN MONOIDAL MODEL CATEGORIES JOHN E. HARPER 1. Introduction There are many interesting situations in which algebraic structure can be de- scribed by operads [1, 12, 13, 14, 17, 20, 27, 32, 33, 34, 35]. Let (C, , k) * *be a symmetric monoidal closed category (Section 2) with all small limits and colimi* *ts. It is possible to define two types of operads (Definition 6.1) in this setting,* * as well as algebras and modules over these operads. One type, called -operad, is based* * on finite sets and incorporates symmetric group actions; the other type, called no* *n- operad, is based on ordered sets and has no symmetric group contribution. (In t* *his paper we use the term -operad for non- operad, where = O for "Ordered.") Given an operad O, we are interested in the possibility of doing homotopy theory in the categories of O-modules and O-algebras, which in practice means putting a Quillen model structure on these categories of modules and algebras. In this setting, O-algebras are left O-modules concentrated at 0 (Section 6.3). Of course, to get started we need some kind of homotopy theoretic structure on C itself; this structure should mesh appropriately with the monoidal structure * *on C. The basic assumption is this. Basic Assumption 1.1. From now on in this paper we assume that (C, , k) is a symmetric monoidal closed category (Section 2) with all small limits and colimi* *ts, that C is a cofibrantly generated model category (Definition 11.5) in which the* * gener- ating cofibrations and acyclic cofibrations have small domains, and that with r* *espect to this model structure (C, , k) is a monoidal model category (Definition 11.7* *). The main theorem is this. Theorem 1.2. Assume that C satisfies Basic Assumption 1.1 and in addition satisfies the monoid axiom (Definition 13.25). Let O be an -operad in C. Then the category of left O-modules and the category of O-algebras both have natural model category structures. The weak equivalences and fibrations in these model structures are inherited in an appropriate sense from the weak equivalences and fibrations in C. Remark 1.3. Given any -operad O, there is an associated -operad O . , such that algebras over O . are the same as algebras over O (Section 7). It follows easily from the above theorem that if O0is a -operad which is a retract of O .* * , then the category of algebras over O0has a natural model category structure. The above remark shows how to handle algebras over certain -operads. We can do a lot better if C satisfies a strong cofibrancy condition. In setting up* * the machinery for Theorems 1.2 and 1.4, we introduce model category structures on 1 2 JOHN E. HARPER the category of (symmetric) sequences in C (Definition 3.1) and on the category* * of (symmetric) arrays in C (Definition 13.1). Theorem 1.4. Assume that C satisfies Basic Assumption 1.1 and in addition that every symmetric array (resp. symmetric sequence) in C is cofibrant in the model category structure described below (Theorems 13.2 and 12.2). Then for any - operad O in C, the category of left O-modules (resp. O-algebras) has a natural * *model category structure. The weak equivalences and fibrations in these model structu* *res are inherited in an appropriate sense from the weak equivalences and fibrations* * in C. 1.1. Some examples of interest. The hypotheses of these theorems may seem restrictive, but in fact they allow, especially in the case of Theorem 1.2, for* * many interesting examples including the case (sSet, x, *) of simplicial sets [5, 9, * *15, 37], the case (Chk, , k) of unbounded chain complexes over a commutative ring with unit [22, 29], and the case (Sp , ^, S) of symmetric spectra [24]. In a related* * paper [16], we improve Theorem 1.2 to -operads for the case (Sp , ^, S) of symmetric spectra. 1.2. Relationship to previous work. One of the main theorems of Schwede and Shipley [39] is that the category of monoids in (C, , k) has a natural model c* *ategory structure, provided the monoid axiom (Definition 13.25) is satisfied. Theorem 1* *.2 improves this result to left modules and algebras over any -operad. One of the main theorems of Hinich [18, 19] is that for unbounded chain com- plexes over a field of characteristic zero, the category of algebras over any * *-operad has a natural model category structure. Theorem 1.4 improves this result to the category of left modules, and also provides (Section 14) a simplified conceptual proof of Hinich's original result. In this rational case our theorem is this. Theorem 1.5. Let k be a field of characteristic zero and let (Chk, , k) be the symmetric monoidal closed category of unbounded chain complexes over k. Let O be any -operad or -operad. Then the category of left O-modules (resp. O-algebras) has a natural model category structure. The weak equivalences are the objectwise homology isomorphisms (resp. homology isomorphisms) and the fibrations are the objectwise dimensionwise surjections (resp. dimensionwise surjections). Another theorem of Hinich [18] is that for unbounded chain complexes over a commutative ring with unit, the category of algebras over any -operad of the f* *orm O . for some -operad O, has a natural model category structure. Theorem 1.2 improves this result to the category of left modules. Our theorem is this. Theorem 1.6. Let k be a commutative ring with unit and let (Chk, , k) be the symmetric monoidal closed category of unbounded chain complexes over k. Let O be any -operad. Then the category of left O-modules (resp. O-algebras) has a natural model category structure. The weak equivalences are the objectwise homo* *logy isomorphisms (resp. homology isomorphisms) and the fibrations are the objectwise dimensionwise surjections (resp. dimensionwise surjections). One of the main theorems of Elmendorf and Mandell [6] is that the category of simplicial multifunctors from a small multicategory (enriched over simplicial s* *ets) to the category of symmetric spectra has a natural simplicial model category st* *ruc- ture. Their proof involves a filtration in the underlying category of certain p* *ushouts HOMOTOPY THEORY OF MODULES OVER OPERADS 3 of algebras. We have benefitted from their paper and our proofs of Theorems 1.2 and 1.4 exploit similar filtrations (Section 13). In the special case of unboun* *ded chain complexes, the analysis of certain pushouts reduces to an analysis of cer- tain coproducts and filtration constructions are not required. We have included* * a shortened proof for this special case (Section 14). The framework presented in this paper for doing homotopy theory in the cate- gories of modules and algebras over an operad is largely influenced by Rezk [38* *]. Acknowledgments. The author would like to thank Bill Dwyer for his constant encouragement and invaluable help and advice. The author is grateful to Emmanuel Farjoun for a stimulating and enjoyable visit to Hebrew University of Jerusalem* * in spring 2006 and for his invitation which made this possible, and to Paul Goerss* * and Mike Mandell for helpful comments and suggestions at a Midwest Topology Semi- nar. Parts of this paper were completed while the author was a visiting researc* *her at the Thematic Program on Geometric Applications of Homotopy Theory at the Fields Institute for Mathematics, Toronto. 2.Preliminaries on group actions Here, for reference purposes, we collect certain basic properties of group ac* *tions and adjunctions involving group actions. Some of the statements we organize into propositions. Their proofs are left to the reader. Remark 2.1. The model category assumptions on C stated in Basic Assumption 1.1 will not be needed until Section 12. 2.1. Symmetric monoidal closed categories. By Basic Assumption 1.1, (C, , k) is a symmetric monoidal closed category with all small limits and colimits. In particular, C has an initial object ; and a terminal object *. See [30, VI* *I] for monoidal categories and [30, VII.7] for symmetric monoidal categories. By closed we mean there exists a functor Copx C-! C, (Y, Z) 7-! Map(Y, Z), which we call mapping object (or cotensor object), which fits into isomorphisms (2.2) homC(X Y, Z) ~=hom C(X, Map(Y, Z)), natural in X, Y, Z. Remark 2.3. This condition is stronger than only requiring each functor - Y: C* *-! C to have a specified right adjoint Map(Y, -): C-! C. 2.2. Group actions and G-objects. The symmetric monoidal closed structure on C induces a corresponding structure on certain diagram categories. Definition 2.4. Let G be a finite group. CGopis the category with objects the f* *unc- tors X : Gop-! Cand morphisms their natural transformations. CG is the category with objects the functors X : G-! Cand morphisms their natural transformations. The diagram category CGop(resp. CG) is isomorphic to the category of objects in C with a specified right action of G (resp. left action of G). 4 JOHN E. HARPER Proposition 2.5. Let G be a finite group. Then (CGop, , k) has a symmetric monoidal closed structure induced from the symmetric monoidal closed structure on (C, , k). In particular, there are isomorphisms hom CGop(X Y, Z) ~=hom CGop(X, Map(Y, Z)) natural in X, Y, Z. The proposition remains true when CGop is replaced by CG. We usually leave such corresponding statements to the reader. 2.3. Copower constructions. If X is a finite set, let's define |X| to be the nu* *mber of elements in X. Definition 2.6. Let X be a finite set and A 2 C. The copowers A . X 2 C and X . A 2 C are defined by the same construction: a a A . X := A, X . A := A, X X the coproduct in C of |X|-copies of A. Sometimes extra structure on the set X induces extra structure on the objects A . X and X . A. For example, if G is a finite group, then A . G and G . A have* * op naturally occurring right and left actions of G, respectively, and hence A.G 2 * *CG and G . A 2 CG. Remark 2.7. In the literature, copower is sometimes indicated by a tensor produ* *ct symbol, but because several tensor products already appear in this paper, we are using the usual dot notation as in [30]. When C = sSetthere are natural isomorphisms A . G ~=A x G, when C = Chk there are natural isomorphisms A . G ~=A k[G], and when C = Sp there are natural isomorphisms A . G ~=A ^ G+. Since left Kan extensions may be calcu- lated objectwise in terms of copower, the copower construction appears in sever* *al adjunctions below. 2.4. G-orbits and G-fixed points. Definition 2.8. Let G be a finite group. If Y : Gopx G-! Cand Z : G x Gop-!C are functors, then YG 2 C and ZG 2 C are defined by YG := coendY, ZG := endZ. The universal properties satisfied by these coends and ends are convenient wh* *en working with YG and ZG , but the reader may take the following calculations as definitions. There are natural isomorphisms, YG ~=colim( Gop_diag//_Gopx Gop~=Gopx G_Y__//C), ZG ~=lim( Gop_diag//_Gopx Gop~=G x GopZ__//C). HOMOTOPY THEORY OF MODULES OVER OPERADS 5 2.5. Adjunctions. Proposition 2.9. Let G be a finite group, H G a subgroup, and lo:pH-! G the inclusion of groups. Let G1, G2 be finite groups and A2 2 CG2 . There are adjunctions _-.HG//_ op op___-_A2___// op (2.10) C _____//CHoplimoo_CGoo_, CG1 oo________C(G1xG2) , l* Map(A2,-)G2 with left adjoints on top. In particular, there are isomorphisms homCGop(A .H G, B)~=homCHop(A, B), hom CGop(A . (H\G),~B)=homC(A, BH ), hom CGop(A . G,~B)=homC(A, B), hom C(G1xG2)op(A1 A2,~X)=homCGop1(A1, Map(A2, X)G2), natural in A, B and A1, X. Remark 2.11. The restriction functor l* is sometimes dropped from the notation, as in the natural isomorphisms in Proposition 2.9. 