HOMOTOPY THEORY OF MODULES OVER OPERADS IN SYMMETRIC SPECTRA JOHN E. HARPER 1. Introduction Operads parametrize simple and complicated algebraic structures and naturally arise in several areas of algebraic topology, homotopy theory, and homological * *alge- bra [1, 13, 18, 24, 30, 31]. The symmetric monoidal category of symmetric spect* *ra [21] provides a simple and convenient model for the classical stable homotopy c* *at- egory, and is an interesting setting where such algebraic structures naturally * *arise. Given an operad O in symmetric spectra, we are interested in the possibility of doing homotopy theory in the categories of O-modules and O-algebras in symmet- ric spectra, which in practice means putting a Quillen model structure on these categories of modules and algebras. In this setting, O-algebras are the same as left O-modules concentrated at 0 (Section 3.4). This paper establishes a homoto* *py theory for modules and algebras over operads in symmetric spectra. The main theorem is this. Theorem 1.1. Let O be an operad in symmetric spectra. Then the category of left O-modules and the category of O-algebras both have natural model category struc- tures. The weak equivalences and fibrations in these model structures are inher* *ited in an appropriate sense from the stable weak equivalences and the stable flat p* *ositive fibrations in symmetric spectra. Remark 1.2. For ease of notation purposes, we have followed Schwede [37] in usi* *ng the term flat (e.g., stable flat model structure) for what is called S (e.g., s* *table S-model structure) in [21, 36, 39]. The theorem remains true when the stable flat positive model structure on sy* *m- metric spectra is replaced by the stable positive model structure. This follows immediately from the proof of Theorem 1.1 since every stable (positive) cofibra* *tion is a stable flat (positive) cofibration. The theorem is this. Theorem 1.3. Let O be an operad in symmetric spectra. Then the category of left O-modules and the category of O-algebras both have natural model category struc- tures. The weak equivalences and fibrations in these model structures are inher* *ited in an appropriate sense from the stable weak equivalences and the stable positi* *ve fibrations in symmetric spectra. In section 5 we prove that a morphism of operads which is an objectwise stab* *le equivalence induces an equivalence between the corresponding homotopy categories of modules (resp. algebras). The theorem is this. Theorem 1.4. Suppose O is an operad in symmetric spectra and let LtO (resp. AlgO) be the category of left O-modules (resp. O-algebras) with the model struc* *ture 1 2 JOHN E. HARPER of Theorem 1.1 or 1.3. If f : O-! O0is a map of operads, then the adjunctions __f*_// _f*_//_ (1.5) LtOoo___LtO0, AlgO oo___AlgO0, f* f* are Quillen adjunctions with left adjoints on top and f* the forgetful functor.* * If furthermore, f is an objectwise stable equivalence, then the adjunctions (1.5)a* *re Quillen equivalences, and hence induce equivalences on the homotopy categories. The properties of the stable flat model structure on symmetric spectra are f* *un- damental to the results of this paper. For some of the good properties, see [2* *1, Theorem 5.3.7 and Corollary 5.3.10]. The stable flat positive model structure, compared to the stable flat model structure, arises very clearly in our argumen* *ts. See, for example, Proposition 4.26 and its proof, the following of which is a s* *pecial case of particular interest. Proposition 1.6. If i : X- !Yis a cofibration between cofibrant objects in symm* *et- ric spectra with the stable flat positive model structure and t 1, then X^t-!* * Y ^t is a cofibration of t-diagrams in symmetric spectra with the stable flat posit* *ive model structure, and hence with the stable flat model structure. 1.1. Relationship to previous work. One of the theorems of Shipley [39] is that the category of commutative monoids in symmetric spectra has a natural model structure inherited from the stable flat positive model structure on symm* *etric spectra. Theorem 1.1 improves this result to left modules and algebras over any operad O in symmetric spectra. One of the theorems of Elmendorf and Mandell [6] is that for symmetric spect* *ra the category of algebras over any operad O in simplicial sets has a natural mod* *el structure inherited from the stable positive model structure on symmetric spect* *ra. Theorem 1.3 improves this result to left modules and algebras over any operad O* * in symmetric spectra. Their proof involves a filtration in the underlying category* * of certain pushouts of algebras. We have benefitted from their paper and our proofs of Theorems 1.1 and 1.3 exploit similar filtrations. Another of the theorems of Elmendorf and Mandell [6] is that a morphism of operads in simplicial sets which is an objectwise weak equivalence induces a Qu* *illen equivalence between categories of algebras over operads. Theorem 1.4 improves t* *his result to left modules and algebras over operads in symmetric spectra. Our approach to studying modules and algebras over operads is largely influe* *nced by Rezk [35]. 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 Seminar. 2.Symmetric spectra The purpose of this section is to recall some basic definitions and properti* *es of symmetric spectra. A useful introduction to symmetric spectra is given in the original paper [21]; see also the development given in [37]. Define the sets n* * := MODULES OVER OPERADS IN SYMMETRIC SPECTRA 3 {1, . .,.n} for each n 0, where 0 := ; denotes the empty set. Let S1 denote the simplicial circle [1]=@ [1] and for each n 0 define Sn := (S1)^n the n-f* *old smash power of S1, where S0 := [0]+ = [0] q [0]. Definition 2.1. Let n 0. o n is the category with exactly one object n and morphisms the bijections of sets. o S* is the category of pointed simplicial sets and their maps. o S*n is the category of functors X : n-! S*and their natural transforma- tions. In other words, an object in S*n is a pointed simplicial set X equipped with* * a basepoint preserving left action of the symmetric group n and a morphism in S*n is a map f : X- !Yin S* such that f is n-equivariant. 2.1. Symmetric spectra. Recall the following definition from [21, Section 1.2]. Definition 2.2. A symmetric spectrum X consists of the following: (1)a sequence of objects Xn 2 S*n (n 0), and (2)a sequence of maps oe : S1^ Xn-! Xn+1in S* (n 0), (3)such that the iterated maps oep: Sp ^Xn-! Xn+pare px n-equivariant for p 1 and n 0. Here, oep := oe(S1^ oe) . .(.Sp-1^ oe) is the compo* *sition of the maps i^oe Si^ S1^ Xn+p-1-iS____//Si^ Xn+p-i. The maps oe are the structure maps of the symmetric spectrum. A map of symmetric spectra f : X- !Yis (1)a sequence of maps fn: Xn-! Yn in S*n (n 0), (2)such that the diagram S1^ Xn __oe//_Xn+1 S1^ fn|| |fn+1| fflffl|oe fflffl| S1^ Yn ____//_Yn+1 commutes for each n 0. Denote by Sp the category of symmetric spectra and their maps; the null object is denoted by *. The sphere spectrum S is the symmetric spectrum defined by Sn := Sn, with left n-action given by permutation and structure maps oe : S1^ Sn-! Sn+1the natural isomorphisms. 2.2. Symmetric spectra as modules over the sphere spectrum. The pur- pose of this subsection is to recall the description of symmetric spectra as mo* *dules over the sphere spectrum. A similar tensor product construction will appear when working with left modules and algebras over operads. Definition 2.3. Let n 0. o is the category of finite sets and their bijections. o S* is the category of functors X : -! S*and their natural transforma- tions. 