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 Omodules and Oalgebras in symmet
ric spectra, which in practice means putting a Quillen model structure on these
categories of modules and algebras. In this setting, Oalgebras are the same as
left Omodules 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
Omodules and the category of Oalgebras 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
Smodel 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
Omodules and the category of Oalgebras 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 Omodules (resp. Oalgebras) 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 tdiagrams 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 nf*
*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 nequivariant.
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 nequivariant
for p 1 and n 0. Here, oep := oe(S1^ oe) . .(.Sp1^ oe) is the compo*
*sition
of the maps
i^oe
Si^ S1^ Xn+p1iS____//Si^ Xn+pi.
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
fflffloe 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 naction 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 Xcopies 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[ss1(1)] ^Y [ss1(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 Smodule.
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 Xcopies 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[ss1(1)] ^. .^.At[ss1(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[ss1(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 (twosided) 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 Omodule is an object in (SymSeq, O*
*, I)
with a left action of O and a morphism of left Omodules is a map in SymSeqwhich
respects the left Omodule 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 Oalgebra is an object in Sp with a
left action of the monad O : Sp ! Sp and a morphism of Oalgebras is a map in
Sp which respects the left action of the monad O : Sp ! Sp.
It is easy to verify that an Oalgebra is the same as an object X 2 Sp with*
* a
left Omodule structure on ^X, and if X and X0 are Oalgebras, then a morphism
of Oalgebras is the same as a map f : X !X0in Sp such that ^f: ^X!X^0is a
morphism of left Omodules. In other words, an algebra over an operad O is the
same as a left Omodule which is concentrated at 0.
Giving a symmetric sequence Y a left Omodule 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 Oalgebra 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 Oalgebra 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 tary opera*
*tions.
Definition 3.19. Let O be an operad.
o LtOis the category of left Omodules and their morphisms.
o AlgOis the category of Oalgebras and their morphisms.
The category AlgO embeds in LtO as the full subcategory of left Omodules
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 A1 oo__A0_oo__A1_oo_and B1 oo__B0_oo___B1oo_are reflexive coequal
izer diagrams in SymSeq, then their objectwise circle product
A1 O B1oo___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_A1oo___O O O O A0oo__OoOoO_O A1
________  
d0_______d1_____mOididOm mOididOm
fflffl___fflffl___fflfflfflfflfflfflfflffl
O O_A1oo______COCOCA0oo______OoOoA1_C_CC_
______ _____________________________________________*
*_________________
s0___m________jOid_m_____________m________________jOid________*
*______________________
__fflffl___ __fflffl________fflffl_______________________*
*_____
A1 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 A1 the structure of a left Omodule 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 selfcontained, 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 Omodules 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 selfcontained, 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 OA!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
s1 := j O id: A! O O A
s1 := j O idO:idO OA!O O O O A
in the underlying category SymSeq which satisfy the relations
d0d0 = d0d1, d0s1 = id, d1s1 = s1d0.
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. tqx qX ~(tq)~Qqq1___//Qq1
i* 
fflffl fflffl
t. tqx qX ~(tq)~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.1x1id~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.2x1id~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.1x2id~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 Qt1tcan be thought of as a tequivariant ver
sion of the colimit of a punctured tcube [16]. There is a natural isomorphism
Y ~t=Qtt1~=(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 selfcontained, 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] ~ tQtt1f*_//At1
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*
fflfflfflffl fflfflfflffld0 fflfflfflffl
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 lefthand 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 lefthand 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 lefthand 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 warmup 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__________//At1

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
*_________//_Bt1
 
 
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
~= fflfflfflffl
B0 ____//_B1___//_B2___//._._._//colimtBt_____//_A q (O O Y )
    _ 
    , 
 j1 fflffljfflffl2j3 fflffl fflffl
A0 ____//_A1___//_A2___//._._._//colimtAt________colimtAt
_
in SymSeq; the morphism of filtered diagrams induces a map ,. We claim_that the
righthand_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=
' t1and __'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 Qtt1*
*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 ~ tQtt1!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=At1~=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 lefthand (resp. righthand) 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=At1~=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
nobjects in pointed simplicial sets.
Definition 6.1. Let n 0.
o The mixed nequivariant 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 nequivariant 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 __Snm^( m =H . K) .fopr n > m,
(S ^X[r])n~= nm m * for n m,
8
< G . n . nmx m Snm ^ ( 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
 _______  ________  
 ______  _______  
fflfflfflffl___fflffl_fflffl____fflffl fflffl
Y _____//B Y [r]n___//_B[r]n (__SE[r])n___//_E[r]n
The lefthand solid commutative diagram in SymSeqGohaspa lift if and only if the
righthand 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 rp* 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 righthand vertica*
*l map
26 JOHN E. HARPER
is a weak equivalence. By assumption, the lefthand 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 lefthand diagram is a pushout diagram
X ____//_A Qtt1(i)___//Qtt1(j)
i j  
fflffl fflffl fflffl fflffl
Y ____//_B Y ~t_______//_B ~t
in SymSeq, then the corresponding righthand 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
fflfflfflfflsfflfflr
X ____//_Y___//_Z
in SymSeq. Then the corresponding diagram
_s _r
Qtt1(i)___//Qtt1(j)__//Qtt1(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 . ntmxHxtSntm ^( [k] )+ . tpfor n > tm,
(Y )[r] n~=: tm .Hxt ( [k]xt)+ . ftpor n = tm,
* for n < tm.
__ ~t ae n . xHxt__Sntm^( [k]xt)+ . ftpor n > tm,
S^ (Y )[r] n~= ntm * for n tm,
8 xt
t < n . ntmxHxtSntm ^@( [k] )+ . tpfor n > tm,
Qt1[r] n~=: tm .Hxt @( [k]xt)+ . ftpor n = tm,
* for n < tm.
__ t ae n . xHxt__Sntm^@( [k]xt)+ . ftpor n > tm,
S ^ Qt1[r] n~= ntm * for n tm,
__ ~ __
and for r 6= tp we have (Y ~t)[r] = * = S^ (Y t)[r] and Qtt1[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 fflfflfflffl
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
Qtt1(g*)___//_Qtt1(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!Qtt1(iq) is a cofibration in SymSeq t, and by above we
know that each Qtt1(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____//Qtt1(i0)_//_Qtt1(i1i0)//_Qtt1(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
Qtt1(i)! Qtt1(ji)! Qtt1(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 ~ tQtt1(i0)//_B ~ tQtt1(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 ~ tQtt1(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 Omodule 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. tqx qX^(tq)^Qqq1____//Qq1
i* 
fflffl fflffl
t. tqx qX^(tq)^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] ^ tQtt1f*_//_At1
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 ^ tQtt1!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 ! Sp
preserves weak equivalences between cofibrant objects, and hence its tot*
*al
left derived functor exists.
32 JOHN E. HARPER
G
(b) If Z 2 Sp is cofibrant, then the functor
Gop
 ^GZ : Sp ! Sp
preserves weak equivalences.
Proposition 7.8. If the map i : X !Yin Proposition 7.4 is a generating acyclic
cofibration in Sp with the stable flat positive model structure, then each map*
* jt is
a monomorphism and a weak equivalence. In particular, the map j is a monomor
phism and a weak equivalence.
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Institut de g'eom'etrie, alg`ebre et topologie, EPFL, CH1015 Lausanne, Swit*
*zerland
Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, U*
*SA
Email address: john.edward.harper@gmail.com