3.Sequences and symmetric sequences In preparation for defining operads, we consider sequences and symmetric se- quences of objects in C. We introduce a symmetric monoidal structure on SymSeq, and a symmetric monoidal structure ^ on Seq. Both of these are rel- atively simple; is a form of the symmetric monoidal structure that is used in the construction of symmetric spectra [23, 24], while ^ is defined in a way that is very similar to the definition of the graded tensor product of chain complex* *es. These monoidal products possess appropriate adjoints, which can be interpreted as mapping objects. For instance, there are objects Map (B, C) and Map ^(Y, Z) which fit into isomorphisms hom(A B, C)~=hom(A, Map (B, C)), hom(X ^Y, Z)~=hom(X, Map^(Y, Z)), natural in the symmetric sequences A, B, C and the sequences X, Y, Z. 3.1. Sequences and symmetric sequences. Let's define the sets n := {1, . .,.n} for each n 0, where 0 := ; denotes the empty set. When regarded as a totally ordered set, n is given its natural ordering. Definition 3.1. Let n 0. o is the category of finite sets and their bijections. is the category * *of totally ordered finite sets and their order preserving bijections. oA symmetric sequence in C is a functoroAp: op-! C. A sequence in C is a functor X : op-! C. SymSeq := C is the category of symmetricose-p quences in C and their natural transformations. Seq:= C is the category of sequences in C and their natural transformations. oA (symmetric) sequence A is concentrated at n if A[s] = ; for all s 6= n. 6 JOHN E. HARPER 3.2. Small skeletons and equivalent categories. The indexing categories for symmetric sequences and sequences are not small, but they have small skeletons, which will be useful for calculations. Definition 3.2. o n is the category with exactly one object n and morphisms the bijections of sets. n is the category with exactly one object n and morphisms the identity map. o 0 is the category with objects the sets n for n 0 and morphisms the bijections of sets. 0is the category with objects the totally ordered se* *ts n for n 0 and morphisms the identity maps. oA small symmetric sequence in C is a functor A : 0op-!C.0oApsmall sequence in C is a functor X : 0op-!C. SymSeq0:= C is the cate- gory of small0symmetricosequencespin C and their natural transformations. Seq0 := C is the category of small sequences in C and their natural transformations. ` ` The indexing categories 0op~= n 0 opnand 0op~= n 0 opnare coproducts of categories, hence giving functors A : 0op-!Cand X : 0op-!Cis the same as giving collections {A[n]}n 0 and {X[n]}n 0 of objects in C such that each A[n] * *is equipped with a right action of the symmetric group n. The inclusions i : 0-! and i : 0-! are equivalences of categories, __j_//_ __j_//_ (3.3) ooi__ 0, ooi__ 0. In particular, i has a left adjoint j with corresponding unit j : id-!ijand cou* *nit " : ji-!idisomorphisms. Remark 3.4. Giving the equivalence of categories j : -! 0is the same as giving a choice of bijection jT: T -!|T|= ij(T ) (i.e., a choice of ordering) for each* * finite set T . The equivalences of categories (3.3)induce equivalences of categories, (3.5) SymSeq ____//_SymSeq0,oo_Seq___//Seq0.oo_ 3.3. Ordered and unordered tensor products. Given any non-empty totally ordered finite set N (i.e., a non-empty finite set N equipped with a bijection j : N- !|N|) and a collection of objects {Xn}n2N in C indexed by N, there is a naturally occurring ordered tensor product n2N Xn 2 C defined by n2N Xn := Xj-1(1). . .Xj-1(|N|). Similarly, given any non-empty finite set T and a collection of objects {Xt}t* *2T in C indexed by T , we would like to define a naturally occuring unordered tens* *or product t2TXt 2 C. One approach is to simply choose an ordering of T (i.e., choose a bijection j : T -!|T|) and declare that t2TXt:= Xj-1(1). . .Xj-1(|T|). This approach is fine, but an isomorphic and more intrinsic definition would be to replace the choice of ordering with a choice of colimit. The idea is, instea* *d of choosing a particular ordering of the unordered set T , take all possible order* *ings HOMOTOPY THEORY OF MODULES OVER OPERADS 7 of T , coproduct the corresponding ordered tensor products together, and identi* *fy them. The construction is this. Definition 3.6. Let T be a non-empty finite set. o T is the category with exactly one object T and morphisms the bijections of sets. Let iT: T-! denote the inclusion of categories. oThe category of orderings of T is the over category Ord(T ) := iT # |T|; * *i.e., the objects are the bijections j : T -!|T|and the morphisms ' : j-! j0 are the commutative diagrams ' T ___~=_//_T j|~=| ~=j0|| fflffl| |fflffl |T|______|T|. Let T be a non-empty finite set and {Xt}t2Ta collection of objects in C index* *ed by T . There is a functor X : Ord(T-)!C defined objectwise by ( X)(j : T -!|T|) := Xj-1(1). . .Xj-1(|T|) which is useful for defining unordered tensor products. Definition 3.7. Let T be a non-empty finite set and {Xt}t2Ta collection of obje* *cts in C indexed by T . The unordered tensor product t2TXt2 C is defined by t2TXt:= colim( X). Remark 3.8. Since every object in Ord(T ) is terminal, for each bijection j : T* * -!|T| there is a natural isomorphism, t2TXt~=Xj-1(1). . .Xj-1(|T|). 4.Tensor products for (symmetric) sequences Sequences and symmetric sequences have naturally occuring tensor products. Definition 4.1. Let A1, . .,.At be symmetric sequences and let X1, . .,.Xt be sequences. The tensor products A1 . . .At2 SymSeqand X1^ . .^.Xt2 Seqare the left Kan extensions of objectwise tensor along coproduct of sets, ( op)xt_A1x...xAt//_Cxt__//C ( op)xtX1x...xXt//_Cxt___//C `| `| | | fflffl| A1 ...At fflffl| X1^...^Xt op ___left_Kan_extension//_C, op ___left_Kan_extension//_C. Remark 4.2. The reader may wish to drop the hat in the tensor product notation ^ for sequences. We have included it only to avoid confusion later, since both * *the tensor product of sequences and of symmetric sequences appear in this paper, and sometimes in the same formula (Section 7). 8 JOHN E. HARPER 4.1. Calculations. This gives a conceptual definition of the tensor products, b* *ut the reader may take either of the following calculations as a definition. Proposition 4.3. Let A1, . .,.At be symmetric sequences and S 2 , with s := |S|. Let X1, . .,.Xt be sequences and M 2 , with m := |M|. There are natural isomorphisms, a (4.4) (A1 . . .At)[S]~= A1[ss-1(1)] . . .At[ss-1(t)], ss:S-!itn Set (4.5) ~= a A1[s1] . . .At[st] . s, s1+...+st=s s1x...x st a (4.6) (X1^ . .^.Xt)[M]~= X1[ss-1(1)] . . .Xt[ss-1(t)], ss:M-!itn OrdSet (4.7) ~= a X1[m1 ] . . .Xt[mt ], m1+...+mt=m Remark 4.8. Giving a map of sets ss : S-! tis the same as giving an ordered partition (I1, . .,.It) of S. Whenever ss is not surjective, at least one Ij wi* *ll be the empty set 0. Proof.Left to the reader. 4.2. Tensor powers. It will be useful to extend the definition of tensor powers A t and X ^nto situations in which the integers t and n are replaced, respectiv* *ely, by a finite set T or a finite ordered set N. The calculations in Proposition 4* *.3 suggest how to proceed. We introduce here the suggestive bracket notation used by Rezk [38]. Definition 4.9. Let A be a symmetric sequence and S, T 2 . Let X be a sequence and M, N 2 . The tensor powers A T 2 SymSeq and X ^N 2 Seqare defined objectwise by a (4.10) (A T )[S] := A[S, T ] := t2TA[ss-1(t)], T 6= ; , ss:S-!iTn Set a (A ;)[S] := A[S, ;] := k, ss:S-!i;n Set a (X ^N)[M] := X[M, N] := n2N X[ss-1(n)], N 6= ;, ss:M-!iNn OrdSet a (X ^;)[M] := X[M, ;] := k. ss:M-!i;n OrdSet We will use the abbreviations A 0 := A ; and X ^0:= X ^;. Remark 4.11. The reader will notice there is no hat appearing in the bracket no- tation X[M, N] and X[M, ;] for sequences. Remark 4.12. The tensor products indexed by T (resp. indexed by N) are regarded as unordered (resp. ordered) (Section 3.3). Set is the category of sets and the* *ir maps. OrdSetis the category of totally ordered sets and their order preserving maps. HOMOTOPY THEORY OF MODULES OVER OPERADS 9 The above constructions give functors SymSeqx opx -! C, (A, S, T ) 7-! A[S, T ], Seqx opx -! C, (X, M, N) 7-! X[M, N], SymSeq x SymSeq-! SymSeq, (A, B) 7-! A B, Seqx Seq-!Seq, (X, Y ) 7-! X ^Y. Observe that the unit for the tensor product on SymSeqand the unit for the tensor product ^ on Seq, both denoted "1", are given by the same formula ae 1[S] := k,;for,|S|o=t0,herwise. It is useful to make some simple calculations. Proposition 4.13. Let A, B be symmetric sequences. There are natural isomor- phisms, A 1 ~=A, A 0 ~=A[-, 0] ~=1, A B ~=B A A ; ~=;, A 1 ~=A[-, 1] ~=A, (A t)[0] ~=A[0, t] ~=A[0] t, t 0. Proof.These are verified directly from Definition 4.9. Similar calculations are true for sequences. 4.3. Mapping objects for (symmetric) sequences. Let B, C be symmetric sequences and T 2 . Let Y, Z be sequences and N 2 . There are functors x op-! C, (S, S0) 7-! Map(B[S], C[T q S0]), x op-! C, (M, M0) 7-! Map(Y [M], Z[N q M0]), which are useful for defining the mapping objects of (SymSeq, , 1) and (Seq, ^* *, 1). Definition 4.14. Let B, C be symmetric sequences and T 2 . Let Y, Z be se- quences and N 2 . The mapping objects Map (B, C) 2 SymSeqand Map^(Y, Z) 2 Seqare defined objectwise by the ends Map (B, C)[T ]:= Map (B, C[T q -]) , Map ^(Y, Z)[N]:= Map (Y, Z[N q -]) . 4.4. Universal properties. Hence Map (B, C) satisfies objectwise the universal property (4.15) f[S]____________0Map(B[S],0C[T_q_S])___S_____________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *____________33__________________@ ______________________________________________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *____________________| _______________________________(id,(idqi)*)||______________________* *____________________________________o[S]_____________________________________* *_________________________________| ________________________________fflffl|_______________~f____________* *______________________|| .____________________//_____________Map(B,MC)[Ta]p(B[S],iC[T|q S0]) __________________________________________9!________________________* *___OO| __________________________________________________o[S0]___________* *__________________________________________|| * * * _______________________________________________________________* *_____________________________________________________________________________* *________________________(i|,id)| ___________________________________________________________* *_____________________________________________________________________________* *____________________________________________________| f[S0] __________..___________________________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *_______________**____________Map(B[S0],SC[T0q|S0]) 10 JOHN E. HARPER that each wedge f factors uniquely through the terminal wedge o of Map (B, C)[T* * ]. The mapping objects Map ^(Y, Z) satisfy objectwise a similar universal property. These constructions give functors SymSeqopx SymSeq-! SymSeq, (B, C) 7-! Map (B, C), Seqopx Seq-!Seq, (Y, Z) 7-! Map ^(Y, Z). Proposition 4.16. Let A, B, C be symmetric sequences and let X, Y, Z be se- quences. There are isomorphims (4.17) hom(A B, C)~=hom(A, Map (B, C)), (4.18) hom(X ^Y, Z)~=hom(X, Map^(Y, Z)), natural in A, B, C and X, Y, Z. Proof.Consider (4.17). Use the calculation (4.4)and the universal property (4.1* *5) with the natural correspondence (2.2)to verify that giving a map A B-! C is the same as giving a map A-! Map (B, C), and that the resulting correspondence is natural. Use a similar argument for the case of sequences. 4.5. Monoidal structures. Proposition 4.19. (SymSeq, , 1) and (Seq, ^, 1) have the structure of symmetric monoidal closed categories with all small limits and colimits. Proof.To verify the symmetric monoidal structure, use (4.4)to describe the re- quired natural isomorphisms and to verify the appropriate diagrams commute. Proposition 4.16 verifies the symmetric monoidal structure is closed. Limits and colimits are calculated objectwise. Argue similarly for sequences. 4.6. Calculations. Definition 4.14 gives objectwise a conceptual interpretation* * of the mapping objects, but the reader may take the following calculations as defi* *ni- tions. Proposition 4.20. Let B, C be symmetric sequences and T 2 , with t := |T |. Let Y, Z be sequences and N 2 , with n := |N|. There are natural isomorphisms, Y (4.21) Map (B, C)[T~]= Map (B[s], C[t + s]) s, s 0 Y (4.22) Map ^(Y, Z)[N]~= Map(Y [m], Z[n + m]). m 0 Proof.To verify (4.21), use the universal property (4.15)and restrict to a small skeleton to obtain natural isomorphisms ` Q ____//_Q 0' Map(B[S], C[T q S])__//_ Map(B[S], C[T q S ]) Map (B, C)[T~]=lim S i:S0-!S . in 0 in 0 This verifies (4.21). Argue similarly for the case of sequences. HOMOTOPY THEORY OF MODULES OVER OPERADS 11 5. Circle products for (symmetric) sequences We describe a circle product O on SymSeqand a related circle product ^Oon Seq. These are monoidal products which are not symmetric monoidal, and they figure in the definitions of -operad and -operad respectively (Definition 6.1). Perh* *aps surprisingly, these monoidal products possess appropriate adjoints, which can be interpreted as mapping objects. For instance, there are objects Map O(B, C) and Map ^O(Y, Z) which fit into isomorphisms hom(A O B, C)~=hom(A, MapO(B, C)), hom (X ^OY, Z)~=hom(X, Map^O(Y, Z)), natural in the symmetric sequences A, B, C and the sequences X, Y, Z. The material in this section is largely influenced by Rezk [38]. Earlier work exploiting circle product O for symmetric sequences includes [10, 11, 40]; more recent work includes [7, 8, 25, 26]. The circle product ^Ois used in [2] for wo* *rking with -operads and their algebras. 