4 JOHN E. HARPER o If X 2 S*, define Xn := X[n] the functor X evaluated on the set n. o An object X 2 S* is concentrated at n if Xr = * for all r 6= n. If X is a finite set, define |X| to be the number of elements in X. Definition 2.4. Let X be a finite set and A in S*. The copowers A . X and X . A in S* are defined as follows: a a A . X := A ~=A ^X+ , X . A := A ~=X+ ^A, X X the coproduct in S* of |X|-copies of A. Definition 2.5. Let X, Y be objects in S*. The tensor product X Y 2 S* is the left Kan extension of objectwise smash along coproduct of sets, x _XxY_//S* x S*^_//_S* |` | fflffl| X Y ____left_Kan_extension//_S* Useful details on Kan extensions and their calculation are given in [26, X];* * in particular, see [26, X.4]. The following is a calculation of tensor product, wh* *ose proof is left to the reader. Proposition 2.6. Let X, Y be objects in S* and N 2 , with n := |N|. There are natural isomorphisms, a (X Y )n ~=(X Y )[N]~= X[ss-1(1)] ^Y [ss-1(2)], ss:N-!i2n Set (2.7) ~= a n . Xn1^ Yn2. n1+n2=n n1x n2 Remark 2.8. The coproduct is in the category S*. Setis the category of sets and their maps. The following is proved in [21, Section 2.1] and verifies that tensor produc* *t in the category S* inherits many of the good properties of smash product in the catego* *ry S*. Proposition 2.9. (S*, , S0) has the structure of a closed symmetric monoidal category. All small limits and colimits exist and are calculated objectwise. Th* *e unit S0 2 S* is given by S0[n] = * for each n 1 and S0[0] = S0. The sphere spectrum S has two naturally occurring maps S S-! S and S0-! S in S* which give S the structure of a commutative monoid in (S*, , S0). Furthe* *r- more, any symmetric spectrum X has a naturally occurring map m : S X-! X which gives X a left action of S in (S*, , S0). The following is proved in [2* *1, Section 2.2] and provides a useful interpretation of symmetric spectra. Proposition 2.10. Define the category 0:= qn 0 n, a skeleton of . (a)The sphere spectrum S is a commutative monoid in (S*, , S0). (b) The category of symmetric spectra is equivalent to the category of left * *S- modules in (S*, , S0). MODULES OVER OPERADS IN SYMMETRIC SPECTRA 5 (c)The category of0symmetric spectra is isomorphic to the category of left * *S- modules in (S* , , S0). In this paper we will not distinguish between these equivalent descriptions * *of symmetric spectra. 2.3. Smash product of symmetric spectra. The smash product X ^Y 2 Sp of symmetric spectra X and Y is defined as the colimit i m id j (2.11) X ^Y := X SY := colim X Y oo___XooS_ Y . id m Note that since S is a commutative monoid, a left action of S on X determines a right action m : X S-! X which gives X the structure of an (S, S)-bimodule. Hence the tensor product X SY has the structure of a left S-module. The following is proved in [21, Section 2.2] and verifies that smash product* *s of symmetric spectra inherit many of the good properties of smash products of poin* *ted simplicial sets. Proposition 2.12. (Sp , ^, S) has the structure of a closed symmetric monoidal category. All small limits and colimits exist and are calculated objectwise. Recall that by closed we mean there exists a functor which we call mapping object (or function spectrum), op Sp x Sp -! Sp , (Y, Z) 7-! Map(Y, Z), which fits into isomorphisms (2.13) hom (X ^Y, Z) ~=hom (X, Map(Y, Z)), natural in symmetric spectra X, Y, Z. These mapping objects will arise when we introduce mapping sequences associated to circle products. 3. Modules and algebras over operads In this section we recall certain definitions and constructions involving sy* *mmetric sequences and modules and algebras over operads. A useful introduction to opera* *ds and their algebras is given in [24]. See also the original article [30]; other * *accounts include [2, 8, 11, 17, 29, 32, 41]. The circle product introduced in Section 3.* *2 goes back to [10, 40] and more recently appears in [7, 9, 12, 22, 23, 35]. A fuller * *account of the material in this section is given in [16] for the general context of a m* *onoidal model category, which was largely influenced by the development in [35]. 3.1. Symmetric sequences. Definition 3.1. Let n 0 and G be a finite group. o A symmetric sequence in Sp is a functor A : op-! Sp. SymSeq is the category of symmetric sequences in Sp and their natural transformations; the null object is denoted by *. o SymSeqG is the category of functors X : G-! SymSeq and their natural transformations. o A symmetric sequence A is concentrated at n if A[r] = * for all r 6= n. 6 JOHN E. HARPER 3.2. Tensor product and circle product of symmetric sequences. Definition 3.2. Let X be a finite set and A in Sp . The copowers A . X and X . A in Sp are defined as follows: a a A . X := A ~=A ^X+ , X . A := A ~=X+ ^A, X X the coproduct in Sp of |X|-copies of A. Definition 3.3. Let A1, . .,.At be symmetric sequences. The tensor products A1~ . .~.At 2 SymSeq are the left Kan extensions of objectwise smash along co- product of sets, xt ^ ( op)xt_A1x...xAt//_Sp ____//_Sp |`| | fflffl| A1~...~At op______left_Kan_extension//_Sp, This definition of tensor product in SymSeqis conceptually the same as the d* *efi- nition of tensor product in S* given in Definition 2.5. The following is a calc* *ulation of tensor product, whose proof is left to the reader. Proposition 3.4. Let A1, . .,.Atbe symmetric sequences and R 2 , with r := |R|. There are natural isomorphisms, a (A1~ . .~.At)[R]~= A1[ss-1(1)] ^. .^.At[ss-1(t)], ss:R-!itn Set (3.5) ~= a A1[r1] ^. .^.At[rt] . r, r1+...+rt=r r1x...x rt It will be useful to extend the definition of tensor powers A~t to situation* *s in which the integers t are replaced by a finite set T . Definition 3.6. Let A be a symmetric sequence and R, T 2 . The tensor powers A~T 2 SymSeq are defined objectwise by a (3.7) (A~T )[R] := ^ t2TA[ss-1(t)], T 6= ; , ss:R-!iTn Set a (A~;)[R] := S. ss:R-!i;n Set Note that there are no functions ss : R-! ;in Setunless R = ;. We will use the abbreviation A~0 := A~;. Definition 3.8. Let A, B be symmetric sequences, R 2 , and define r := |R|. The circle product (or composition product) A O B 2 SymSeq is defined objectwise by the coend a ~ (3.9) (A O B)[R] := A ^ (B ~-)[R] ~= A[t] ^ t(B t)[r]. t 0 MODULES OVER OPERADS IN SYMMETRIC SPECTRA 7 Definition 3.10. Let B, C be symmetric sequences, T 2 , and define t := |T |. The mapping sequence MapO(B, C) 2 SymSeqand the mapping object Map ~(B, C) 2 SymSeq are defined objectwise by the ends Y ~ MapO(B, C)[T ]:= Map ((B ~T)[-], C) ~= Map ((B t)[r], C[r]) r, r 0 Y (3.11) Map ~(B, C)[T ]:= Map (B, C[T q -]) ~= Map (B[r], C[t + r]) r. r 0 These mapping sequences and mapping objects are part of closed monoidal cat- egory structures on symmetric sequences and fit into isomorphisms hom (A O B, C)~=hom(A, MapO(B, C)), hom(A ~B, C)~=hom (A, Map~(B, C)), natural in symmetric sequences A, B, C. The mapping sequences also arise in de- scribing modules and algebras over operads (3.18). Proposition 3.12. (a)(SymSeq, ~, 1) has the structure of a closed symmetric monoidal category. All small limits and colimits exist and are calculated objectwise. The u* *nit 1 2 SymSeq is given by 1[n] = * for each n 1 and 1[0] = S. (b) (SymSeq, O, I) has the structure of a closed monoidal category with all * *small limits and colimits. Circle product is not symmetric. The (two-sided) un* *it I 2 SymSeq is given by I[n] = * for each n 6= 1 and I[1] = S. 3.3. Symmetric sequences build functors. The category Sp embeds in SymSeq as the full subcategory of symmetric sequences concentrated at 0, via the funct* *or -^: Sp -! SymSeqdefined objectwise by ae (3.13) Z^[R] := Z, for |R| = 0, *, otherwise. Definition 3.14. Let O be a symmetric sequence and Z 2 Sp . The corresponding functor O : Sp -! Sp is defined objectwise by, a O(Z) := O O (Z) := O[t] ^ tZ^t ~=(O O ^Z)[0]. t 0 3.4. Modules and algebras over operads. Definition 3.15. An operad is a monoid object in (SymSeq, O, I) and a morphism of operads is a morphism of monoid objects in (SymSeq, O, I). Similar to the case of any monoid object, we study operads because we are interested in the objects they act on. A useful introduction to monoid objects * *and monoidal categories is given in [26, VII]. Definition 3.16. Let O be an operad. A left O-module is an object in (SymSeq, O* *, I) with a left action of O and a morphism of left O-modules is a map in SymSeqwhich respects the left O-module structure. Each operad O determines a functor O : Sp -! Sp (Definition 3.14) together with natural transformations m : OO -! Oand j : id-!Owhich give the functor O : Sp -! Sp the structure of a monad (or triple) in Sp . One perspective offer* *ed 8 JOHN E. HARPER in [24, I.3] is that operads determine particularly manageable monads. A useful introduction to monads and their algebras is given in [26, VI]. Recall the foll* *owing definition from [24, I.2 and I.3]. Definition 3.17. Let O be an operad. An O-algebra is an object in Sp with a left action of the monad O : Sp -! Sp and a morphism of O-algebras is a map in Sp which respects the left action of the monad O : Sp -! Sp. It is easy to verify that an O-algebra is the same as an object X 2 Sp with* * a left O-module structure on ^X, and if X and X0 are O-algebras, then a morphism of O-algebras is the same as a map f : X- !X0in Sp such that ^f: ^X-!X^0is a morphism of left O-modules. In other words, an algebra over an operad O is the same as a left O-module which is concentrated at 0. Giving a symmetric sequence