5.1. Circle products (or composition products). Let A, B be symmetric se- quences and S 2 . Let X, Y be sequences and M 2 . There are functors opx -! C, (T 0, T ) 7-! A[T 0] B[S, T ], opx -! C, (N0, N) 7-! X[N0] Y [M, N], which are useful for defining circle products of sequences and of symmetric se- quences. Definition 5.1. Let A, B be symmetric sequences and S 2 . Let X, Y be se- quences and M 2 . The circle products (or composition products) AOB 2 SymSeq and X ^OY 2 Seqare defined objectwise by the coends (A O B)[S]:= A (B - )[S] = A B[S, -], (X ^OY )[M]:= X (Y ^-)[M] = X Y [M, -]. Remark 5.2. The reader may wish to drop the hat in the circle product notation ^Ofor sequences. We have included it only to avoid confusion later, since both * *the circle product of sequences and of symmetric sequences appear in this paper, and sometimes in the same formula (Section 7). 5.2. Universal properties. Hence AOB satisfies objectwise the universal property (5.3) T A[T ]OOB[S,_T_]_________________________________f[T]______* *_____________________________________________________________________________* *______________________________________________________________________________ | ____________________________________________* *_____________________________________________________________________________* *_________________ | ,* [id,id]| i[T]_________________________________________* *________________________________________________ | || __OEOE____________________________~~____* *___________________________________________________________~ |,| A[T 0] B[S, T ] (A O B)[S]f___//___________.JJ____ | __@@_____9!_____________________________* *___________________________________ | id [id,,]| i[T0]________________________________________* *____________________________________________ | | ___________________________________________* *__________________________________________________________________ fflffl| fflffl|_____________________________________________0* *_____________________________________________________________________________* *_____________________________________________________________________________* *_________ T 0 A[T 0] B[S,_T_0]_____f[T_]________________ that each wedge f factors uniquely through the initial wedge i of (A O B)[S]. A similar universal property is satisfied objectwise by the circle products X ^OY* * . These 12 JOHN E. HARPER constructions give functors SymSeqx SymSeq-! SymSeq, (A, B) 7-! A O B, Seqx Seq-!Seq, (X, Y ) 7-! X ^OY. 5.3. Calculations. Definition 5.1 gives objectwise a conceptual interpretation * *of the circle products, but the reader may take the following calculations as defi* *nitions. Proposition 5.4. Let A, B be symmetric sequences and S 2 , with s := |S|. Let X, Y be sequences and M 2 , with m := |M|. There are natural isomorphisms, a a (5.5) (A O B)[S]~= A[t] t(B t)[s] ~= A[t] tB[s, t], t 0 t 0 a ^ a (X ^OY )[M]~= X[n] (Y n)[m] ~= X[n] Y [m, n]. n 0 n 0 Proof.Use the universal property (5.3)and restrict to a small skeleton to obtain natural isomorphisms _ ` ____//_` ! A[T 0] B[S,_T_]//_ A[T ] B[S, T ] (A O B)[S] ~=colim ,:T-! T0 T . in 0 in 0 This verifies (5.5). The other case is similar. Observe that the unit for the circle product O on SymSeq and the unit for the circle product ^Oon Seq, both denoted "I", are given by the same formula ae I[S] := k,;for,|S|o=t1,herwise. Definition 5.6. Let A be a symmetric sequence, X a sequence, and Z 2 C. The corresponding functors A O (-): C-! Cand X ^O(-): C-! Care defined objectwise by, a A O (Z):= A[t] tZ t, t 0 a X ^O(Z):= X[t] Z t. t 0 The category C embeds in SymSeq(resp. Seq) as the full subcategory of symmet- ric sequences (resp. sequences) concentrated at 0, via the functor ^-: C-! SymS* *eq (resp. ^-: C-! Seq) defined objectwise by ae (5.7) Z^[S] := Z, for |S| = 0, ;, otherwise. It is useful to make some simple calculations. HOMOTOPY THEORY OF MODULES OVER OPERADS 13 Proposition 5.8. Let A, B be symmetric sequences and Z 2 C. There are natural isomorphisms, ; O A~=;, I O A ~=A, A O I ~=A, ae (A O ;)[S]~= A[0],;for,|S|o=t0,herwise, ae (A O ^Z)[S]~=A O (Z),for;|S|,=o0,therwise, (A O B)[0]~=A O (B[0]). Proof.These can be verified directly from Proposition 5.4. Similar calculations are true for sequences. 5.4. Properties of tensor and circle products. It is useful to understand how tensor products and circle products interact. Proposition 5.9. Let A, B, C be symmetric sequences, X, Y, Z be sequences, and t 0. There are natural isomorphisms (A B) O C~=(A O C) (B O C), (X ^Y ) ^OZ~=(X ^OZ) ^(Y ^OZ), (5.10) (B t) O C~=(B O C) t, (Y ^t) ^OZ~=(Y ^OZ)^t. Proof.Using (4.5)and (5.5), there are natural isomorphisms a (A B) O C~= (A B)[s] sC s s 0 ~=a a A[s1] B[s2] s1x s2C (s1+s2) si0 s1+s2=s j i j ~= a A[s1] s1C s1 a B[s2] s2C s2 s1 0 s2 0 ~=(A O C) (B O C). The argument for sequences is similar. Proposition 5.11. Let A, B be symmetric sequences and let X, Y be sequences. Suppose Z 2 C and t 0. There are natural isomorphisms (A B) O (Z)~=(A O (Z)) (B O (Z)), (X ^Y ) ^O(Z)~=(X ^O(Z)) (Y ^O(Z)), (B t) O (Z)~=(B O (Z)) t, (Y ^t) ^O(Z)~=(Y ^O(Z)) t. Proof.Argue as in the proof of Proposition 5.9, or use the embedding (5.7)to deduce it as a special case. 14 JOHN E. HARPER Proposition 5.12. Let A, B, C be symmetric sequences and let X, Y, Z be se- quences. There are natural isomorphisms (A O B) O C~=A O (B O C), (X ^OY ) ^OZ~=X ^O(Y ^OZ). Proof.Using (5.5)and (5.10), there are natural isomorphisms a A O (B O C)~= A[t] t(B O C) t t 0 ~=a A[t] t(B t) O C t 0 ~=a a A[t] t(B t)[s] sC s s 0t 0 ~=(A O B) O C. Argue similarly for the case of sequences. Proposition 5.13. Let A, B be symmetric sequences and let X, Y be sequences. Suppose Z 2 C. There are natural isomorphisms (A O B) O (Z) ~=A O B O (Z) , (X ^OY ) ^O(Z) ~=X ^OY ^O(Z) . Proof.Argue as in the proof of Proposition 5.12, or use the embedding (5.7)to deduce it as a special case. 5.5. Mapping sequences. Let B, C be symmetric sequences and T 2 . Let Y, Z be sequences and N 2 . There are functors x op-! C, (S, S0) 7-! Map(B[S, T ], C[S0]), x op-! C, (M, M0) 7-! Map(Y [M, N], Z[M0]), which are useful for defining mapping objects of (SymSeq, O, I) and (Seq, ^O, I* *). Definition 5.14. Let B, C be symmetric sequences and T 2 . Let Y, Z be se- quences and N 2 . The mapping sequences MapO(B, C) 2 SymSeqand Map^O(Y, Z) 2 Seqare defined objectwise by the ends Map O(B, C)[T:]= Map ((B T )[-], C) = Map (B[-, T ], C) , Map ^O(Y, Z)[N]:= Map ((Y ^N)[-], Z) = Map (Y [-, N], Z) . 5.6. Universal properties. Hence MapO(B, C) satisfies objectwise the universal property (5.15) f[S]____________Map0(B[S,0T_],_C[S])_S______________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *______________________________33@ _____________________________________________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *____| ______________________________([id,id],i*)||______________________* *_____________________________________o[S]____________________________________* *_________________________________| _______________________________fflffl|_______________~f____________* *_________________________|| .____________________//_____________MapO(B,MC)[Ta]p(B[S,iT|], C[S0]) __________________________________________9!_______________________* *_______OO| ___________________________________________________o[S0]_________* *_____________________________________________|| _______________________________________________________________* *_____________________________________________________________________________* *_________________([i,id],id)|| __________________________________________________________* *_____________________________________________________________________________* *_____________________________________________________________|| f[S0] _________..___________________________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *__________________**______________Map(B[S0,ST0], C[S0]) HOMOTOPY THEORY OF MODULES OVER OPERADS 15 that each wedge f factors uniquely through the terminal wedge o of MapO(B, C)[T* * ]. A similar universal property is satisfied objectwise by the mapping objects Map* *^O(Y, Z). These constructions give functors SymSeqopx SymSeq-! SymSeq, (B, C) 7-! MapO(B, C), Seqopx Seq-!Seq, (Y, Z) 7-! Map^O(Y, Z). Proposition 5.16. Let A, B, C be symmetric sequences and let X, Y, Z be se- quences. There are isomorphims (5.17) hom(A O B, C)~=hom(A, MapO(B, C)), (5.18) hom (X ^OY, Z)~=hom(X, Map^O(Y, Z)), natural in A, B, C and X, Y, Z. Proof.Use the universal properties (5.3)and (5.15)with the natural correspon- dence (2.2)to verify that giving a map A O B-! C is the same as giving a map A-! Map O(B, C), and that the resulting correspondence is natural. Argue simi- larly for the case of sequences. 5.7. Monoidal structures. Proposition 5.19. (SymSeq, O, I) and (Seq, ^O, I) have the structure of monoidal closed categories with all small limits and colimits. Circle product is not sym* *metric. Proof.To verify the monoidal structure, use (5.5)along with properties of from Proposition 4.19 to describe the required natural isomorphisms and to verify the appropriate diagrams commute. Proposition 5.16 verifies the monoidal structure * *is closed. Limits and colimits are calculated objectwise. Argue similarly for the * *case of sequences. 5.8. Calculations. Definition 5.14 gives objectwise a conceptual interpretation* * of the mapping sequences, but the reader may take the following calculations as de* *fi- nitions. Proposition 5.20. Let B, C be symmetric sequences and T 2 , with t := |T |. Let Y, Z be sequences and N 2 , with n := |N|. There are natural isomorphisms, (5.21) Y Y Map O(B, C)[T~]= Map((B t)[s], C[s]) s ~= Map(B[s, t], C[s]) s, s 0 s 0 Y ^ Y Map ^O(Y, Z)[N]~= Map ((Y n)[m], Z[m]) ~= Map(Y [m, n], Z[m]). m 0 m 0 Proof.Use the universal property (5.15)for MapO(B, C)[T ] and restrict to a sma* *ll skeleton to obtain natural isormorphisms ` Q ____//_Q ' 0 Map (B[S, T ], C[S])//_ Map (B[S, T ], C[S ]) Map O(B, C)[T~]=lim S i:S0-!S . in 0 in 0 This verifies (5.21). The case for sequences is similar. It is useful to make some simple calculations. 16 JOHN E. HARPER Proposition 5.22. Let B, C be symmetric sequences, T 2 , and Z 2 C. There are natural isomorphisms, MapO(B, *)~=*, ae MapO(;, C)[T~]= C[0],*for,|To|t=h0,erwise, Map O(B, C)[0]~=C[0], Y Map O(B, C)[1]~= Map(B[s], C[s]) s, s 0 Map O(Z^, C)[T~]=Map(Z |T|, C[0]). Proof.These can be verified directly from Proposition 5.20. Similar calculations are true for sequences. 5.9. Circle products as Kan extensions. Circle products can also be under- stood as Kan extensions. Let F be the category of finite sets and their maps an* *d let OrdF be the category of totally ordered finite sets and their order preserving * *maps. Then circle products are left Kan extensions of - * - along projection onto sou* *rce, op A*B .!. op X*Y Iso(F.!.) __________//C Iso(OrdF ) __________//C proj|| proj|| fflffl|AOB fflffl|X ^OY op left_Kan_extension//_C, op _left_Kan_extension//_C. The functors - * - are defined objectwise by (A * B)(ss : S !:T=)A[T ] t2TB[ss-1(t)], (X * Y )(ss : M ! N):= X[N] n2NY [ss-1(n)]. Remark 5.23. The tensor products indexed by T (resp. indexed by N) are regarded as unordered (resp. ordered) (Section 3.3). 6. Operads, modules, and algebras In this section we define operads and the objects they act on. 6.1. Operads. Definition 6.1. oA -operad is a monoid object in (SymSeq, O, I) and a morphism of - operads is a morphism of monoid objects in (SymSeq, O, I). oAn -operad is a monoid object in (Seq, ^O, I) and a morphism of -operads is a morphism of monoid objects in (Seq, ^O, I). These two types of operads were originally defined by May [32]; the -operad has symmetric groups and the -operad is based on ordered sets and is called a non- operad [27, 32]. HOMOTOPY THEORY OF MODULES OVER OPERADS 17 Example 6.2. More explicitly, for instance, a -operad is a symmetric sequence O together with maps m : O O O-!O and j : I-! Oin SymSeq which make the diagrams jOid idOj O O O O OmOid//_O O O I O O____//_O OoOo__O O I |idOm| |m| |~=| m|| ~=|| fflffl|m fflffl| fflffl| fflffl| fflffl| O O O________//O O _________O________O_ commute. If O and O0 are -operads, then a morphism of -operads is a map f : O-! O0in SymSeqwhich makes the diagrams j O O O__m__//_O O oo___I |fOf| |f| |f| |||| fflffl|m fflffl|fflffl|||j O0O O0 ____//_O0 O0 oo___I commute. 