Y a left O-module structure is the same as giving a morphism of operads (3.18) m : O-! Map O(Y, Y.) Similarly, giving an object X 2 Sp 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 in [30] of an O-algebra structure on X, w* *here Map O(X^, ^X) is called the endomorphism operad of X, and motivates the suggest* *ion in [24, 30] that O[t] should be thought of as parameter objects for t-ary opera* *tions. Definition 3.19. Let O be an operad. o LtOis the category of left O-modules and their morphisms. o AlgOis the category of O-algebras and their morphisms. The category AlgO embeds in LtO as the full subcategory of left O-modules concentrated at 0, via the functor ^-: AlgO-! LtOdefined objectwise by (3.13). Proposition 3.20. Let O be an operad in symmetric spectra. There are adjunc- tions OO-_//_ OO(-)//_ (3.21) SymSeq oo___LtO, Sp oo___AlgO, U U with left adjoints on top and U the forgetful functor. Proof.The unit I for circle product is the initial operad, hence there is a uni* *que map of operads f : I- !O. The desired adjunctions are the following special cas* *es f* // f* // SymSeq= LtI_____LtO,oo_ Sp = AlgI_____AlgO,oo_ f* f* of change of operads adjunctions. od0o_ Definition 3.22. Let C be a category. A pair of maps of the form X0 oo__X1_ d1 in C is called a reflexive pair if there exists s0: X0-! X1in C such that d0s0 * *= id and d1s0 = id. A reflexive coequalizer is the coequalizer of a reflexive pair. MODULES OVER OPERADS IN SYMMETRIC SPECTRA 9 The following proposition is proved in [35, Proposition 2.3.5], and allows u* *s to calculate certain colimits in modules and algebras over operads by working in t* *he underlying category. It is also proved in [16] and is closely related to [5, Pr* *oposition 7.2]. Since it plays a fundamental role in several of the main arguments in th* *is paper, we have included a proof below. Proposition 3.23. Let O be an operad in symmetric spectra. Reflexive coequalize* *rs and filtered colimits exist in LtO and AlgO, and are preserved (and created) by* * the forgetful functors. First we consider the following proposition which is proved in [35, Lemma 2.* *3.4]. It is also proved in [16] and follows from the proof of [5, Proposition 7.2] or* * the arguments in [12, Section 1] as we indicate below. Proposition 3.24. (a)If A-1 oo__A0_oo__A1_oo_and B-1 oo__B0_oo___B1oo_are reflexive coequal- izer diagrams in SymSeq, then their objectwise circle product A-1 O B-1oo___A0O B0oo___A1OoB1o_ is a reflexive coequalizer diagram in SymSeq. (b) If A, B: D-! SymSeqare filtered diagrams, then objectwise circle product* * of their colimiting cones is a colimiting cone. In particular, there are na* *tural isomorphisms colimd2D(Ad O Bd) ~=(colimd2DAd) O (colimd2DBd) in SymSeq. Proof.Consider part (a). The corresponding statement for smash products of sym- metric spectra follows from the proof of [5, Proposition 7.2] or the argument a* *ppear- ing between Definition 1.8 and Lemma 1.9 in [12, Section 1]. Using this together with (3.7)and (3.9), the statement for circle products easily follows by verify* *ing the universal property of a colimit. Consider part (b). It is easy to verify th* *e cor- responding statement for smash products of symmetric spectra, and the statement for circle products easily follows as in part (a). Proof of Proposition 3.23.Suppose A0oo___A1oo_is a reflexive pair in LtOand con- sider the solid commutative diagram 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 O_A-1oo______COCOCA0oo______OoOoA1_C_CC_ ______ __|______________|_____________________________* *_________________ s0___m________jOid_|m_____________m|________________jOid________* *______________________ __fflffl___ __fflffl|________fflffl|_______________________* *_____ A-1 oo_________A0 oo_________oA1o_ in SymSeq, with bottom row the reflexive coequalizer diagram of the underlying reflexive pair in SymSeq. By Proposition 3.24, the rows are reflexive coequaliz* *er diagrams and hence there exist unique dotted arrows m, s0, d0, d1 in SymSeq whi* *ch make the diagram commute. By uniqueness, s0 = jOid, d0 = mOid, and d1 = idOm. It is easy to verify that m gives A-1 the structure of a left O-module and that* * the bottom row is a reflexive coequalizer diagram in LtO; it is easy to check the d* *iagram 10 JOHN E. HARPER lives in LtOand that the colimiting cone is initial with respect to all cones i* *n LtO. The case for filtered colimits is similar. The next proposition is proved in [35, Proposition 2.3.5]. It verifies the e* *xistence of all small colimits in left modules and algebras over an operad, and provides* * one approach to their calculation. The proposition also follows from the argument in [5, Proposition 7.4]. To keep the paper relatively self-contained, we have incl* *uded a proof at the end of Section 6. Proposition 3.25. Let O be an operad in symmetric spectra. All small colimits exist in LtO and AlgO. If A : D-! LtOis a small diagram, then colimA in LtO may be calculated by a reflexive coequalizer of the form i oo___ j colimA ~=colim O O colimd2DAdoo_O O colimd2D(O O Ad) in the underlying category SymSeq; the colimits appearing inside the parenthesi* *s are in the underlying category SymSeq. The proof of the following is left to the reader. Proposition 3.26. Let O be an operad in symmetric spectra. All small limits exi* *st in LtO and AlgO, and are preserved (and created) by the forgetful functors. 4.Model structures The purpose of this section is to prove Theorems 1.1 and 1.3, which establish certain model category structures on left modules and algebras over an operad. Model categories provide a setting in which one can do homotopy theory, and in particular, provide a framework for constructing and calculating derived functo* *rs. A useful introduction to model categories is given in [4]; see also the origina* *l articles [34, 33] and the more recent [15, 19, 20]. When we refer to the extra structure* * of a monoidal model category, we are using [38, Definition 3.1]; an additional condi* *tion involving the unit is assumed in [25, Definition 2.3] which we will not require* * in this paper. In this paper, our primary method of establishing model structures is to use* * a small object argument together with the extra structure enjoyed by a cofibrantly generated model category ([19, Chapter 11], [20, Section 2.1], [38, Section 2])* *. The reader unfamiliar with the small object argument may consult [4, Section 7.12] * *for a useful introduction, followed by the (possibly transfinite) versions describe* *d in [19, Chapter 10], [20, Section 2.1], and [38, Section 2]. In [38, Section 2] an account of these techniques is provided which will be * *suffi- cient for our purposes; our proofs of Theorems 1.1 and 1.3 will reduce to verif* *ying the conditions of [38, Lemma 2.3(1)]. This verification amounts to a homotopical analysis of certain pushouts (Section 4.1) which lies at the heart of this pape* *r. The reader may contrast this with a path object approach explored in [2], which amounts to verifying the conditions of [38, Lemma 2.3(2)]; compare also [17, 41* *]. A first step is to recall just enough notation so that we can describe and w* *ork with the stable flat (positive) model structure on symmetric spectra, and the corres* *pond- ing projective model structures on the diagram categories SymSeq and SymSeqG, for G a finite group. The functors involved in such a description are easy to u* *nder- stand when defined as the left adjoints of appropriate functors, which is how t* *hey naturally arise in this context. MODULES OVER OPERADS IN SYMMETRIC SPECTRA 11 For each m 0 and subgroup H m denote by l : H- ! mthe inclusion of groups and define the evaluation functor evm: S*-! S*m objectwise by evm(X) := Xm . There are adjunctions _m.H-//_ ____//_ S* ____//_SH*oo_S*moo_oo_S* limH l* evm with left adjoints on top. Define GHm: S*-! S*to be the composition of the three top functors, and define limHevm : S*-! S*to be the composition of the three bottom functors; we have dropped the restriction functor l* from the notation. * *It is easy to check that if K 2 S*, then GHm(K) is the object in S* which is conce* *ntrated at m with value m .H K. Consider the forgetful functor Sp -! S*. It follows fr* *om Proposition 2.10 that there is an adjunction _S_-//_ S* oo___Sp with left adjoint on top. For each p 0, define the evaluation functor Evp : SymSeq-! Sp objectwise by Evp(A) := A[p], and for each finite group G, consider the forgetful functor SymSeq G-! SymSeq. There are adjunctions _Gp_//_ _G.-_// Sp Evoo_SymSeqoo___SymSeqG p with left adjoints on top. It is easy to check that if X 2 Sp , then Gp(X) is t* *he symmetric sequence concentrated at p with value X . p. Putting it all together, there are adjunctions GHm S - Gp G.- (4.1) S* ____//_S*oo_//_Spoo__//SymSeqoo_//_SymSeqGoo_ limHevm Evp with left adjoints on top. We are now in a good position to describe several us* *e- ful model structures. It is proved in [39] that the following two model catego* *ry structures exist on symmetric spectra. Definition 4.2. (a)The stable flat model structure on Sp has weak equivalences the stable equivalences, cofibrations the retracts of (possibly transfinite) compos* *itions of pushouts of maps S GHm@ [k]+- !S GHm [k]+ (m 0, k 0, H m subgroup), and fibrations the maps with the right lifting property with respect to * *the acyclic cofibrations. (b) The stable flat positive model structure on Sp has weak equivalences the stable equivalences, cofibrations the retracts of (possibly transfinite)* * com- positions of pushouts of maps S GHm@ [k]+- !S GHm [k]+ (m 1, k 0, H m subgroup), and fibrations the maps with the right lifting property with respect to * *the acyclic cofibrations. 12 JOHN E. HARPER It follows immediately from the above description that every stable flat pos* *itive cofibration is a stable flat cofibration. Several useful properties of the stab* *le flat model structure are proved in [21, Section 5.3]; here, we remind the reader of Remark 1.2. The stable model structure on Sp is defined by fixing H in Definition 4.2(a* *) to be the trivial subgroup. This is one of several model category structures that * *is proved in [21] to exist on symmetric spectra. The stable positive model structure on Sp is defined by fixing H in Definit* *ion 4.2(b) to be the trivial subgroup. This model category structure is proved in [* *28] to exist on symmetric spectra. It follows immediately that every stable (positi* *ve) cofibration is a stable flat (positive) cofibration. These model structures on symmetric spectra enjoy several good properties, including that smash products of symmetric spectra mesh nicely with each of the model structures defined above. More precisely, each model structure above is cofibrantly generated in which the generating cofibrations and acyclic cofibrat* *ions have small domains, and that with respect to each model structure (Sp , ^, S) is a monoidal model category. If G is a finite group, it is easy to check that the diagram categories SymS* *eq and SymSeqG inherit corresponding projective model category structures, where the weak equivalences (resp. fibrations) are the objectwise weak equivalences (* *resp. objectwise fibrations). We refer to these model structures by the names above (e.g., the stable flat positive model structure on SymSeqG). Each of these model structures is cofibrantly generated in which the generating cofibrations and ac* *yclic cofibrations have small domains. Furthermore, with respect to each model struct* *ure (SymSeq, , 1) is a monoidal model category; this is proved in [16], but can ea* *sily be verified directly using (3.11). Proof of Theorem 1.1.Consider SymSeqand Sp both with the stable flat positive model structure. We will prove that the model structure on LtO (resp. AlgO) is created by the adjunction _OO-_// i OO(-)//_ j SymSeqoo___LtO resp. Sp oo___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. fibration) in SymSeq. Similarly, define a map f in AlgOto be* * a weak equivalence (resp. fibration) if U(f) is a weak equivalence (resp. fibrati* *on) in Sp . Define a map f in LtO(resp. AlgO) to be a cofibration if it has the left * *lifting property with respect to all acyclic fibrations in LtO(resp. AlgO). Consider the case of LtO. We want to verify the model category axioms (MC1)- (MC5) in [4]. By Propositions 3.25 and 3.26, we know that (MC1) is satisfied, and verifying (MC2) and (MC3) is clear. The (possibly transfinite) small object arguments described in the proof of [38, Lemma 2.3] reduce the verification of * *(MC5) to the verification of Proposition 4.3 below. The first part of (MC4) is satisf* *ied by definition, and the second part of (MC4) follows from the usual lifting and ret* *ract argument, as described in the proof of [38, Lemma 2.3]. This verifies the model category axioms. By construction, the model category is cofibrantly generated. Argue similarly for the case of AlgOby considering left O-modules concentrated * *at 0. MODULES OVER OPERADS IN SYMMETRIC SPECTRA 13 Proof of Theorem 1.3.Consider SymSeq and Sp both with the stable positive model structure. We will prove that the model structure on LtO (resp. AlgO) is created by the adjunction _OO-_// i OO(-)//_ j SymSeqoo___LtO resp. Sp oo___AlgO U U with left adjoint on top and U the forgetful functor. Define a map f in LtO to * *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 Sp . Define a map* * f in LtO (resp. AlgO) to be a cofibration if it has the left lifting property with r* *espect to all acyclic fibrations in LtO(resp. AlgO). The model category axioms are verified exactly as in the proof of Theorem 1.* *1; (MC5) is verified by Proposition 4.3 below since every cofibration in SymSeq (r* *esp. Sp ) with the stable positive model structure is a cofibration in SymSeq(resp.* * Sp ) with the stable flat positive model structure. 4.1. Homotopical analysis of certain pushouts. The purpose of this section is to prove the following proposition which we used in the proofs of Theorems 1* *.1 and 1.3. The constructions developed here will also be important for homotopical analyses in other sections of this paper. Proposition 4.3. Let O be an operad in symmetric spectra, A 2 LtO, and i : X- !Y a generating acyclic cofibration in SymSeq with the stable flat positive model * *struc- ture. Consider any pushout diagram in LtO of the form, f (4.4) O O X ___________//_A |idOi| |j| |fflffl fflffl| O O Y_____//A q(OOX)(O O Y ). Then j is a monomorphism and a weak equivalence. Symmetric arrays arise naturally when calculating certain coproducts and pus* *houts of left modules and algebras over operads (Propositions 4.6 and 4.18). Definition 4.5. o A symmetric array in Sp is a symmetric sequence in SymSeq; i.e. a functor A : op-! SymSeq. op op op o SymArray:= SymSeq ~= Sp x is the category of symmetric ar- rays in Sp and their natural transformations. First we analyze certain coproducts of modules over operads. The following proposition is proved in [16] in the more general context of monoidal model cat- egories, and was motivated by a similar argument given in [14, Section 2.3] and [27, Section 13] in the context of algebras over an operad. Since the propositi* *on is important to several results in this paper, and in an attempt to keep the pa* *per relatively self-contained, we have included a proof below. 14 JOHN E. HARPER Proposition 4.6. Let O be an operad in symmetric spectra, A 2 LtO, and Y 2 SymSeq . Consider any coproduct in LtO of the form (4.7) A q (O O Y ). There exists a symmetric array OA and natural isomorphisms a ~ A q (O O Y ) ~= OA[q] ~ qY q q 0 in the underlying category SymSeq. If q 0, then OA[q] is naturally isomorphic* * to a colimit of the form ` ` ~ d0 ` ' poo___oo_ ~p OA[q] ~=colim p 0O[p + q] ^ pA d1 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. First we make the following observation. Proposition 4.8. Let O be an operad in symmetric spectra and A 2 LtO. Then omOido_ (4.9) A oom__O O AoidOmOoO_O O A is a reflexive coequalizer diagram in LtO. Proof.We use a split fork argument. The unit map j : I- !Oinduces a map s0 := idO j O:idO O-A!O O O O Ain LtO. Relabeling the three maps in (4.9)as d0 := m, d0 := m O id, d1 := idO m, it is easy to verify that d0s0 = idand d1s0* * = id. Hence the pair of maps is a reflexive pair in LtO, and by Proposition 3.23 it is enough to verify that (4.9)is a coequalizer diagram in the underlying category SymSeq . The unit map j : I- !Oalso induces maps s-1 := j O id: A-! O O A s-1 := j O idO:idO O-A!O O O O A in the underlying category SymSeq which satisfy the relations d0d0 = d0d1, d0s-1 = id, d1s-1 = s-1d0. Using these relations, it is easy to check that (4.9)is a coequalizer diagram in SymSeq by verifying the universal property of colimits. Proof of Proposition 4.6.The objectwise coproduct of two reflexive coequalizer * *di- agrams is a reflexive coequalizer diagram, hence by Proposition 4.8 the coprodu* *ct (4.7)may be calculated by a reflexive coequalizer in LtOof the form, i d0 j A q (O O Y ) ~=colim (O O A) q (O O Yo)o_(OoOoO_O A) q (O O Y.) d1 The maps d0 and d1 are induced by maps m : O O O-! Oand m : O O A-!A, respectively. By Proposition 3.23, this reflexive coequalizer may be calculated* * in MODULES OVER OPERADS IN SYMMETRIC SPECTRA 15 the underlying category SymSeq. There are natural isomorphisms, (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 in the underlying category SymSeq. The maps d0 and d1 similarly factor in the underlying category SymSeq. Remark 4.10. We have used the natural isomorphisms a ~ ~ (A q Y )~t ~= p+q. px q A p ~Y q, p+q=t in the proof of Proposition 4.6. Definition 4.11. 