6.2. Modules over operads. Similar to the case of any monoid object, we intro- duce operads because we are interested in the objects they act on. Compare the following definition with [29, Chapter VII]. Definition 6.3. Let Q, R, S be -operads (resp. -operads). oA left R-module is an object in (SymSeq, O, I) (resp. an object in (Seq, * *^O, I)) with a left action of R and a morphism of left R-modules is a map which respects the left R-module structure. oA right S-module is an object in (SymSeq, O, I) (resp. an object in (Seq,* * ^O, I)) with a right action of S and a morphism of right S-modules is a map which respects the right S-module structure. oAn (R, S)-bimodule is an object in (SymSeq, O, I) (resp. an object in (Se* *q, ^O, I)) with compatible left R-module and right S-module structures and a mor- phism of (R, S)-bimodules is a map which respects the (R, S)-bimodule structure. Example 6.4. More explicitly, for instance, if R is a -operad, then a left R-m* *odule is a symmetric sequence B together with a map m : R O B-!B in SymSeq which makes the diagrams jOid (6.5) R O R O BmOid//_R O B I O B____//R O B |idOm| m|| ~=|| |m| fflffl|m fflffl| fflffl| fflffl| R O B_______//_B B _________B commute. If B and B0are left R-modules, then a morphism of left R-modules is a map f : B-! B0in SymSeqwhich makes the left-hand diagram (6.6) R O B _m__//_B R O B O SmOid//_B O S idOf|| f|| |idOm| m|| fflffl|m fflffl| fflffl|m fflffl| R O B0____//_B0 R O B_______//_B 18 JOHN E. HARPER commute. If R and S are -operads, then an (R, S)-bimodule is an object in (SymSeq, O, I) with left R-module and right S-module structures which make the right-hand diagram in (6.6)commute. 6.3. Algebras over operads. An algebra over an operad O is a left O-module which is concentrated at 0. Definition 6.7. Let O be an operad. oAn O-algebra is an object X 2 C with a left O-module structure on ^X. oLet X and X0be O-algebras. A morphism of O-algebras is a map f : X-! X0 in C such that ^f: ^X-!X^0is a morphism of left O-modules. Giving a symmetric sequence Y a left O-module structure is the same as giving a morphism of operads m : O-! Map O(Y, Y.) Similarly, giving an object X 2 C an O-algebra structure is the same as giving a morphism of operads m : O-! Map O(X^,.^X) This is the original definition given by May [32] of an O-algebra structure on * *X, where MapO(X^, ^X) is called the endomorphism operad of X (see Proposition 5.22* *), and motivates the suggestion in [27, 32] that O[t] should be thought of as obje* *cts of parameters for t-ary operations. Algebras over an operad can also be described as objects in C with a left ac- tion of a particular monad (or triple). Each -operad O determines a functor O(-) : C-! Cdefined objectwise by a O(X) := O O (X) = O[t] tX t, t 0 along with natural transformations m : OO -! Oand j : id-!Owhich make ap- propriate associativity and unit diagrams commute. Giving an O-algebra structure on X is the same as giving a left action of O(-) on X; i.e., a map m : O(X)-! X in C which makes the appropriate associativity and unit diagrams commute. From this perspective, operads correspond to special functors in such a way that cir* *cle product corresponds to composition, but because these functors have such simple descriptions in terms of symmetric sequences, operads are easier to work with t* *han arbitrary functors. 7.Freely adding -actions Proposition 7.1. There is an adjunction _-._//_ Seq oo___SymSeq * with left adjoint on top and : -! the forgetful functor. Proof.There is a chain of adjunctions _-._//_ ____//_ Seq_____//Seq0oo_SymSeq0oo_ooSymSeq_, with left adjoints on top; here, (A. )[s] := A[s]. s. Composition gives the des* *ired adjunction, with left adjoint also denoted - . . HOMOTOPY THEORY OF MODULES OVER OPERADS 19 7.1. Properties of the left adjoint. It is useful to understand how tensor prod- ucts and circle products behave with respect to - . . Proposition 7.2. Let A, B be sequences and t 0. There are natural isomor- phisms, (7.3) (A . ) (B .~=)(A ^B) . , (7.4) (B . ) t~=(B ^t) . , (7.5) (A . ) O (B .~=)(A ^OB) . , in SymSeq. Proof.To verify (7.3), let s 0 and consider the natural isomorphisms, a (A . ) (B . )~[s]= (A[s1] . s1) (B[s2] . s2) . s1x s2 s s1+s2=s ~= (A ^B) . [s]. To verify (7.5), consider the natural isomorphisms, a ^ (A . ) O (B . ) ~= (A . )[t] t(B t) . ~=(A ^OB) . . t 0 7.2. Naturally occurring maps. Proposition 7.6. Let B 2 Seqand C 2 SymSeq. There is a map i : B ^OC-!(B . ) O C in Seq, natural in B, C. Proof.Each M 2 may be regarded as an unordered set; use this together with (4.4)and (4.6)to look for naturally occurring maps iM : (B ^OC)[M]-!(B . ) O C [M], for each M 2 . 7.3. Adjunctions. Proposition 7.7. Let O be an -operad. There are adjunctions _-._//_ _____// LtOoo___LtO. AlgOoo___AlgO. U U with left adjoints on top and U the forgetful functor. The right-hand adjunctio* *n is an isomorphism of categories. Proof.Use Propositions 7.2 and 7.6. 8.Limits and colimits for modules and algebras We have introduced modules and algebras over an operad, but to work with them, we need to understand how to build basic constructions such as limits and colimits. The material in this section is largely influenced by Rezk [38]. 20 JOHN E. HARPER 8.1. Reflexive coequalizers and filtered colimits. od0o_ Definition 8.1. A pair of maps of the form X0 oo__X1_in C is called a reflexive d1 pair if there exists s0: X0-! X1in C such that d0s0 = idand d1s0 = id. A reflex* *ive coequalizer is the coequalizer of a reflexive pair. Reflexive coequalizers will be useful for building colimits in the categories* * of modules and algebras over an operad. Proposition 8.2. (a)Suppose X-1 oo__X0_oo__X1_oo_and Y-1oo___Y0oo__Y1_oo_are reflexive co- equalizer diagrams in C. Then their objectwise tensor product X-1 Y-1oo___X0 Y0 oo___X1oY1o_ is a reflexive coequalizer diagram in C. (b)Suppose X, Y: D-! Care filtered diagrams. Then objectwise tensor product of their colimiting cones is a colimiting cone. In particular, there are * *natural isomorphisms colimd2D(Xd Yd) ~=(colimd2DXd) (colimd2DYd). in C. Proof.Consider (a) and the diagram X0 Y1 oo___X1oY1o_ || || || || fflffl|fflffl|fflffl|fflffl| X0 Y0 oo___X1oY0.o_ Look for a naturally occurring cone into X-1 Y-1, and verify this cone is init* *ial with respect to all cones. Use relations satisfied by the reflexive pairs and n* *ote that tensoring with any X 2 C preserves colimiting cones. Verification of (b) is sim* *ilar to (a), except instead of properties satisfied by reflexive pairs, use properties * *satisfied by filtered diagrams. Hence objectwise tensor product of diagrams in (C, , k) respects certain col* *im- iting cones. Objectwise circle product of diagrams in (SymSeq, O, I) and (Seq, * *^O, I) behave similarly. Proposition 8.3. (a)Suppose A-1 oo__A0_oo__A1oo_and B-1 oo__B0_oo__B1oo_are reflexive co- equalizer diagrams in SymSeq. Then their objectwise circle product (8.4) A-1 O B-1oo___A0O B0oo___A1OoB1o_ is a reflexive coequalizer diagram in SymSeq. (b)Suppose A, B: D-! SymSeqare filtered diagrams. Then objectwise circle product of their colimiting cones is a colimiting cone. In particular, th* *ere are natural isomorphisms colimd2D(AdO Bd) ~=(colimd2DAd) O (colimd2DBd). in SymSeq. HOMOTOPY THEORY OF MODULES OVER OPERADS 21 (c)For sequences, the corresponding statements in (a) and (b) remain true; i.e., when (SymSeq, O, I) is replaced by (Seq, ^O, I). Proof.Consider (a). We want to verify that (8.4)is a colimiting cone, hence suf* *fi- cient to verify it is initial with respect to all cones. Use the universal prop* *erty (5.3) together with (4.10)and Proposition 8.2(a). Verification of (b) is similar, exc* *ept use Proposition 8.2(b). Proposition 8.5. (a)Suppose A-1oo___A0oo___A1oo_is a reflexive coequalizer diagram in SymSeq and Z-1 oo__Z0_oo__Z1oo_is a reflexive coequalizer diagram in C. Then their objectwise evaluation A-1 O (Z-1)oo__A0O (Z0)oo___A1Oo(Z1)o_ is a reflexive coequalizer diagram in C. (b)Suppose A : D-! SymSeqand Z : D-! Care filtered diagrams. Then ob- jectwise evaluation of their colimiting cones is a colimiting cone. In pa* *rtic- ular, there are natural isomorphisms colimd2DAdO (Zd) ~=(colimd2DAd) O (colimd2DZd). in C. (c)For sequences, the corresponding statements in (a) and (b) remain true; i.e., when (SymSeq, O, I) is replaced by (Seq, ^O, I). Proof.Argue as a special case of Proposition 8.3 using the embedding in (5.7). 8.2. Free-forgetful adjunctions. Definition 8.6. Let O, R, S be operads. oLtO is the category of left O-modules and their morphisms. oRtO is the category of right O-modules and their morphisms. oBi(R,S)is the category of (R, S)-bimodules and their morphisms. oAlgO is the category of O-algebras and their morphisms. It will be useful to establish the following free-forgetful adjunctions. Proposition 8.7. (a)Let O, R, S be -operads. There are adjunctions OO-_//_ _-OO_// RO-OS//_ SymSeq oo___LtO, SymSeqoo___RtO, SymSeqoo___Bi(R,S), U U U with left adjoints on top and U the forgetful functor. (b)Let O, R, S be -operads. There are adjunctions _O_^O-//_ -O^O//_ R_^O-O^S//_ Seqoo___LtO, Seq oo___RtO, Seqoo___Bi(R,S), U U U with left adjoints on top and U the forgetful functor. Proof.To verify the first adjunction of (a), it is enough to show there are iso* *mor- phisms homLtO(O O B, C) ~=hom SymSeq(B, UC) 22 JOHN E. HARPER natural in B, C. For this, it is enough to verify the universal_property that g* *iven any map f : B-! UCin SymSeq, there exists a unique map f: O O B-!C in LtO such that f factors through the map B ~=idO BjOid//_O O B = U(O O B) __ via the map Uf . The other cases are similar. There are similar free-forgetful adjunctions for algebras over an operad. Proposition 8.8. Let O be a -operad and O0be an -operad. There are adjunc- tions OO(-)//_ O0^O(-)//_ C oUo__AlgO, CooU__AlgO0, with left adjoints on top and U the forgetful functor. Proof.Argue as in the proof of Proposition 8.7. 8.3. Construction of colimits. Proposition 8.9. Let O, R, S be operads. Reflexive coequalizers and filtered co* *l- imits exist in LtO, RtO, Bi(R,S), and AlgO, and are preserved (and created) by * *the forgetful functors in Propositions 8.7 and 8.8. Proof.Let O be a -operad and consider the case of left O-modules. Suppose A0 oo__A1_oo_is a reflexive pair in LtOand consider the solid commutative diagr* *am O O O_O_A-1oo__O O O O A0oo__OoOoO_O A1 ________ || || d0_______d1_____mOid|idOm| mOid|idOm| fflffl___fflffl___fflffl|fflffl|fflffl|fflffl| O OCA-1_oo_____OCOCA0oo______OoOoA1_C_CC_ ______ __|_____________|_______________________________* *________________ s0__m_________jOid|m____________|m__________________jOid________* *______________________ __fflffl___ __fflffl|_______fflffl|_________________________* *____ A-1 oo_________A0 oo_________A1oo_ in SymSeq, with bottom row the reflexive coequalizer diagram of the underlying reflexive pair in SymSeq. By Proposition 8.3, the rows are reflexive coequaliz* *er diagrams and hence there exist unique dotted arrows m, s0, d0, d1 in SymSeqwhich make the diagram commute. By uniqueness, s0 = jOid, d0 = mOid, and d1 = idOm. Verify that m gives A-1 the structure of a left O-module and also verify that t* *he bottom row is a reflexive coequalizer diagram in LtO. First check the diagram l* *ives in LtO, then check the colimiting cone is initial with respect to all cones in * *LtO. The case for filtered colimits is similar. The cases for RtO, Bi(R,S), and AlgO* *can be argued similarly. Argue similarly for -operads. Proposition 8.10. Let O, R, S be operads. All small colimits exist in LtO, RtO, Bi(R,S), and AlgO. Proof.Let O be a -operad and consider the case of left O-modules. Suppose A : D-! LtOis a small diagram. Want to show that colimA exists. This colimit HOMOTOPY THEORY OF MODULES OVER OPERADS 23 may be calculated by a reflexive coequalizer in LtOof the form, i (mOid)*oo_ j colimA ~=colim colimd2D(O O(Ad)colimd2D(OiOdOOOmAd))*oo_, provided the indicated colimits appearing in this reflexive pair exist in LtO. * *The underlying category SymSeq has all small colimits, and left adjoints preserve c* *ol- imiting cones, hence there is a commutative diagram (mOid)* colimd2D(O OoAd)o_colimd2D(OoOoO_O Ad) (idOm)* ~=|| |~=| fflffl| fflffl| O O colimd2DAdoo_OoOocolimd2D(O_O Ad) in LtO. The colimits in the bottom row exist since they are in the underlying category SymSeq(we have dropped the notation for the forgetful functor U), hence the colimits in the top row exist in LtO. Therefore colimA exists. The cases for RtO, Bi(R,S), and AlgOcan be argued similarly. Argue similarly for -operads. Example 8.11. For instance, if O is a -operad and A, B 2 LtO, then the coprodu* *ct A q B in LtOmay be calculated by a reflexive coequalizer of the form i oo___ j A q B ~=colim O O (A q B)oo__O O (O O A) q (O O B) in the underlying category SymSeq. The coproducts appearing inside the parenthe- ses are in the underlying category SymSeq. 8.4. Colimits for right modules. Colimits in right modules over an operad are particularly simple. Proposition 8.12. Let O be an operad. The forgetful functors from right O- modules in Proposition 8.7 preserve (and create) all small colimits. Proof.Let O be a -operad and suppose A : D-! RtOis a small diagram. By the proof of Proposition 8.10, colimA may be calculated in the underlying category by the colimit in SymSeq of the reflexive pair in the top row of the commutative diagram oo___ colimd2DAd OoOo_ colimd2D(AdO O) O O ~=|| |~=| fflffl| fflffl| colimd2D(AdOoO)o__colimd2D(AdOoOoO_O), and since the functor - O O: SymSeq-! SymSeqpreserves colimiting cones, this is the same as calculating the colimit of the bottom row in SymSeq, which is the colimit of the underlying diagram of A in SymSeq. We have dropped the notation for the forgetful functor U. Argue similarly for -operads. 24 JOHN E. HARPER 8.5. Construction of limits. Proposition 8.13. Let O, R, S be operads. All small limits exist in LtO, RtO, Bi(R,S), and AlgO, and are preserved (and created) by the forgetful functors in Propositions 8.7 and 8.8. Proof.This can be argued similar to the proof of Proposition 8.9. 9.Basic constructions for modules In this section we present some basic constructions for modules. The material in this section is largely influenced by Rezk [38]. Compare the following defin* *ition with [29, Chapter VI.5]. 9.1. Circle products (mapping sequences) over an operad. Definition 9.1. Let R be a -operad (resp. -operad), A a right R-module, and B a left R-module. Define A OR B 2 SymSeq(resp. A ^ORB 2 Seq) by the reflexive coequalizer i d0 j A OR B:= colim A O Boo__A_OoRoO_B , d1 i i d0 jj resp. A ^ORB:= colim A ^OBoo__Ao^ORo^OB_, d1 with d0 induced by m : A O R-!A and d1 induced by m : R O B-!B (resp. d0 induced by m : A ^OR-!Aand d1 induced by m : R ^OB-!B). Definition 9.2. Let S be a -operad (resp. -operad) and let B and C be right S- modules. Define MapOS(B, C) 2 SymSeq(resp. Map ^OS(B, C) 2 Seq) by the equalizer i d0 j MapOS(B, C):= lim MapO(B, C)____//_//_MapO(B O S,,C) d1 i i d0 jj resp. Map^OS(B, C):= lim Map^O(B, C)___//_//_Map^O(B ^OS,,C) d1 with d0 induced by m : B O S-!B and d1 induced by m : C O S-!C (resp. d0 induced by m : B ^OS-!Band d1 induced by m : C ^OS-!C). 9.2. Adjunctions. Compare the following adjunctions with [29, Chapter VI.8]. Proposition 9.3. Let Q, R, S be -operads. There are isomorphisms, (9.4) homRtS(A O B, C)~=hom(A, MapOS(B, C)), (9.5) hom(A OR B, C)~=homRtR(A, MapO(B, C)), (9.6) hom (Q,S)(A OR B,~C)=hom(Q,R)(A, MapOS(B, C)), natural in A, B, C. Remark 9.7. In (9.4), A is a symmetric sequence, and both B and C have right S-module structures. In (9.5), A has a right R-module structure, B has a left R- module structure, and C is a symmetric sequence. In (9.6), A has a (Q, R)-bimod* *ule structure, B has a (R, S)-bimodule structure, and C has a (Q, S)-bimodule struc- ture. HOMOTOPY THEORY OF MODULES OVER OPERADS 25 Proof.The natural correspondence (9.6)implies both (9.4)and (9.5), and its proof can be built up from those for (9.4)and (9.5). Use the natural correspondences (5.17)together with the commutative diagrams satisfied by each object as defined in Section 6. There is a corresponding statement for -operads. Proposition 9.8. Let Q, R, S be -operads. There are isomorphisms, homRtS(A ^OB, C)~=hom(A, Map^OS(B, C)), hom (A ^ORB, C)~=homRtR(A, Map^O(B, C)), hom (Q,S)(A ^ORB,~C)=hom(Q,R)(A, Map^OS(B, C)), natural in A, B, C. Proof.Argue as in the proof of Proposition 9.3. 9.3. Cancellation and associativity properties. Proposition 9.9. Let R be a -operad (resp. -operad), A a right R-module, and B a left R-module. There are natural isomorphisms A OR R ~=A R OR B ~=B, resp. A ^ORR ~=A R ^ORB ~=B . Proof.Suppose R is a -operad. We want to verify A is naturally isomorphic to a particular coequalizer. Look for a naturally occurring cone into A and verify i* *t is initial with respect to all cones. The other cases are similar. Proposition 9.10. Let R, S be -operads (resp. -operads), A a right R-module, B an (R, S)-bimodule, and C a left S-module. There are natural isomorphisms (A OR B) OS C~=A OR (B OS C), resp. (A ^ORB) ^OSC~=A ^OR(B ^OSC) . Proof.Use Proposition 8.3. 9.4. Change of operads adjunction. Proposition 9.11. Let f : R-! Sbe a morphism of -operads (resp. -operads). There is an adjunction __f*_// LtRoo___LtS, f* with left adjoint f* := S OR - (resp. f* := S ^OR-) and f* the forgetful functo* *r. In particular, there are isomorphisms hom LtS(S OR A, B)~=homLtR(A, f*(B)), * resp.hom LtS(S ^ORA, B)~=homLtR(A, f (B)) , natural in A, B. Proof.Look for natural transformations j : id-!f*f*and " : f*f*-!idand verify * f*" that f*_f*j//_f*f*f*"f*//_f*and f*_jf_//_f*f*f*__//f*each factor the identity. 26 JOHN E. HARPER 10.Basic constructions for algebras Here we present some corresponding constructions for algebras. 10.1. Circle products over an operad. Definition 10.1. Let R be a -operad (resp. -operad), A a right R-module, and Z an R-algebra. Define A OR (Z) 2 C (resp. A ^OR(Z) 2 C) by the reflexive coequalizer i d0 j A OR (Z):= colim A O (Z)oo__(AoOoR)_O (Z), d1 i i d0 jj resp.A ^OR(Z):= colim A ^O(Z)oo__(Ao^OR)o^O(Z)_, d1 with d0 induced by m : A O R-!Aand d1 induced by m : R O (Z)-!Z(resp. d0 induced by m : A ^OR-!Aand d1 induced by m : R ^O(Z)-!Z). 10.2. Cancellation and associativity properties. Proposition 10.2. Let R be a -operad (resp. -operad) and Z an R-algebra. There are natural isomorphisms R OR (Z) ~=Z, resp. R ^OR(Z) ~=Z . Proof.Argue as a special case of Proposition 9.9 by taking B := ^Z. Proposition 10.3. Let R, S be -operads (resp. -operads), A a right R-module, B an (R, S)-bimodule, and Z an S-algebra. There are natural isomorphisms (A OR B) OS (Z)~=A OR B OS (Z) , i j resp.(A ^ORB) ^OS(Z)~=A ^ORB ^OS(Z) . Proof.Argue as a special case of Proposition 9.10 by taking C := ^Z. 10.3. Change of operads adjunction. Proposition 10.4. Let f : R-! Sbe a morphism of -operads (resp. -operads). There is an adjunction f* // AlgR_____AlgS,oo_ f* with left adjoint f* := S OR (-) (resp. f* := S ^OR(-)) and f* the forgetful fu* *nctor. In particular, there are isomorphisms homAlgS(S OR (A),~B)=homAlgR(A, f*(B)), * resp. homAlgS(S ^OR(A),~B)=homAlgR(A, f (B)) , natural in A, B. Proof.Argue as in the proof of Proposition 9.11. HOMOTOPY THEORY OF MODULES OVER OPERADS 27 11. Model categories and definitions In this section we establish some notation; the definitions appearing below a* *re only intended to make precise Basic Assumption 1.1. We assume the reader is familiar with model categories. A useful introduction is given in [4], from whi* *ch we have taken the model category axioms listed below. See also the original articl* *es by Quillen [36, 37], and the more recent [3, 15, 21, 22]. In this paper, our primary method of constructing model categories from exist* *ing ones involves the additional structure of a cofibrantly generated model category together with (possibly transfinite) small object arguments. Schwede and Shipley provide an account of these techniques in [39, Section 2] which will be suffici* *ent for our purposes. The reader unfamiliar with the small object argument may consult [4, Section 7.12] for a useful introduction; after which the (possibly transfin* *ite) versions in [21, 22, 39] appear quite natural. 11.1. Model categories. Definition 11.1. In C, a map i : A-! Bhas the left lifting property (LLP) with respect to a map p : X-! Yif every solid commutative diagram of the form (11.2) A _____//X>>___ ____ i||,____p||____ fflffl|fflffl|____ B _____//Y has a lift ,. In C, a map p : X-! Yhas the right lifting property (RLP) with re* *spect to a map i : A-! Bif every solid commutative diagram of the form (11.2)has a lift ,. Definition 11.3. A model category is a category C with three distinguished sub- categories of C: oW the subcategory of weak equivalences oFibthe subcategory of fibrations oCof the subcategory of cofibrations each of which contains all objects of C. A map which is both a fibration (resp. cofibration) and a weak equivalence is called an acyclic fibration (resp. acyc* *lic cofibration). An object X 2 C is called cofibrant (resp. fibrant) if ;-! X is* * a cofibration (resp. X-! * is a fibration). We require the following axioms: (MC1) Finite limits and colimits exist in C. (MC2) If f and g are maps in C such that gf is defined and if two of the three maps f, g, gf are weak equivalences, then so is the third. (MC3) If f is a retract of g and g is a fibration, cofibration, or weak equival* *ence, then so is f. (MC4) Cofibrations have the LLP with respect to acyclic fibrations. Acyclic cof* *i- brations have the LLP with respect to fibrations. (MC5) Any map f can be factored in two ways: (i) f = pi, where i is a cofibrati* *on and p is an acyclic fibration, and (ii) f = pi, where i is an acyclic cof* *ibration and p is a fibration. Remark 11.4. The definition above describes what was originally called a "close* *d" model category [37]; following [4] we have dropped the term "closed" in this pa* *per. 28 JOHN E. HARPER 11.2. Cofibrantly generated model categories. To construct model category structures on (symmetric) sequences and on modules and algebras over an operad, we will require some extra conditions on the model category structure of C. Definition 11.5. A model category C is cofibrantly generated if it has all small limits and colimits and there exists a set I of cofibrations and a set J of acy* *clic cofibrations satisfying certain properties [39, Definition 2.2]; these properti* *es will imply that I and J completely determine the model category structure. Remark 11.6. The maps in I (resp. J) are called generating cofibrations (resp. generating acyclic cofibrations). The properties referred to in Definition 11.5* * are useful for creating a model category structure on the target D of a left adjoin* *t, from an existing cofibrantly generated model structure on the source C. Sometimes the single left adjoint is replaced by a set of left adjoints with the same target. 11.3. Monoidal model categories. Definition 11.7. A monoidal model category is a model category C with a sym- metric monoidal closed structure (C, , k) such that the following axiom is sat* *isfied: (ENR) If j : A-! Bis a cofibration and p : X-! Yis a fibration, then the pullba* *ck corner map Map (B, X)____//_Map(A, X) xMap(A,YM)ap(B, Y ) is a fibration that is an acyclic fibration if either j or p is a weak eq* *uivalence. Remark 11.8. A model category with a symmetric monoidal closed structure (C, ,* * k) satisifies the axiom (ENR) if and only if it satisfies the axiom: (ENR')If i : K-! Land j : A-! Bare cofibrations, then the pushout corner map ` L A K A K B ____//_L B is a cofibration that is an acyclic cofibration if either i or j is a weak equivalence. In particular, Definition 11.7 is equivalent to the definition by Schwede and S* *hip- ley [39, Section 3] of a monoidal model category. Lewis and Mandell [28] refer * *to (ENR) as the enrichment axiom, and also include an additional condition involvi* *ng the unit of which we will not require in this paper. 12.Model categories for G-objects and (symmetric) sequences So far in this paper, except in Section 1, we have only used the property that (C, , k) is a symmetric monoidal closed category with all small limits and col* *imits. In this section, we begin to make use of the model category assumptions on (C, * * , k) described in Basic Assumption 1.1. In this section, we establish natural model catgory structures on G-objects a* *nd (symmetric) sequences, and investigate how well tensor product and circle produ* *ct mesh with this model category structure. 12.1. Model category structure for G-objects. Theorem 12.1. Let G be a finite group. Then the category CGopof right G objects has a natural model category structure. The weak equivalences are the objectwise weak equivalences and the fibrations are the objectwise fibrations. The model s* *truc- ture is cofibrantly generated. HOMOTOPY THEORY OF MODULES OVER OPERADS 29 Proof.The model category structure on CGopis created by the adjunction _-.G//_op C ooU__CG with left adjoint on top and U the forgetful functor. Define a map f in CGopto * *be a weak equivalence (resp. fibration)oifpUf is a weak equivalence (resp. fibrati* *on) in C. Define a map f in CG toobepa cofibration if it has the LLP with respect * *to all acyclic fibrations in CG . To verify this gives a model category structure* *, argue as in the proof of Theorem 12.2. By construction, the model category structure * *is cofibrantly generated. 12.2. Model category structure for (symmetric) sequences. Theorem 12.2. The symmetric monoidal closed categories (SymSeq, , 1) of sym- metric sequences and (Seq, ^, 1) of sequences have natural model category struc- tures. The weak equivalences are the objectwise weak equivalences and the fibr* *a- tions are the objectwise fibrations. The model structures are cofibrantly gener* *ated and give (SymSeq, , 1) and (Seq, ^, 1) the structure of monoidal model categor* *ies. Proof.The model category structure on SymSeqis created by the set of adjunctions -._s//_op____//_ ____//_ C oUo__C s oo___SymSeq0oo___SymSeq, s 0, in*s with left adjoints on top, U the forgetful functor, and inclusion ins: ops-! 