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 (4.12) t. t-qx qX ~(t-q)~Qqq-1___//Qq-1 i*|| || fflffl| fflffl| t. t-qx qX ~(t-q)~Y ~q_____//Qtq in SymSeq t. We sometimes denote Qtqby Qtq(i) to emphasize in the notation the map i : X- !Y. The maps pr*and i* are the obvious maps induced by i and the appropriate projection maps. Remark 4.13. For instance, to construct Q32, first construct Q21via the pushout diagram ~= (4.14) 2. 1x 1X ~X ____//_ 2. 2 X ~2__//_X ~2 id.|1x|1id~i || fflffl| fflffl| 2. 1x 1X ~Y ____________________//Q21 in SymSeq 2, then construct Q31by the pushout diagram ~= 3. 2x 1X ~2~X ____//_ 3. 3 X ~3__//_X ~3 id.|2x|1id~i || fflffl| fflffl| 3. 2x 1X ~2~Y ____________________//Q31 16 JOHN E. HARPER in SymSeq 3, and finally construct Q32by the pushout diagram pr* (4.15) 3. 1x 2X ~Q21_____//_Q31 id.|1x|2id~i* || fflffl| fflffl| 3. 1x 2X ~Y ~2 ____//_Q32 in SymSeq 3. The map i* in (4.15)is induced via (4.14)by the two maps X ~2-!Y ~2, 2. 1x 1X ~Y -! 2. 1x 1Y ~Y -! 2. 2 Y ~2~=Y ~2. The pushout diagram (4.16) 3. 1x 1x 1X ~X ~X ____//_ 3. 1x 2X ~X ~2 | | | | fflffl| fflffl| 3. 1x 1x 1X ~X ~Y ______// 3. 1x 2X ~Q21 in SymSeq 3 is obtained by applying 3 . 1x 2 X ~- to (4.14); the map pr*in (4.15)is induced via (4.16)by the two maps 3. 1x 2X ~X ~2-! 3. 3 X ~3~=X ~3-!Q31, 3. 1x 1x 1X ~X ~Y -! 3. 2x 1X ~2~Y -! Q31. Remark 4.17. The construction Qt-1tcan be thought of as a t-equivariant ver- sion of the colimit of a punctured t-cube [16]. There is a natural isomorphism Y ~t=Qtt-1~=(Y=X)~t. The following proposition is proved in [16] in the more general context of m* *onoidal model categories, and was motivated by a similar construction given in [6, sect* *ion 12] in the context of simplicial multifunctors of symmetric spectra. Since seve* *ral results in this paper require both the proposition and its proof, and in an eff* *ort to keep the paper relatively self-contained, we have included a proof below. Proposition 4.18. Let O be an operad in symmetric spectra, A 2 LtO, and i : X- !Yin SymSeq. Consider any pushout diagram in LtO of the form, f (4.19) O O X ___________//_A |idOi| |j| |fflffl fflffl| O O Y_____//A q(OOX)(O O Y ). The pushout in (4.19)is naturally isomorphic to a filtered colimit of the form i j1 j2 j3 j (4.20) A q(OOX)(O O Y ) ~=colim A0 _____//A1___//_A2___//_. . . MODULES OVER OPERADS IN SYMMETRIC SPECTRA 17 in the underlying category SymSeq, with A0 := OA[0] ~=A and At defined induc- tively by pushout diagrams in SymSeq of the form (4.21) OA[t] ~ tQtt-1f*_//At-1 |id~|ti* |jt| fflffl| ,t fflffl| OA[t] ~ tY ~t_____//At Proof.It is easy to verify that the pushout in (4.19)may be calculated by a ref* *lexive coequalizer in LtOof the form i _i j A q(OOX)(O O Y ) ~=colim A q (O O Y )oo__A_qo(OoO_X) q (O O Y.) f By Proposition 3.23, this reflexive coequalizer may be calculated in the underl* *ying category SymSeq. Hence it suffices to reconstruct this coequalizer in SymSeq vi* *a a suitable filtered colimit in SymSeq. A first step is to understand what it mean* *s to give a cone in_SymSeq_out of this diagram. The maps iand f are induced by maps idO i* and idO f* which fit into the commutative diagram od0o_ (4.22) A q O O (X q Y )oo__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| A q (O O Y )oo_______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 SymSeq induced by i : X- !Yand f : X- !A in SymSeq. Here we have used the same notation for both f and its adjoint (3.21). By Proposition 3.23, the pushout in (4.19)may be calculated by the colimit of the left-hand column of (4.22)in_the underlying category SymSeq. By (4.22)and Proposition 4.6, f induces maps fq,p which make the diagrams ` ` ijinqi,p j A q O O (X q Y ) ~= oo___ OA[p + q] ~ x X ~p~Y ~q q 0p 0 p_ q __ |_f ______ | fq,p____ fflffl| ij fflffl____ ` inq i j A q (O O Y ) ~= oo_____________ OA[q] ~ Y ~q t 0 q _ in SymSeq commute. Similarly, i induces maps iq,pwhich make the diagrams ` ` ijinqi,p j A q O O (X q Y ) ~= oo___ OA[p + q] ~ x X ~p~Y ~q q 0p 0 p_ q __ |_i ______ | iq,p____ fflffl| ij fflffl____ ` inp+q i j A q (O O Y ) ~= oo_________ OA[p + q] ~ Y ~(p+q) t 0 p+q 18 JOHN E. HARPER in SymSeq commute. We can now describe more explicitly what it means to give a cone in SymSeq out of the left-hand column of (4.22)._Let_' : A q (O O Y-)!. be a morphism in SymSeq and define 'q := 'inq. Then 'i= 'f if and only if the diagrams _f q,p ~q (4.23) OA[p + q] ~ px qX ~p~Y ~q____//OA[q] ~ qY |_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. The next step is to reconstruct the colimit of the left-hand column of (4.22) in SymSeq via a suitable filtered colimit in SymSeq. The diagrams (4.23)suggest how to proceed. We will describe two filtration constructions that calculate t* *he pushout (4.19)in the underlying category SymSeq. The purpose of presenting the filtration construction (4.25)is to provide motivation and intuition for the fi* *ltration construction (4.21)that we are interested in. Since (4.25)does not use the glue* *ing construction in Definition 4.11 it is simpler to verify that (4.20)is satisfied* * and provides a useful warm-up for working with (4.21). For each t 1, there are natural isomorphisms a ~ ~ (4.24) (X q Y )~t- Y ~t~= p+q. px q X p~Y q. p+q=t q 0, p>0 Here, (X q Y )~t - Y ~tdenotes the coproduct of all factors in (X q Y )~t except Y ~t. Define A0 := OA[0] ~=A and for each t 1 define Atby the pushout diagram h i f* (4.25) OA[t] ~ t(X q Y )~t- Y ~t__________//At-1 | |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. We want to use (4.24), (4.25)and (4.23)to verify that (4.20)is satisfied;* * it is sufficient to verify the universal property of colimits. By Proposition 4.6* *, the coproduct A q (O O Y ) is naturally isomorphic to a filtered colimit of the form i j A q (O O Y ) ~=colim B0 ____//_B1___//_B2___//_. . . in the underlying category SymSeq, with B0 := OA[0] and Bt defined inductively by pushout diagrams in SymSeq of the form *_________//_Bt-1 | | | | fflffl| fflffl| OA[t] ~ tY ~t____//_Bt MODULES OVER OPERADS IN SYMMETRIC SPECTRA 19 For each t 1, there are naturally occurring maps Bt-! At, induced by the ap- propriate ,iand jimaps in (4.25), which fit into the commutative diagram A q O O (X q Y ) _i|_| ||f ~= fflffl|fflffl| B0 ____//_B1___//_B2___//._._._//colimtBt_____//_A q (O O Y ) || | | | _ | || | | | , | || j1 fflffl|jfflffl|2j3 fflffl| fflffl| A0 ____//_A1___//_A2___//._._._//colimtAt________colimtAt _ in SymSeq; the morphism of filtered diagrams induces a map ,. We claim_that the right-hand_column_is_a_coequalizer diagram in SymSeq. To verify that , satisfies , i= ,f, by (4.23)it is enough to check that the diagrams _f q,p ~q OA[p + q] ~ px qX ~p~Y ~q____//OA[q] ~ qY |_iq,p |_ | |,inq fflffl| _,inp+q fflffl| OA[p + q] ~ p+qY ~(p+q)______//_colimtAt commute for every q 0 and p > 0; this is easily verified using (4.24)and (4.2* *5), and is_left_to the reader. Let ' : A q (O O Y-)!.be a morphism in SymSeq such that 'i= 'f. We want to_verify that there exists a unique map __': colimtAt-!.in SymSeq_such that ' = __',. Consider the corresponding maps 'iin (4.23)and define '_0:= '0. For each t 1, the maps 'i induce maps __'t: At-!s.uch that __'tjt= ' t-1and __'t,t = 't. In particular, the maps __'tinduce a map __': colimtAt-!. in SymSeq. Using (4.23)it_is an easy exercise (which the reader should verify) that __'satisfies ' = __',and that __'is the unique such map. Hence the filtrat* *ion construction (4.25)satisfies (4.20). One drawback of (4.25)is that it may be di* *fficult to analyze homotopically. A hint at how to improve_the_construction is given by the observation that the collection of maps fq,pand iq,psatisfy many compatibil* *ity relations. To obtain a filtration construction we can homotopically analyze, t* *he idea is to replace (X q Y )~t- Y ~tin (4.25)with the glueing construction Qtt-1* *in Definition 4.11 as follows. Define A0 := OA[0] ~=A and for each t 1 define At by the pushout diagram_ (4.21)in_SymSeq. The maps f* and i* are induced by the appropriate maps fq,p and iq,p. Arguing exactly as above for the case of (4.25), it is easy to use t* *he diagrams (4.23)to verify that (4.20)is satisfied. The only difference is that * *the naturally occurring maps Bt-! At are induced by the appropriate ,i and ji maps in (4.21)instead of in (4.25). The following proposition illustrates some of the good properties of the sta* *ble flat positive model structure on SymSeq. The statement in part (b) is motivated by [6, Lemma 12.7] in the context of symmetric spectra with the stable positive model structure. We defer the proof to Section 6. 20 JOHN E. HARPER op Proposition 4.26. Let B 2 SymSeq t and t 1. If i : X- !Yis a cofibration between cofibrant objects in SymSeq with the stable flat positive model structu* *re, then (a)X ~t-!Y ~tis a cofibration in SymSeq twith the stable flat positive model structure, and hence with the stable flat model structure, (b) the map B ~ tQtt-1-!B ~ tY ~tis a monomorphism. We will prove the following proposition in Section 6. Propositiono4.27.pLet G be a finite group and consider SymSeq, SymSeqG, and SymSeq G each with the stable flat model structure. op (a)If B 2 SymSeqG , then the functor B ~G- : SymSeqG-! SymSeq preserves weak equivalences between cofibrant objects, and hence its tot* *al left derived functor exists. (b) If Z 2 SymSeqG is cofibrant, then the functor op - ~GZ : SymSeqG -! SymSeq preserves weak equivalences. We are now in a good position to give a homotopical analysis of the pushout * *in Proposition 4.3. Proposition 4.28. If the map i : X- !Y in Proposition 4.18 is a generating acyclic cofibration in SymSeq with the stable flat positive model structure, th* *en each map jt is a monomorphism and a weak equivalence. In particular, the map j is a monomorphism and a weak equivalence. Proof.The generating acyclic cofibrations in SymSeq have cofibrant domains. By Proposition 4.26, each jtis a monomorphism. We know At=At-1~=OA[t] ~ t(Y=X)~t and that *-! Y=X is an acyclic cofibration in SymSeq with the stable flat posit* *ive model structure. It follows from Propositions 4.26 and 4.27 that jtis a weak eq* *uiv- alence. Proof of Proposition 4.3.By assumption, the map i : X- !Yis a generating acyclic cofibration in SymSeq with the stable flat positive model structure, hence Prop* *osi- tion 4.28 finishes the proof. 5. Relations between homotopy categories The purpose of this section is to prove Theorem 1.4, which establishes an eq* *uiva- lence between certain homotopy categories of modules (resp. algebras) over oper* *ads. Our argument is a verification of the conditions in [4, Theorem 9.7] for an adj* *unc- tion to induce an equivalence between the corresponding homotopy categories, and amounts to a homotopical analysis (Section 5.1) of the unit of the adjunction. Proof of Theorem 1.4.Let f : O-! O0be a morphism of operads and consider the case of left modules. We know (1.5)is a Quillen adjunction since the forgetful functor f* preserves fibrations and acyclic fibrations. Assume furthermore that* * f is a weak equivalence in the underlying category SymSeqwith the stable flat pos* *itive model structure; let's verify the Quillen adjunction (1.5)is a Quillen equivale* *nce. By [4, Theorem 9.7], it is enough to verify: for cofibrant Z 2 LtO and fibrant MODULES OVER OPERADS IN SYMMETRIC SPECTRA 21 B 2 LtO0, a map , : Z- !f*Bis a weak equivalence in LtOif and only if its adjoi* *nt map j : f*Z-! Bis a weak equivalence in LtO0. Noting that , factors as f*j * Z _____//f*f*Z___//_f B together with Proposition 5.1 below finishes the proof. Argue similarly for the* * case of algebras by considering left modules concentrated at 0. 5.1. Homotopical analysis of the unit of the adjunction. The purpose of this subsection is to prove the following proposition which we used in the proo* *f of Theorem 1.4. Our argument is motivated by the proof of [6, Theorem 12.5]. Proposition 5.1. Let f : O-! O0be a morphism of operads and consider LtO with the stable flat positive model structure. If Z 2 LtOis cofibrant and f is * *a weak equivalence in the underlying category SymSeq with the stable flat positive mod* *el structure, then the natural map Z- !f*f*Z is a weak equivalence in LtO. First we make the following observation. Proposition 5.2. Consider SymSeq with the stable flat positive model structure. If W 2 SymSeq is cofibrant, then the functor - O W: SymSeq-! SymSeq preserves weak equivalences. Proof.Let A-! B be a weak equivalence in SymSeq; we want to verify A[t] ^ t(W ~t)[r]-! B[t] ^ t(W ~t)[r] is a weak equivalence in Sp with the stable flat model structure for each r, t* * 0. By Proposition 4.26 we know W ~tis cofibrant in SymSeq t with the stable flat model structure for each t 1. By considering symmetric sequences concentrated at 0, Proposition 4.27 finishes the proof. Proof of Proposition 5.1.Let X- !Y be a generating cofibration in SymSeq with the stable flat positive model structure, and consider the pushout diagram (5.3) O O X ____//_Z0 | | | | fflffl| fflffl| O O Y____//_Z1 in LtO. For each W 2 SymSeq consider the natural maps (5.4) Z0q (O O W )-! f*f* Z0q (O O W ) , (5.5) Z1q (O O W )-! f*f* Z1q (O O W ) , and note that the left-hand (resp. right-hand) diagram O O X_____//_Z0q (O O W ) =: A O0O X ______//f*Z0q (O0O W ) =: A0 | | | | | | | | fflffl| fflffl| fflffl| fflffl| O O Y_____//Z1q (O O W ) ~=A1 O0O Y ____//_f*Z1q (O0O W ) ~=f*A1 is pushout diagram in LtO (resp. LtO0). Assume (5.4)is a weak equivalence for every cofibrant W 2 SymSeq; let's verify (5.5)is a weak equivalence for every 22 JOHN E. HARPER cofibrant W 2 SymSeq. Suppose W 2 SymSeq is cofibrant. By Proposition 4.18 there are corresponding filtrations A0 ____//_A1___//__A2__//_._._.//_colimtAt_____A1 ___ _____ | | |,0| _,1______,2_____ | | fflffl| fflffl___fflffl___ fflffl|~= fflffl| A00____//_A01__//_A02__//._._._//colimtA0t__//f*f*A1 , together with induced maps ,t(t 1) which make the diagram in SymSeqcommute. By assumption we know ,0 is a weak equivalence, and to verify (5.5)is a weak eq* *uiv- alence, it is enough to check that ,tis a weak equivalence for each t 1. Sinc* *e the horizontal maps are monomorphisms and we know At=At-1~=OA[t] ~ t(Y=X)~t, it is enough to verify that A q (O O (Y=X))___//_A0q (O0O (Y=X)) is a weak equivalence, which is the same as verifying that Z0q (O O W ) q (O O (Y=X))-! f*f* Z0q (O O W ) q (O O (Y=X)) is a weak equivalence. Noting that W q (Y=X) is cofibrant finishes the argument that (5.5)is a weak equivalence. Consider a sequence Z0 ____//_Z1___//Z2____//. . . of pushouts of maps as in (5.3). Assume Z0 makes (5.4)a weak equivalence for every cofibrant W 2 SymSeq; we want to show that for Z1 := colimkZk the natural map (5.6) Z1 q (O O W )-! f*f* Z1 q (O O W ) is a weak equivalence for every cofibrant W 2 SymSeq. Consider the diagram Z0q (O O W )_________//Z1q (O O W_)________//Z2q (O O W_)_____//_. . . | | | | | | fflffl| fflffl| fflffl| f*f* Z0q (O O W )____//_f*f* Z1q (O O W_)__//_f*f* Z2q (O O W_)__//_. . . in LtO. The horizontal maps are monomorphisms and the vertical maps are weak equivalences, hence the induced map (5.6)is a weak equivalence. Noting that eve* *ry cofibration O O *-! Z in LtOis a retract of a (possibly transfinite) compositio* *n of pushouts of maps as in (5.3), starting with Z0 = O O *, together with Propositi* *on 5.2, finishes the proof. 