0* *op. The right-hand adjunction is the equivalence of categories (3.5). Define a map * *f in SymSeq to be a weak equivalence (resp. fibration) if U(in*s(f)) is a weak equiv* *alence (resp. fibration) in C for every s 0. Define a map f in SymSeqto be a cofibra* *tion if it has the LLP with respect to all acyclic fibrations in SymSeq. We want to verify the model category axioms (MC1)-(MC5). We already know (MC1) is satisfied, and verifying (MC2) and (MC3) is clear. The arguments in the proof of [39, Lemma 2.3] use (possibly transfinite) small object arguments to v* *erify (MC5). The first part of (MC4) is satisfied by definition, and the second part * *of (MC4) follows from the usual lifting and retract argument, as described in the * *proof of [39, Lemma 2.3]. This verifies the model category axioms. By construction, t* *he model category is cofibrantly generated. By Theorem 12.4, this gives (SymSeq, * *, 1) the structure of a monoidal model category. The model category structure on Seqis created by the set of adjunctions C ____//_CUopsoo_//_Seq0oo_//_Seqoo_, s 0, in*s with left adjoints on top, U the forgetful functor, and inclusion ins: ops-! 0* *op. Argue as in the SymSeqcase. Remark 12.3. Since the right adjoints in this proof all commute with filtered c* *ol- imits, the smallness conditions needed for the (possibly transfinite) small obj* *ect arguments in [39, Lemma 2.3] are satisfied. Also, condition (1) of [39, Lemma 2.3] is easily verified since colimits in SymSeq and Seq are computed objectwise in the underlying category, and since acyclic cofibrations in C are preserved u* *nder coproducts, pushouts, and transfinite compositions. 30 JOHN E. HARPER 12.3. Pushout corner map for tensor products. Here we verify that tensor products of symmetric sequences mesh nicely with the model structure. Theorem 12.4. (a)In symmetric sequences (resp. sequences), if i : K-! Land j : A-! Bare cofibrations, then the pushout corner map ` L A K A K B ____//_L B, ` resp. L ^A K ^AK ^B ____//_L ^B, is a cofibration that is an acyclic cofibration if either i or j is a wea* *k equiv- alence. (b)In symmetric sequences (resp. sequences), if j : A-! Bis a cofibration and p : X-! Yis a fibration, then the pullback corner map Map (B, X)____//_Map(A, X) xMap (A,YM)ap (B, Y,) ^ ^ ^ resp. Map (B, X)____//_Map(A, X) xMap^(A,YM)ap (B, Y,) is a fibration that is an acyclic fibration if either j or p is a weak eq* *uivalence. We prove this theorem in section 12.8. 12.4. Pushout corner map for circle products. These model structures also mesh nicely with circle product, provided an additional cofibrancy condition is satisfied. A version of the following theorem is given by Rezk [38] for symmetr* *ic sequences of simplicial sets, using a model category structure with fewer weak equivalences. Theorem 12.5. Let A be a cofibrant symmetric sequence (resp. cofibrant se- quence). (a)In symmetric sequences (resp. sequences), if i : K-! Land j : A-! Bare cofibrations, then the pushout corner map ` L O A KOA K O B___//_L O,B ` resp. L ^OA K ^OAK ^OB___//L ^OB, is a cofibration that is an acyclic cofibration if either i or j is a wea* *k equiv- alence. (b)In symmetric sequences (resp. sequences), if j : A-! Bis a cofibration and p : X-! Yis a fibration, then the pullback corner map Map O(B, X)____//MapO(A, X) xMapO(A,YM)apO(B,,Y ) ^O ^O resp. Map ^O(B, X)___//Map(A, X) xMap^O(A,YM)ap(B, Y,) is a fibration that is an acyclic fibration if either j or p is a weak eq* *uivalence. We prove this theorem in section 12.8. HOMOTOPY THEORY OF MODULES OVER OPERADS 31 12.5. Fixed points and the pullback corner map. Proposition 12.6. Let G be a finite group and H G a subgroup. In CGop, suppose j : A-! Bis a cofibration and p : X-! Yis a fibration. Then in C the pullback corner map Map(B, X)H_____//Map(A, X)H xMap(A,Y )HMap(B, Y )H is a fibration that is an acyclic fibration if either j or p is a weak equivale* *nce. Proof.Suppose j : A-! Bis a cofibration and p : X-! Yis an acyclic fibration. Let i : C-! Dbe a cofibration in C. We want to verify the pullback corner map satisfies the right lifting property with respect to i. (12.7) C ______________//_Map(B, X)H | ____55________ | ____________| | ___________ | |fflffl_____________ fflffl| D ____//_Map(A, X)H xMap(A,Y )HMap(B, Y )H . The solid commutativeodiagramp(12.7)in C has a lift if and only if the solid di* *agram (12.8)in CG has a lift, (12.8) C . (H\G)_____________//_Map(B,4X)4__ _______ | ___________|__ | _____________ | fflffl|_______ fflffl| D . (H\G)____//_Map(A, X) xMap(A,YM)ap(B, Y ). if and only if the solid diagram (12.9)in CGophas a lift. (12.9) A ___________________//_Map(D . (H\G), X) | _____33__________ | _______________| | ________________ | fflffl|__________________ fflffl| B ____//_Map(C . (H\G), X) xMap(C.(H\G),YM)ap(D . (H\G), Y ). Hence it is sufficient to verify that theoright-handpvertical map in (12.9)is an acyclic fibration in C, and hence in CG . The map i . id: C . (H\G)-!D . (H\G) is isomorphic in C to a coproduct of cofibrations in C, hence is itself a cofib* *ration in C, and the (ENR) axiom finishes the argument for this case. The other cases * *are similar. 12.6. Calculations for mapping sequences. Proposition 12.10. Let B and X be symmetric sequences and t 1. Then for each s 0 there is a natural isomorphism in C, Y Map (B[s, t], X[s]) s ~= Map (B[s1] . . .B[st], X[s]) s1x...x st. s1+...+st=s Proof.Use the calculation in Proposition 4.3. 32 JOHN E. HARPER 12.7. Tensor products and (acyclic) cofibrations for G-objects. Proposition 12.11. Let G1, . .,.Gn be finite groups. (a)Suppose for ko=p1, . .,.n that jk: Ak-! Bkis a cofibration between cofibr* *ant objects in CGk . Then the induced map j1 . . .jn: A1 . . .An-!B1 . . .Bn is a cofibration in C(G1x...xGn)opthat is an acyclic cofibration if each * *jk is a weak equivalence. op (b)Suppose for k = 1, . .,.n that Ak is a cofibrant objectoinpCGk . Then A1 . . .An is a cofibrant object in C(G1x...xGn). Remark 12.12. By the right-hand adjunction in (2.10), - Ak preserves initial objects. In particular, if A1, . .,.An in the statementoofp(a) are all initial * *objects, then A1 . . .An is an initial object in C(G1x...xGn). Proof.For each n, statement (b) is a special case of statement (a), hence it is sufficient to verify (a). By induction on n, it is enough to verify the case n * *= 2. Supposeoforpk = 1, 2 that jk: Ak-! Bkis a cofibration between cofibrant objects in CGk . The induced map j1 j2: A1 A2- !B1 B2 factors as j1 id id j2 A1 A2 ____//_B1 A2___//_B1 B2, hence it is sufficient to verify each of theseoispa cofibration in C(G1xG2)op. * *Consider any acyclic fibration p : X-! Yin C(G1xG2) . We want to show that j1 id has the LLP with respect to p. A1 A2 ____//_X;;___A1___//Map(A2,9X)G29_ | ____|_____ | ______|_____ | ______| | _______ | fflffl|__fflffl|__fflffl|___fflffl|______ B1 A2 ____//_Y B1_____//Map(A2, Y )G2 The left-hand solid commutative diagramoinpC(G1xG2)ophas a lift if and only ifop the right-hand solid diagram in CG1 has a lift. We know A2 is cofibrant in CG2 , hence by Proposition 12.6 the right-hand solid diagramohaspa lift, finishing the argument thatoj1pidis a cofibration in C(G1xG2) . Similarly, id j2 is a cofibra* *tion in C(G1xG2) . The case for acyclic cofibrations is similar. The following proposition is also useful. Proposition 12.13. Let G1 and G2 beofinitepgroups. Suppose for k = 1, 2 that jk: Ak-! Bkis a cofibration in CGk . Then the pushout corner map (12.14) B1 A2qA1 A2A1 B2 ____//_B1 B2 is a cofibration in C(G1xG2)opthat is an acyclic cofibration if either j1 or j2* * is a weak equivalence. op Proof.Suppose for k = 1, 2 that jk: Ak-! Bkisoapcofibration in CGk . Consider any acyclic fibration p : X-! Yin C(G1xG2) . We want to show that the pushout HOMOTOPY THEORY OF MODULES OVER OPERADS 33 corner map (12.14)has the LLP with respect to p. The solid commutative diagram B1 A2qA1 A2A1 B2 ____//_X66_______ _____ | ________|_ | __________ | fflffl|_____ fflffl| B1 B2 __________//_Y in C(G1xG2)ophas a lift if and only if the solid diagram A1________________//Map(B2,4X)G24__ ________ | ____________|__ | _____________ | fflffl|______________ fflffl| B1_____//Map(A2, X)G2 xMap(A2,Y )G2Map(B2, Y )G2 op Gop in CG1 has a lift. We know A2-! B2is a cofibration in C 2 , henceoPropositionp1* *2.6 finishes the argument that (12.14)is a cofibration in C(G1xG2) . The other cases are similar. 12.8. Proofs for the pushout corner map theorems. Proof of Theorem 12.4.Statements (a) and (b) are equivalent. This can be verifi* *ed using the natural correspondence (4.17)together with the various lifting charac* *ter- izations [4, Proposition 3.13] satisfied by any closed model category. Hence it* * is sufficient to verify statement (b). Suppose j : A-! Bis an acyclic cofibration and p : X-! Yis a fibration. We want to verify each pullback corner map Map (B, X)[t]___//_Map(A, X)[t] xMap (A,Y )[t]Map(B, Y )[t], is an acyclic fibration in C. By Proposition 4.20 it is sufficient to show each* * map Map(B[s], X[t + s]) s | | fflffl| Map(A[s], X[t + s]) sxMap(A[s],Y [t+Ms])asp(B[s], Y [t + s]) s is an acyclic fibration in C. Proposition 12.6 completes the proof for this cas* *e. The other cases are similar. Proof of Theorem 12.5.Statements (a) and (b) are equivalent. This can be verifi* *ed using the natural correspondence (5.17)together with the various lifting charac* *ter- izations [4, Proposition 3.13] satisfied by any closed model category. Hence it* * is sufficient to verify statement (b). Suppose j : A-! Bis an acyclic cofibration between cofibrant objects and p : * *X-! Y is a fibration. We want to verify each pullback corner map Map O(B, X)[t]__//_MapO(A, X)[t] xMapO(A,Y )[t]MapO(B, Y )[t], 34 JOHN E. HARPER is an acyclic fibration in C. If t = 0, this map is an isomorphism by a calcula* *tion in Proposition 5.22. If t 1, by Proposition 5.20 it is sufficient to show eac* *h map Map(B[s, t], X[s]) s | | fflffl| Map(A[s, t], X[s]) sxMap(A[s,t],Y [s])Msap(B[s, t], Y [s]) s is an acyclic fibration in C. By Propositions 12.10 and 12.6, it is enough to v* *erify each map A[s1] . . .A[st]_//_B[s1] . . .B[st], is an acyclic cofibration in C( s1x...x st)op. Proposition 12.11 completes the * *proof for this case. The other cases are similar. 12.9. A special case. Remark 12.15. Consider Theorem 12.5. It is useful to note that L O ; and K O ; may not be isomorphic, and similarly Map O(;, X) and Map O(;, Y ) may not be isomorphic. On the other hand, Theorem 12.5 reduces the proof of the following proposition to a trivial inspection at the emptyset 0. Proposition 12.16. Let B be a cofibrant symmetric sequence (resp. cofibrant sequence). (a)In symmetric sequences (resp. sequences), if i : K-! Lis a cofibration, then the induced map K O B-! L O B , resp. K ^OB-! L ^OB , is a cofibration that is an acyclic cofibration if i is a weak equivalenc* *e. (b)In symmetric sequences (resp. sequences), if p : X-! Yis a fibration, then the induced map ^O ^O Map O(B, X)-! Map O(B, Y ) , resp. Map (B, X)-! Map (B, Y ) , is a fibration that is an acyclic fibration if p is a weak equivalence. Proof.Statements (a) and (b) are equivalent. This can be verified using the nat* *ural correspondence (5.17)together with the various lifting characterizations satisf* *ied by any closed model category. Hence it is sufficient to verify (b). Suppose B is cofibrant and p : X-! Yis an acyclic fibration. We want to verify each induced map MapO(B, X)[t]-! Map O(B, Y )[t] is an acyclic fibration in C. Theorem 12.5(b) implies this for t 1. For t =* * 0, it is enough to note that X[0]-! Y [0] is an acyclic fibration. The other case* * is similar. 13.Proofs In this section we give proofs of Theorems 1.2 and 1.4. When working with left modules over an operad, we are led naturally to replace (C, , k) with (SymSeq,* * , 1) as the underlying monoidal model category, and hence to working with symmetric arrays. HOMOTOPY THEORY OF MODULES OVER OPERADS 35 13.1. Arrays and symmetric arrays. Definition 13.1. oA symmetric array in C is a symmetric sequence in SymSeq; i.e. a functor A : op-! SymSeq. An array in C is a sequence in Seq; i.e. a functor A : op-! Seq. op op op oSymArray := SymSeq ~=C x is the category of symmetricoarrayspopop in C and their natural transformations. Array:= Seq ~=C x is the category of arrays in C and their natural transformations. All of the statements and constructions which were previously described in te* *rms of (C, , k) are equally true for (SymSeq, , 1) and (Seq, ^, 1), and we usuall* *y cite and use the appropriate statements and constructions without further comment. Theorem 13.2. The categories SymArrayof symmetric arrays and Arrayof arrays have natural model category structures. The weak equivalences are the objectwise weak equivalences and the fibrations are the objectwise fibrations. The model s* *truc- tures are cofibrantly generated. Proof.This is a special case of Theorem 12.2 with (C, , k) replaced by (SymSeq* *, , 1) and (Seq, ^, 1). 