6. Proofs The purpose of this section is to prove Propositions 4.26 and 4.27; we have * *also included a proof of Proposition 3.25 at the end of this section. First we estab* *lish a characterization of stable flat cofibrations. MODULES OVER OPERADS IN SYMMETRIC SPECTRA 23 6.1. Stable flat cofibrations. The purpose of this subsection is to prove Propo- sition 6.5, which identifies stable flat cofibrations in SymSeqG, for G a finit* *e group. It is proved in [39] that the following model category structure exists on l* *eft n-objects in pointed simplicial sets. Definition 6.1. Let n 0. o The mixed n-equivariant model structure on S*n has weak equivalences the underlying weak equivalences of simplicial sets, cofibrations the re* *tracts of (possibly transfinite) compositions of pushouts of maps n=H . @ [k]+- ! n=H . [k]+ (k 0, H n subgroup), and fibrations the maps with the right lifting property with respect to * *the acyclic cofibrations. Furthermore, it is proved in [39] that this model structure is cofibrantly g* *enerated in which the generating cofibrations and acyclic cofibrations have small domain* *s, and that the cofibrations are the monomorphisms. It is easy to prove that the diagram category of ( oprx G)-shaped diagrams in S*n appearing in the following proposition inherits a corresponding projective model structure. This propositi* *on, whose proof is left to the reader, will be needed for identifying stable flat c* *ofibrations in SymSeqG. Proposition 6.2. LetoGpbe a finite group and consider any n, r 0. The dia- gram category S*n r xG inherits a corresponding projective model structure fr* *om the mixed n-equivariant model structure on S*n. The weak equivalences (resp. fibrations) are the underlying weak equivalences (resp. fibrations) in S*n and * *the cofibrations are the monomorphisms such that oprx G acts freely on the simplic* *es of the codomain not in the image. __ __ __ Definition 6.3. Define S 2 Sp such that Sn := Sn for n 1 and S0 := *. The structure maps are the_naturally_occurring ones such that there exists a map of symmetric spectra i : S-! Ssatisfying in = idfor each n 1. The following calculation, which follows easily from 2.7 and 2.11, will be n* *eeded for characterizing stable flat cofibrations in SymSeqG below. Calculation 6.4. Let m, p 0, H m a subgroup, and K a pointed simplicial set. Define X := G . Gp(S GHmK) 2 SymSeqG. Here, X is obtained by applying the indicated functors in (4.1)to K. Then for r = p we have __ aeG . n . x __Sn-m^( m =H . K) .fopr n > m, (S ^X[r])n~= n-m m * for n m, 8 < G . n . n-mx m Sn-m ^ ( m =H . K) .fopr n > m, X[r]n ~=: G . ( m =H . K) . fpor n = m, * for n < m. __ and for r 6= p we have X[r] = * = S^ X[r]. The following characterization of stable flat cofibrations in SymSeqG is mot* *ivated by [21, Proposition 5.2.2]; we benefitted from the discussion and corresponding characterization in [37] of cofibrations in Sp with the stable flat model stru* *cture. Proposition 6.5. Let G be a finite group. 24 JOHN E. HARPER (a)A map f : X- !Y in SymSeqG with the stable flat model structure is a cofibration if and only if the induced maps X[r]0-! Y [r]0,r 0, n = 0, __ (S ^Y [r])n q(_S^X[r])nX[r]n-! Y [r]n,r 0, n 1, opxG are cofibrations in S*n r with the model structure of Proposition 6* *.2. (b) A map f : X- !Yin SymSeqG with the stable flat positive model structure is a cofibration if and only if the maps X[r]0-! Y [r]0, r 0, are isom* *or- phisms, and the induced maps __ (S ^Y [r])n q(_S^X[r])nX[r]n-! Y [r]n,r 0, n 1, opxG are cofibrations in S*n r with the model structure of Proposition 6* *.2. Proof.It suffices to prove part (a). Consider any f : X- !Yin SymSeqG with the stable flat model structure. We want a sufficient condition for f to be a cofib* *ration. The first step is to rewrite a lifting problem as a sequential lifting problem. __ X _____//E>>___X[r]n___//_E[r]n;;__(S Y [r])n//_Y [r]n | ____|___ | _____|___ | | | _____|_ | _______| | | fflffl|fflffl|___fflffl|_fflffl|____fflffl| fflffl| Y _____//B Y [r]n___//_B[r]n (__SE[r])n___//_E[r]n The left-hand solid commutative diagram in SymSeqGohaspa lift if and only if the right-hand sequence of lifting problems in S*n r xG has a solution, if and on* *ly if the sequence of lifting problems __ X[r]n____//_E[r]n;;__(S ^Y [r])n_//Y [r]n | _____|____ | | | _______| | | fflffl|___fflffl|___ fflffl| fflffl| Y [r]n___//_B[r]n (S ^E[r])n____//E[r]n opxG in S*n r has a solution, if and only if the sequence of lifting problems __ X[r]0____//_E[r]0;;__(S ^Y [r])n q(_S^X[r])nX[r]n//_E[r]n55_ _____ _____________ (*)0||_______||___ (*)n|| _____________||__ fflffl|___fflffl|__ fflffl|_____________fflffl| Y [r]0___//_B[r]0 Y [r]n_____________//B[r]n (n 1) has a solution. If each (*)n is a cofibration then f has the left lifting prop* *erty with respect to all acyclic fibrations, and hence f is a cofibration. Converse* *ly, suppose f is a cofibration. We want to verify that each (*)n is a cofibration. Every cofibration is a retract of a (possibly transfinite) composition of pusho* *uts of generating cofibrations, and hence by a reduction argument that we leave to the reader, it is sufficient to verify for f a generating cofibration. Let g : K- !* *Lbe a monomorphism in S*, m, p 0, H m a subgroup, and define f : X- !Yin SymSeq Gto be the induced map g* H G . Gp(S GHmK)____//G . Gp(S Gm.L) MODULES OVER OPERADS IN SYMMETRIC SPECTRA 25 Here, the map g* is obtained by applying the indicated functors in (4.1)to the * *map g. We know (*)0 is a cofibration. Consider n 1. By Calculation 6.4: (*)n is an isomorphism for the case r 6= p and for the case (r = p and n 6= m). For the ca* *se (r = p and n = m), (*)n is the map G.( m=H.g). p G . ( m =H . K) .__p_______//G . ( m =H . L) . p Hence in all cases (*)n is a cofibration. 6.2. Proofs. Proof of Proposition 4.27.Consider part (b). Let g : K- !Lbe a monomorphism in S*, m, p 0, H m a subgroup, and consider the pushout diagram (6.6) G . Gp(S GHmK)____//Z0 g*|| || fflffl| fflffl| G . Gp(S GHmL)____//Z1 in SymSeqG. Here, the map g* is obtained by applying the indicated functors in (4.1)to the map g. Consider the functors op (6.7) - ~GZ0: SymSeqG -! SymSeq, op (6.8) - ~GZ1: SymSeqG -! SymSeq, and assume (6.7)preserves weak equivalences;olet'spverify (6.8)preserves weak equivalences. Suppose A-! B in SymSeqG is a weak equivalence. Applying A ~G- to (6.6)gives the pushout diagram A ~Gp(S GHmK)_____//A ~GZ0 (*)|| |(**)| fflffl| fflffl| A ~Gp(S GHmL)_____//A ~GZ1 in SymSeq. Let's check (*) is a monomorphism. This amounts to a calculation: H ae A[r - p] ^(S GH K) . x1 rfor r p A ~Gp(S Gm K) [r] ~= m r-p* for r < p Since the map S GHmK- !S GHmL is a cofibration in Sp with the stable flat model structure, smashing with any symmetric spectrum gives a monomorphism. It follows that (*) is a monomorphism, and hence (**) is a monomorphism. Consid* *er the commutative diagram A ~GZ0 ____//_A ~GZ1___//A ~Gp(S GHm(L=K)) | | | | | | fflffl| fflffl| fflffl| B ~GZ0 ____//_//_B ~GZ1//_B ~Gp(S GHm(L=K)). Since S GHm(L=K) is cofibrant in Sp with the stable flat model structure, smas* *h- ing with it preserves weak equivalences. It follows that the right-hand vertica* *l map 26 JOHN E. HARPER is a weak equivalence. By assumption, the left-hand vertical map is a weak equi* *v- alence, hence the middle vertical map is a weak equivalence and we get that (6.* *8) preserves weak equivalences. Consider a sequence Z0 ____//_Z1___//Z2___//_. . . of pushouts of maps as in (6.6). Assume (6.7)preserves weak equivalences; we wa* *nt to show that for Z1 := colimkZk the functor op - ~GZ1 : SymSeqG -! SymSeq op preserves weak equivalences. Suppose A-! B in SymSeqG is a weak equivalence and consider the diagram A ~GZ0 ____//_A ~GZ1___//_A ~GZ2___//. . . | | | | | | fflffl| fflffl| fflffl| B ~GZ0 ____//_B ~GZ2___//_B ~GZ3___//. . . in SymSeq. The horizontal maps are monomorphisms and the vertical maps are weak equivalences, hence the induced map A ~GZ1 -! B ~GZ1 is a weak equiva- lence. Noting that every cofibration *-! Z in SymSeqG is a retract of a (possib* *ly transfinite) composition of pushouts of maps as in (6.6), starting with Z0 = *, finishes the proof of part (b). Consider part (a). Suppose X- !Y in SymSeqG is a weak equivalence between cofibrant objects; we want to showothatpB ~GX- !B ~GY is a weak equivalence. The map *-! B factors in SymSeqG as *____//_Bc__//_B a cofibration followed by an acyclic fibration, the diagram Bc~ GX ____//_Bc~ GY | | | | fflffl| fflffl| B ~GX _____//_B ~GY commutes, and since three of the maps are weak equivalences, so is the fourth. op Proposition 6.9. Let G be a finite group. If B 2 SymSeqG , then the functor B ~G- : SymSeqG-! SymSeq sends cofibrations in SymSeqG with the stable flat model structure to monomor- phisms. Proof.Let g : K- !Lbe a monomorphism in S*, m, p 0, H m a subgroup, and consider the pushout diagram (6.10) G . Gp(S GHmK)____//Z0 g*|| || fflffl| fflffl| G . Gp(S GHmL)____//Z1 MODULES OVER OPERADS IN SYMMETRIC SPECTRA 27 in SymSeqG. Here, the map g* is obtained by applying the indicated functors in (4.1)to the map g. Applying B ~G- gives the pushout diagram B ~Gp(S GHmK)_____//B ~GZ0 (*)|| |(**)| fflffl| fflffl| B ~Gp(S GHmL)_____//B ~GZ1 in SymSeq. The map (*) is a monomorphism by the same arguments used in the proof of Proposition 4.27, hence (**) is a monomorphism. Noting that every cofi- bration in SymSeqG is a retract of a (possibly transfinite) composition of push* *outs of maps as in (6.10)completes the proof. The following two propositions are exercises left to the reader. Proposition 6.11. Let t 1. If the left-hand diagram is a pushout diagram X ____//_A Qtt-1(i)___//Qtt-1(j) |i| j|| || || |fflffl fflffl| fflffl| fflffl| Y ____//_B Y ~t_______//_B ~t in SymSeq, then the corresponding right-hand diagram is a pushout diagram in SymSeq t. Proposition 6.12. Let t 1 and consider a commutative diagram of the form A __s_//_B_r_//_C |i| |j| k|| fflffl||fflfflsfflffl|r X ____//_Y___//_Z in SymSeq. Then the corresponding diagram _s _r Qtt-1(i)___//Qtt-1(j)__//Qtt-1(k) | | | | | | fflffl| |fflffl fflffl| X ~t_______//_Y ~t_____//_Z ~t __ in SymSeq tcommutes. Furthermore, _r_s= __rsand id= id. The following calculation, which follows easily from (2.7), (2.11), and (3.5* *), will be needed in the proof of Proposition 4.26 below. Calculation 6.13. Let k, m, p 0, H m a subgroup, and t 1. Let the map g : @ [k]+-! [k]+be a generating cofibration for S* and define X- !Y in SymSeq to be the induced map g* H Gp(S GHm@ [k]+)____//_Gp(S Gm [k]+). 28 JOHN E. HARPER Here, the map g* is obtained by applying the indicated functors in (4.1)to the * *map g. For r = tp we have the calculation 8 xt ~t < n . n-tmxHxtSn-tm ^( [k] )+ . tpfor n > tm, (Y )[r] n~=: tm .Hxt ( [k]xt)+ . ftpor n = tm, * for n < tm. __ ~t ae n . xHxt__Sn-tm^( [k]xt)+ . ftpor n > tm, S^ (Y )[r] n~= n-tm * for n tm, 8 xt t < n . n-tmxHxtSn-tm ^@( [k] )+ . tpfor n > tm, Qt-1[r] n~=: tm .Hxt @( [k]xt)+ . ftpor n = tm, * for n < tm. __ t ae n . xHxt__Sn-tm^@( [k]xt)+ . ftpor n > tm, S ^ Qt-1[r] n~= n-tm * for n tm, __ ~ __ and for r 6= tp we have (Y ~t)[r] = * = S^ (Y t)[r] and Qtt-1[r] = * = S^ Qtt-* *1[r]. The following proposition is proved in [3, I.2] and will be useful below for* * verifying that certain induced maps are cofibrations. Proposition 6.14. Let M be a model category and consider a commutative diagram of the form A0 oo___A1_____//A2 | | | | | | fflffl| fflffl||fflffl B0 oo___B1_____//B2 in M. If the maps A0-! B0 and B1qA1A2-! B2 are cofibrations, then the induced map A0qA1 A2-! B0qB1 B2 is a cofibration. Proof of Proposition 4.26.Consider part (a). The argument is by induction on t. Let m 1, H m a subgroup, and k, p 0. Let g : @ [k]+-! [k]+be a generating cofibration for S* and consider the pushout diagram (6.15) Gp(S GHm@ [k]+)______//_Z0 |g*| |i0| |fflffl fflffl| D := Gp(S GHm [k]+)____//Z1 in SymSeqwith Z0 cofibrant. Here, the map g* is obtained by applying the indica* *ted functors in (4.1)to the map g. By Proposition 6.11, the corresponding diagram Qtt-1(g*)___//_Qtt-1(i0) |(*)| |(**)| fflffl| fflffl|~ D ~t________//_Z1 t MODULES OVER OPERADS IN SYMMETRIC SPECTRA 29 is a pushout diagram in SymSeq t. Since m 1, it follows from Proposition 6.5 and Calculation 6.13 that (*) is a cofibration in SymSeq t, and hence (**) is a cofibration. Consider a sequence (6.16) Z0 _i0_//_Z1i1_//Z2_i2//_. . . of pushouts of maps as in (6.15), define Z1 := colimqZq, and consider the natur* *ally occurring map i1 : Z0-! Z1. Using Proposition 6.14 and (4.12), it is easy to verify that each Zq~t-!Qtt-1(iq) is a cofibration in SymSeq t, and by above we know that each Qtt-1(iq)-! Zq~t+1is a cofibration; it follows immediately that * *each Zq~t-!Zq~t+1is a cofibration in SymSeq t, and hence the map Z0~t-!Z1~tis a cofibration. Noting that every cofibration between cofibrant objects in SymSeq with the stable flat positive model structure is a retract of a (possibly trans* *finite) composition of pushouts of maps as in (6.15)finishes the proof for part (a). Co* *nsider part (b). Proceed as above for part (a) and consider the commutative diagram (6.17) Z0~t____//Qtt-1(i0)_//_Qtt-1(i1i0)//_Qtt-1(i2i1i0)//_. . . || | | | || | | | || fflffl| fflffl| fflffl| Z0~t______//Z1~t_______//Z2~t_________//Z3~t______//. . . in SymSeq t. We claim that (6.17)is a diagram of cofibrations. By part (a), the bottom row is a diagram of cofibrations. Using Proposition 6.14 and (4.12), it * *is easy to verify that if i and j are composable cofibrations between cofibrant ob* *jects in SymSeq, then the induced maps Qtt-1(i)-! Qtt-1(ji)-! Qtt-1(j) are cofibrations in SymSeq t; it follows easily that the vertical maps and the * *top row maps are cofibrations. Applying B ~ t- to (6.17)gives the commutative diagram (6.18) B ~ tZ0~t____//_B ~ tQtt-1(i0)//_B ~ tQtt-1(i1i0)//_. . . || | | || | | || fflffl| fflffl| B ~ tZ0~t______//_B ~ tZ1~t______//B ~ tZ2~t_____//_. . . in SymSeq. By Proposition 6.9, (6.18)is a diagram of monomorphisms, hence the induced map B ~ tQtt-1(i1 )-! B ~ tZ1~tis a monomorphism. Noting that every cofibration between cofibrant objects in SymSeq is a retract of a (possib* *ly transfinite) composition of pushouts of maps as in (6.15), together with Propos* *ition 6.12, finishes the proof for part (b). Proof of Proposition 3.25.Suppose A : D-! LtOis a small diagram. We want to show that colimA exists. It is easy to verify, using Proposition 4.8, that this* * colimit may be calculated by a reflexive coequalizer in LtOof the form, i (mOid)*oo_ j colimA ~=colim colimd2D(O O(Ad)icolimd2D(OdOOOmO)Ad)*oo_, 30 JOHN E. HARPER provided that the indicated colimits appearing in this reflexive pair exist in * *LtO. The underlying category SymSeq has all small colimits, and left adjoints preser* *ve colimiting cones, hence there is a commutative diagram (mOid)* colimd2D(O OoAd)o_colimd2D(OoOoO_O Ad) (idOm)* ~=|| |~=| fflffl| fflffl| O O colimd2DAdooO_Ooocolimd2D(O_O Ad) in LtO; the colimits in the bottom row exist since they are in the underlying c* *ategory SymSeq (we have dropped the notation for the forgetful functor U), hence the colimits in the top row exist in LtO. Therefore colimA exists and Proposition 3* *.23 completes the proof. 7.Constructions in the special case of algebras over an operad Some readers may only be interested in the special case of algebras over an operad and may wish to completely avoid working with the circle product and the left O-module constructions. It is easy to translate the constructions and proo* *fs in this paper into the special case of algebras while avoiding the circle produ* *ct notation. Usually, this amounts to replacing (SymSeq, ~) with (Sp , ^), replaci* *ng the left adjoint O O -: SymSeq-! LtOwith the left adjoint O(-) : Sp -! AlgO (Definition 3.14), and then replacing the symmetric array OA in Proposition 4.6 with the symmetric sequence OA in Proposition 7.1 below. We illustrate below with several special cases of particular interest. 7.1. Special cases. Proposition 4.6 has the following special case. Proposition 7.1. Let O be an operad in symmetric spectra, A 2 AlgO, and Y 2 Sp . Consider any coproduct in AlgO of the form (7.2) A q O(Y ). There exists a symmetric sequence OA and natural isomorphisms a A q O(Y ) ~= OA[q] ^ qY ^q q 0 in the underlying category Sp . If q 0, then OA[q] is naturally isomorphic to* * a colimit of the form ` ` ^pod0o_` ^p' OA[q] ~=colim p 0O[p + q] ^ pA ood1_p 0O[p + q] ^ p(O(A)) , in Sp , with d0 induced by operad multiplication and d1 induced by m : O(A)-! A. Definition 4.11 has the following special case. MODULES OVER OPERADS IN SYMMETRIC SPECTRA 31 Definition 7.3. Let i : X- !Ybe a morphism in Sp and t 1. Define Qt0:= X^t and 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*|| || fflffl| fflffl| t. t-qx qX^(t-q)^Y ^q______//Qtq t in Sp . We sometimes denote Qtqby Qtq(i) to emphasize in the notation the map i : X- !Y. The maps pr*and i* are the obvious maps induced by i and the appropriate projection maps. Proposition 4.18 has the following special case. Proposition 7.4. Let O be an operad in symmetric spectra, A 2 AlgO, and i : X- !Yin Sp . Consider any pushout diagram in AlgO of the form, (7.5) O(X) ____f_____//A id(i)|| j|| fflffl| fflffl| O(Y )____//_A qO(X)O(Y ). The pushout in (7.5)is naturally isomorphic to a filtered colimit of the form i j1 j2 j3 j A qO(X)O(Y ) ~=colim A0 _____//A1___//_A2___//_. . . in the underlying category Sp , with A0 := OA[0] ~=A and At defined inductively by pushout diagrams in Sp of the form OA[t] ^ tQtt-1f*_//_At-1 |id^|ti* jt|| fflffl| ,t fflffl| OA[t] ^ tY ^t_____//At Propositions 4.26, 4.27, and 4.28 have the following special cases, respecti* *vely. op Proposition 7.6. Let B 2 Sp t and t 1. If i : X- !Yis a cofibration between cofibrant objects in Sp with the stable flat positive model structure,* * then t (a)X^t-! Y ^tis a cofibration in Sp with the stable flat positive model structure, and hence with the stable flat model structure, (b) the map B ^ tQtt-1-!B ^ tY ^tis a monomorphism. G Gop Proposition 7.7. Let G be a finite group and consider Sp , Sp , and Sp each with the stable flat model structure. Gop (a)If B 2 Sp , then the functor G B ^G- : Sp -! 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