13.2. Model category structures in -operad case. Proof of Theorem 1.4.The model category structure on LtO(resp. AlgO) is created by the adjunction OO-_//_ i OO(-)//_ j SymSeq oo__LtO_ resp. Coo___AlgO U U with left adjoint on top and U the forgetful functor. Define a map f in LtOto be a weak equivalence (resp. fibration) if U(f) is a weak equivalence (resp. fibra* *tion) in SymSeq. Similarly, define a map f in AlgO to be a weak equivalence (resp. fibration) if U(f) is a weak equivalence (resp. fibration) in C. Define a map f* * in LtO (resp. AlgO) to be a cofibration if it has the LLP with respect to all acyc* *lic fibrations in LtO(resp. AlgO). Consider the case of LtO. We want to verify the model category axioms (MC1)- (MC5). We already know (MC1) is satisfied, and verifying (MC2) and (MC3) is clear. The arguments in the proof of [39, Lemma 2.3] use (possibly transfinite) small object arguments to reduce (MC5) to verifying Proposition 13.4 below. The first part of (MC4) is satisfied by definition, and the second part of (MC4) fo* *llows from the usual lifting and retract argument, as described in the proof of [39, * *Lemma 2.3]. This verifies the model category axioms. By construction, the model categ* *ory is cofibrantly generated. Consider the case of AlgO. Argue similar to the case of LtO, except use Propo- sition 13.4 together with Remark 13.7. By construction, the model category is cofibrantly generated. Remark 13.3. Since the forgetful functors in this proof commute with filtered c* *ol- imits, the smallness conditions needed for the (possibly transfinite) small obj* *ect arguments in [39, Lemma 2.3] are satisfied. 36 JOHN E. HARPER 13.3. Analysis of pushouts for -operad case. Proposition 13.4. Let O be a -operad and A 2 LtO. Assume every object in SymArrayis cofibrant. Then every (possibly transfinite) composition of pushouts* * in LtO of the form (13.5) O O X_________//_A |idOi| j|| fflffl| fflffl| O O Y____//_A qOOX O O Y, such that i : X-! Yan acyclic cofibration in SymSeq, is a weak equivalence in t* *he underlying category SymSeq. Remark 13.6. The proof of this proposition verifies the stronger statement that* * the (possibly transfinite) composition of pushouts in LtOis an acyclic cofibration * *in the underlying category SymSeq. Remark 13.7. If X, Y, A are concentrated at 0, then the pushout diagram (13.5) is concentrated at 0. To verify this, use Proposition 5.8 and the construction * *of colimits described in the proof of Proposition 8.10. This subsection is devoted to proving Proposition 13.4, which we cited above * *in the proof of Theorem 1.4. A first step in analyzing the pushouts in (13.5)is an analysis of certain coproducts. The following proposition is motivated by a sim* *ilar construction given in [14, Section 2.3] and [31, Section 13] in the context of * *algebras over an operad. Proposition 13.8. Let O be a -operad, A 2 LtO, and Y 2 SymSeq. Consider any coproduct in LtOof the form (13.9) A q (O O Y ). There exists a symmetric array OA and natural isomorphisms a A q (O O Y ) ~=OA O (Y ) = OA[q] qY q q 0 in the underlying category SymSeq. If Q 2 and q := |Q|, then OA[Q] is natural* *ly isomorphic to a colimit of the form ` ` pod0o_` 'p OA[Q] ~=colim p 0O[p + q] pA ood1_p 0O[p + q] p(O O A) , in SymSeq, with d0 induced by operad multiplication and d1 induced by m : O O A* *-!A. Proof.The coproduct in (13.9)may be calculated by a reflexive coequalizer in LtO of the form, i d0 j A q (O O Y ) ~=colim (O O A) q (O O Yo)o(O_OoOoO_A) q (O O Y,) d1 HOMOTOPY THEORY OF MODULES OVER OPERADS 37 The maps d0 and d1 are induced by maps m : O O O-!O and m : O O A-!A, respectively. There are natural isomorphisms in the underlying category SymSeq, (O O A) q (O O Y~)=O O (A q Y ) ~=a O[t] t(A q Y ) t t 0i j ~=a a O[p + q] pA p qY q, q 0p 0 and similarly, a ia j (O O O O A) q (O O Y ) ~= O[p + q] p(O O A) p qY q. q 0 p 0 The maps d0 and d1 similarly factor in the underlying category SymSeq. Remark 13.10. We have used the natural isomorphisms a (A q Y ) t ~= p+q. px qA p Y q, p+q=t in the proof of Proposition 13.8. Definition 13.11. Let i : X-! Ybe a morphism in SymSeq and t 1. Define Qt0:= X tand Qtt:= Y t. For 0 < q < t define Qtqinductively by the pushout diagrams pr* t t. t-qx qX (t-q)Qqq-1___//_Qq-1 i*|| i*|| fflffl| fflffl| t. t-qx qX (t-q)Y q____//_Qtq in SymSeq t. Remark 13.12. The construction Qt-1tcan be thought of as a t-equivariant versi* *on of the colimit of a punctured t-cube (Proposition 13.23). If the category C is * *pointed, there is a natural isomorphism Y t=Qtt-1~=(Y=X) t. The following proposition provides a useful description of certain pushouts of left modules, and is motivated by a similar construction given in [6, section 1* *2] in the context of simplicial multifunctors of symmetric spectra. Proposition 13.13. Let O be a -operad, A 2 LtO, and i : X-! Yin SymSeq. Consider any pushout diagram in LtOof the form, f (13.14) O O X __________//_A idOi|| || fflffl| fflffl| O O Y_____//A q(OOX)(O O Y ). The pushout in (13.14)is naturally isomorphic to a filtered colimit of the form i j1 j2 j3 j (13.15) A q(OOX)(O O Y ) ~=colim A0 ____//_A1___//A2___//_. . . 38 JOHN E. HARPER in the underlying category SymSeq, with A0 := OA[0] ~=A and At defined induc- tively by pushout diagrams in SymSeq of the form (13.16) OA[t] tQtt-1f*//_At-1. id||ti* jt|| fflffl| ,t fflffl| OA[t] tY t____//_At Proof.The pushout in (13.14)may be calculated by a reflexive coequalizer in LtO of the form i _i j A q(OOX)(O O Y ) ~=colim A q (O O Yo)o__A_qo(OoO_X) q (O O Y ) f _ __ The maps i and f are induced by maps idO i* and idO f* which fit into the commutative diagram od0o_ (13.17) OA O (X q Yo)o__O O (A q X q Yo)o__O O (O O A) q X q Y d1 _i|_| || || |f| idOi*|idOf*| idOi*|idOf*| fflffl|fflffl| fflffl|fflffl|d0 fflffl|fflffl| OA O (Yo)o______O O (A q Yo)o______OoOo(O_O A) q Y ) d1 in LtO, with rows reflexive coequalizer diagrams, and maps i* and f* in SymSeqi* *n- duced by i : X-! Yand f : X-! A. By Proposition 13.8, the coproduct in (13.14) may be calculated by the colimit of the left hand column of (13.17)in the under- lying category SymSeq. We want to reconstruct this colimit via a suitable filte* *red colimit. __ Using (13.17), there exist maps fq,pwhich make the diagrams ` ` ijinq,ip j OA O (X q Y ) ~= oo___ OA[p + q] x X p Y q q 0p 0 _p q __ |_f ______ | 9_fq,p___ fflffl| ij fflffl____ ` inq i j OA O (Y ) ~= oo____________ OA[q] Y q t 0 q _ in SymSeqcommute. Similarly, there exist maps iq,pwhich make the diagrams ` ` ijinq,ip j OA O (X q Y ) ~= oo___ OA[p + q] x X p Y q q 0p 0 _p q __ |_i ______ | 9_iq,p___ fflffl| ij fflffl____ ` inp+q i j OA O (Y ) ~= oo________ OA[p + q] Y (p+q) t 0 p+q HOMOTOPY THEORY OF MODULES OVER OPERADS 39 in SymSeq commute._Let_'_: OA O (Y-)!.be a morphism in SymSeq and define 'q := 'inq. Then 'i= 'f if and only if the diagrams _f OA[p + q] px qX p Y _qq,p//_OA[q] qY q |_iq,p | | 'q| fflffl| 'p+q fflffl|| OA[p + q] p+qY (p+q)_________//. _ __ commute for every p, q 0. Since iq,0= idand fq,0= id, it is sufficient to con* *sider q 0 and p > 0. To motivate the construction (13.16), it is useful to describe a preliminary * *con- struction which also calculates the pushout in (13.14). Define A0 := OA[0] ~=A and for each t 1 define Atby the pushout diagram ` OA[p + q] px qX p Y q f* (13.18) p+q=t __________//At-1 q 0,p>0 | |i*| |jt fflffl| ,t fflffl|| OA[t] tY _t__________________//At __ _ in SymSeq. The maps f* and i* are induced by the appropriate maps fq,pand iq,p. Verify that (13.15)is satisfied.__ The collection of maps fq,pand iq,psatisfy many compatibility relations. This suggests we replace the coproduct in (13.18), which is isomorphic to h i OA[t] t(X q Y ) t- Y t, with an appropriate (possibly smaller) pushout construction. Here, (XqY ) t-Y t means the coproduct of all factors in (X q Y ) t except Y t. Define A0 := OA[0] ~=A and for each t 1 define At by the pushout diagram_ (13.16)in_SymSeq. The maps f* and i* are induced by the appropriate maps fq,p and iq,p. Verify that (13.15)is satisfied. Proposition 13.19. Let O be a -operad, A 2 LtO, and i : X-! Ya cofibration (resp. acyclic cofibration) in SymSeq. Assume every object in SymArrayis cofibr* *ant. Consider any pushout diagram in LtOof the form, (13.20) O O X_________//_A |idOi| j|| fflffl| fflffl| O O Y____//_A qOOX O O Y. Then each map jt: At-1-!Atin the filtration (13.15)is a cofibration (resp. acyc* *lic cofibration) in SymSeq. In particular, j is a cofibration (resp. acyclic cofibr* *ation) in the underlying category SymSeq. Proof.Suppose i : X-! Yis an acyclic cofibration in SymSeq. We want to show each jt: At-1-!Atis an acyclic cofibration in SymSeq. By the construction of jt in Proposition 13.13, it is sufficient to verify each id ti* in (13.16)is an a* *cyclic 40 JOHN E. HARPER cofibration. Suppose p : C-! Dis a fibration in SymSeq. We want to verify id t* *i* has the LLP with respect to p. OA[t] tQtt-1__//_C99__OA[t]__Qtt-1___//C_99__ ____ _______ | ______||__ | ______|_ | _______ | | _______ | fflffl|____ fflffl| fflffl|____fflffl| OA[t] tY t___//_D OA[t] Y t___//_D, The left-hand solid commutative diagramoinpSymSeq has a lift if and only if the right-hand solid diagram in SymSeq t has a lift. Hence it is sufficient to veri* *fy that the solid diagram ; __________________//_Map(Y t, C) | ____44_________ | _____________|(*) | _____________ | fflffl|______________ fflffl| OA[t]____//Map(Qtt-1, C) xMap (Qtt-1,D)Map(Y t, D) op op in SymSeq t has a lift. We know OA[t] is cofibrant in SymSeq t , hence suffi- cient to verify (*) is an acyclic fibration. By Theorem 12.4 it is enough to ve* *rify i*: Qtt-1-!Y tis an acyclic cofibration in the underlying category SymSeq, and Proposition 13.21 finishes the proof. The other case is similar. 13.4. Punctured cube. Proposition 13.21. Let i : X-! Ybe a cofibration (resp. acyclic cofibration) in SymSeq. Then the induced map i*: Qtt-1-!Y tis a cofibration (resp. acyclic cofibration) in the underlying category SymSeq. Before proving this proposition, we establish some notation. Definition 13.22. Let t 2. oCubet is the category with objects the vertices (v1, . .,.vt) 2 {0, 1}t o* *f the unit t-cube; there is at most one morphism between any two objects, and there is a morphism (v1, . .,.vt)-! (v01, . .,.v0t) if and only if vi v0ifor every 1 i t. In particular, Cubetis the cat* *egory associated to a partial order on the set {0, 1}t. oThe punctured cube pCubetis the full subcategory of Cubetwith all objects except the terminal object (1, . .,.1) of Cubet. Let i : X-! Ybe a morphism in SymSeq. It will be useful to introduce an associated functor w : pCubet-!SymSeqdefined objectwise by ae w(v1, . .,.vt) := c1 . . .ct with ci:= X,Y,forfvi=o0,r v i= 1, HOMOTOPY THEORY OF MODULES OVER OPERADS 41 and with morphisms induced by i : X-! Y. In particular, for t = 3 the diagram w looks like, X XOOY ____//_X OYO Y ppp | | pppp | | wwpppp | | Y XOOY X X X ____//_X Y X | ppp pppp | pppp pppp | wwpppp wwppp Y X X ____//_Y Y X. Proposition 13.23. Let i : X-! Ybe a morphism in SymSeq and t 2. There are natural isomorphisms Qtt-1~=colimw : pCubet-!SymSeq in the underlying category SymSeq. Proof.Use the t maps in SymSeqobtained from the map t. 1x t-1X Y (t-1)-!Qtt-1 in Definition 13.11 to define a cone into Qtt-1and verify this cone is initial * *with respect to all cones. Proof of Proposition 13.21.Suppose i : X-! Yis an acyclic cofibration in SymSeq. The colimit of the diagram w : pCubet-!SymSeqmay be computed inductively using pushout corner maps, and hence by Proposition 13.23 there are natural iso- morphisms Q21~=Y X qX X X Y, Q32~=Y Y X q Y XqX XX Y X Y X qX X X Y Y, . . . in the underlying category SymSeq. The same argument provides an inductive con- struction of the induced map i*: Qtt-1-!Y itn the underlying category SymSeq; using the natural isomorphisms in Proposition 13.23, for each t 2 the Qtt-1fit into pushout squares Qt-1t-2Xid_i_//Qt-1t-2Y__ _________________________________________* *__________ |i*|id || _______________________________________* *____ fflffl| fflffl|i*_id_______________________________* *_______ (t-1) ______//Qt ____________________ Y __X_________t-1_ ________________________ ______________________________________________________* *__________________________________i_______ ____________________________________________________* *__________________________________________________________________*_________ __idi__________''_______________________________* *_____________________________________________________________________________* *____________________________________________9!$$______ _______//_________________________________* *_____________________________________________________________________________* *________________________Y t in the underlying category SymSeqwith induced map i*: Qtt-1-!Y tthe indicated pushout corner map. By iterated applications of Theorem 12.4, i* is an acyclic cofibration in SymSeq. The case for cofibrations is similar. Remark 13.24. This construction of i* by iterated pushout corner maps is used in the proof of the main theorem in [39]. 42 JOHN E. HARPER Proof of Proposition 13.4.By Proposition 13.19, each map j is an acyclic cofibr* *a- tion in the underlying category SymSeq. Noting that (possibly transfinite) comp* *o- sitions of acyclic cofibrations are acyclic cofibrations, completes the proof. 13.5. Model category structures in -operad case. In the following subsec- tions we prove Theorem 1.2. The argument and required constructions are related to those of the previous subsections, but different enough to require some expo* *si- tion. The strong cofibrancy condition exploited in Theorem 1.4 is replaced here by the weaker monoid axiom [39], but at the cost of dropping all -actions; i.e* *., working with -operads instead of -operads. Definition 13.25. A monoidal model category satisfies the monoid axiom if every map which is a (possibly transfinite) composition of pushouts of maps in (13.26) {acyclic cofibrations} C is a weak equivalence. Remark 13.27. In this definition, (13.26)is notation for the collection of maps* * of the form f id: K B-! L B such that f : K-! Lis an acyclic cofibration and B 2 C. Proof of Theorem 1.2.The model category structure on LtO(resp. AlgO) is created by the adjunction O_^O-//_ i O_^O(-)//_j Seqoo___LtO resp. C oo___AlgO U U with left adjoint on top and U the forgetful functor. Define a map f in LtOto b* *e a weak equivalence (resp. fibration) if U(f) is a weak equivalence (resp. fibrati* *on) in Seq. Similarly, define a map f in AlgOto be a weak equivalence (resp. fibration) if U(f) is a weak equivalence (resp. fibration) in C. Define a map f in LtO(res* *p. AlgO) to be a cofibration if it has the LLP with respect to all acyclic fibrati* *ons in LtO (resp. AlgO). Consider the case of LtO. We want to verify the model category axioms (MC1)- (MC5). We already know (MC1) is satisfied, and verifying (MC2) and (MC3) is clear. The arguments in the proof of [39, Lemma 2.3] use (possibly transfinite) small object arguments to reduce (MC5) to verifying Proposition 13.29 below. The first part of (MC4) is satisfied by definition, and the second part of (MC4) fo* *llows from the usual lifting and retract argument, as described in the proof of [39, * *Lemma 2.3]. This verifies the model category axioms. By construction, the model categ* *ory is cofibrantly generated. Consider the case of AlgO. Argue similar to the case of LtO, except use Propo- sition 13.29 together with Remark 13.31. By construction, the model category is cofibrantly generated. Remark 13.28. Since the forgetful functors in this proof commute with filtered colimits, the smallness conditions needed for the (possibly transfinite) small * *object arguments in [39, Lemma 2.3] are satisfied. HOMOTOPY THEORY OF MODULES OVER OPERADS 43 13.6. Analysis of pushouts for -operad case. Proposition 13.29. Let O be an -operad and A 2 LtO. Assume (C, , k) satisfies the monoid axiom. Then every (possibly transfinite) composition of pushouts in * *LtO of the form (13.30) O ^OX_________//_A id^Oi|| j|| fflffl| fflffl| O ^OY____//_A qO ^OXO ^OY, such that i : X-! Yis an acyclic cofibration in Seq, is a weak equivalence in t* *he underlying category Seq. Remark 13.31. If X, Y, A are concentrated at 0, then the pushout diagram (13.30) is concentrated at 0. To verify this, argue as in Remark 13.7. This subsection is devoted to proving Proposition 13.29, which we cited above in the proof of Theorem 1.2. Similar to the previous subsections, a first step* * in analyzing the pushouts in (13.30)is an analysis of certain coproducts. The foll* *owing is an -operad version of Proposition 13.8. Proposition 13.32. Let O be an -operad, A 2 LtO, and Y 2 Seq. Consider any coproduct in LtOof the form (13.33) A q (O ^OY ). There exists an array OA and natural isomorphisms a ^ A q (O ^OY ) ~=OA ^O(Y ) = OA[q] ^Y q q 0 in the underlying category Seq. If Q 2 and q := |Q|, then OA[Q] is naturally isomorphic to a colimit of the form ` ` h ^di0o`o_ h ' ^ i colim p 0O[p + q] _p+q_.pxAqpd1opo0O[p_+ q] _p+q_.px(qO ^OA),p in Seq, with d0 induced by operad multiplication and d1 induced by m : O ^OA-!A. Proof.Verify the coproduct in (13.33)may be calculated by a reflexive coequaliz* *er in LtOof the form, i d0 j A q (O ^OY ) ~=colim (O ^OA) q (O ^OYo)o(O_^OOo^OA)oq_(O ^OY.) d1 The maps d0 and d1 are induced by maps m : O ^OO-!O and m : O ^OA-!A, respectively. There are natural isomorphisms in the underlying category Seq, (O ^OA) q (O ^OY~)=O ^O(A q Y ) ~= a O[t] (A q Y )^t t 0 ia h ij ~= a O[p + q] __p+q__. A^p ^ Y ^q, q 0p 0 px q 44 JOHN E. HARPER and similarly, a ia h p+q ^ ij ^ (O ^OO ^OA) q (O ^OY ) ~= O[p + q] _______. (O ^OA) p ^ Y q. q 0 p 0 px q The maps d0 and d1 similarly factor in the underlying category Seq. It is impor* *tant to note that the ordering of all tensor power factors is respected, and that we* * are simply using the symmetric groups in the isomorphisms a ^ ^ (A q Y )^t ~= p+q. px qA p^ Y q p+q=t to build convenient indexing sets for the tensor powers. Definition 13.34. Let i : X-! Ybe a morphism in Seqand t 1. Define Qt0:= X ^tand Qtt:= Y ^t. For 0 < q < t define Qtqinductively by the pushout diagrams pr* t t. t-qx qX ^(t-q)^Qqq-1__//_Qq-1 i*|| i*|| fflffl| fflffl| t. t-qx qX ^(t-q)^Y ^q___//_Qtq in Seq t. The following is an -operad version of Proposition 13.13, and provides a use* *ful description of certain pushouts of left modules. Proposition 13.35. Let O be an -operad, A 2 LtO, and i : X-! Yin Seq. Consider any pushout diagram in LtOof the form, f (13.36) O ^OX___________//_A |id^Oi| || fflffl| fflffl| O ^OY_____//A q(O ^OX)(O ^OY ). The pushout in (13.36)is naturally isomorphic to a filtered colimit of the form i j1 j2 j3 j (13.37) A q(O ^OX)(O ^OY ) ~=colim A0 ____//_A1___//A2___//_. . . in the underlying category Seq, with A0 := OA[0] ~=A and At defined inductively by pushout diagrams in Seqof the form f* (13.38) OA[t] ^Qtt-1___//At-1 id^i*|| |jt| fflffl|,t fflffl| OA[t] ^Y ^t____//_At. Proof.Verify the pushout in (13.36)may be calculated by a reflexive coequalizer in LtOof the form i _i j A q(O ^OX)(O ^OY ) ~=colim A q (O ^OYo)o__A_qo(Oo^OX)_q (O ^OY ) f HOMOTOPY THEORY OF MODULES OVER OPERADS 45 _ __ The maps iand f are induced by maps id^Oi* and id^Of* which fit into the com- mutative diagram od0o_ (13.39) OA ^O(X q Yo)o__O ^O(A q X q Yo)o__O ^O(O ^OA) q X q Y d1 _i|_| || || |f| id^Oi*|id^Of*| id^Oi*|id^Of*| fflffl|fflffl| fflffl|fflffl|d0 fflffl|fflffl| OA ^O(Yo)o______O ^O(A q Yo)o______Oo^O(Oo^OA)_q Y ) d1 in LtO, with rows reflexive coequalizer diagrams, and maps i* and f* in Seqindu* *ced by i : X-! Yand f : X-! A. By Proposition 13.32, the coproduct in (13.36)may be calculated by the colimit of the left hand column of (13.39)in the underlying category Seq. We want to reconstruct_this colimit via a suitable filtered colim* *it. Using (13.39), there exist maps fq,pwhich make the diagrams ` ` ijinqi,p j OA ^O(X q Y ) ~= oo___ OA[p + q] ^ p+q. x X ^p^Y ^q q 0p 0 _ p q __ |_f ______ | 9_fq,p___ fflffl| ij fflffl____ ` inq i j OA ^O(Y ) ~= oo________________ OA[q] ^Y ^q t 0 _ in Seqcommute. Similarly, there exist maps iq,pwhich make the diagrams ` ` ijinqi,p j OA ^O(X q Y ) ~= oo___ OA[p + q] ^ p+q. x X ^p^Y ^q q 0p 0 _ p q __ |_i ______ | 9_iq,p___ fflffl| ij fflffl____ ` inp+q i j OA ^O(Y ) ~= oo_____________ OA[p + q] ^Y ^(p+q) t 0 in Seqcommute._Let_' : OA ^O(Y-)!.be a morphism in Seqand define 'q := 'inq. Then 'i= 'f if and only if the diagrams ^ ^ _fq,p ^ OA[p + q] ^ p+q. px qX p^Y q_____//OA[q] ^Y q |_iq,p |' | | q fflffl| 'p+q fflffl|| OA[p + q] ^Y ^(p+q)_____________//_. _ __ commute for every p, q 0. Since iq,0= idand fq,0= id, it is sufficient to con* *sider q 0 and p > 0. To motivate the construction (13.38), it is useful to describe a preliminary * *con- struction which also calculates the pushout in (13.36). Define A0 := OA[0] ~=A 46 JOHN E. HARPER and for each t 1 define Atby the pushout diagram ` ^ ^ OA[p + q] ^ p+q. px qX p^Y q f* // (13.40) p+q=t __________ At-1 q 0,p>0 | |i*| |jt| fflffl| ,t |fflffl| OA[t] ^Y ^t_______________________//At __ _ in Seq. The maps f* and i* are induced by the appropriate maps fq,pand iq,p. Verify that (13.37)is satisfied.__ The collection of maps fq,pand iq,psatisfy many compatibility relations. This suggests we replace the coproduct in (13.40), which is isomorphic to h i OA[t] ^ (X q Y )^t- Y ^t, with an appropriate pushout construction. Here, (X q Y )^t - Y ^tmeans the coproduct of all factors in (X q Y )^t except Y ^t. Define A0 := OA[0] ~=A and for each t 1 define At by the pushout diagram_ (13.38)in_Seq. The maps f* and i* are induced by the appropriate maps fq,pand iq,p. Verify that (13.37)is satisfied. Proposition 13.41. Let i : X-! Ybe a cofibration (resp. acyclic cofibration) in Seq. Then the induced map i*: Qtt-1-!Y ^tis a cofibration (resp. acyclic cofibration) in the underlying category Seq. Proof.Argue as in the proof of Proposition 13.21, replacing (SymSeq, , 1) with (Seq, ^, 1). Proposition 13.42. Assume that C satisfies Basic Assumption 1.1 and in addition satisfies the monoid axiom. Then (Seq, ^, 1) satisfies the monoid axiom. Proof.Since colimits in Seqare calculated objectwise, use (4.7)together with an argument that the pushout of a coproduct qfffffof a finite set of maps can be written as a finite composition of pushouts of the maps fff. Proof of Proposition 13.29.By Proposition 13.35, j is a (possibly transfinite) * *com- position of pushouts of maps of the form id^i*, and Propositions 13.42 and 13.41 finish the argument. 14. Shortened proof for chain complexes In this section, we include a shortened proof of Theorem 1.4, for the special* * case (Chk, , k) of unbounded chain complexes over a field of characteristic zero. Consider the proof of Theorem 1.4 given in Section 13.2, for the case of LtO;* * for the special case of AlgOsee Remark 13.7. The (possibly transfinite) small object arguments only require the pushouts in Proposition 13.4 to be constructed from a set of generating acyclic cofibrations. In the special case of chain complexes * *[22, Section 2.3], a set of generating acyclic cofibrations for SymSeqmay be chosen * *such that each has the form i : ;-!.DSince acyclic cofibrations are preserved under (possibly transfinite) compositions, in this special case, the proof of Theorem* * 1.4 reduces to the following proposition. HOMOTOPY THEORY OF MODULES OVER OPERADS 47 Proposition 14.1. Let O be a -operad and D 2 SymSeq such that i : ;-! Dis an acyclic cofibration. Assume every object in SymArrayis cofibrant. Consider a* *ny pushout diagram in LtOof the form, O O ;________//_A idOi|| j|| fflffl| fflffl| O O D ____//_A q (O O D). Then j is an acyclic cofibration in the underlying category SymSeq. Remark 14.2. Suppose G is any finite group. Since k is a field of characterist* *ic zero, every k[G]-module is projective. It follows that every symmetric array in* * Chk is cofibrant. Proof.By Proposition 13.8, it is enough to verify OA O (i): OA O (;)-!OA O (D) is an acyclic cofibration in SymSeq. Consider any fibration p : X-! Yin SymSeq. We want to show that OA O (i) has the LLP with respect to p. OA O (;)____//_X;;____OA O_^;_//_^X<<___ ___ | _____|____ | _____|_ | _______p| | _______^p| fflffl|___fflffl|_|fflffl___fflffl|_ OA O (D)____//_Y OA O ^D___//_^Y The left-hand solid commutative diagram in SymSeq has a lift if and only if the right-hand solid diagram in SymArrayhas a lift. Hence it is sufficient to verif* *y that the solid diagram ; ______________//_MapO(D^, ^X) | ____55_________ | __________ (*)| | ___________ | fflffl|___________ fflffl| OA ____//_MapO(^;, ^X) xMapO(^;,^Y)MapO(D^, ^Y) in SymArrayhas a lift. We know OA is cofibrant in SymArray, hence it is suffici* *ent to verify (*) is an acyclic fibration. By Proposition 5.22, it is enough to sho* *w each map Map (D t, X)____//Map(; t, X) xMap (; t,YM)ap(D t, Y ) is an acyclic fibration in SymSeq. 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