HOMOTOPY THEORY OF MODULES OVER OPERADS AND
NON OPERADS IN MONOIDAL MODEL CATEGORIES
JOHN E. HARPER
1. Introduction
There are many interesting situations in which algebraic structure can be de
scribed by operads [1, 12, 13, 14, 17, 20, 27, 32, 33, 34, 35]. Let (C, , k) *
*be a
symmetric monoidal closed category (Section 2) with all small limits and colimi*
*ts.
It is possible to define two types of operads (Definition 6.1) in this setting,*
* as well
as algebras and modules over these operads. One type, called operad, is based*
* on
finite sets and incorporates symmetric group actions; the other type, called no*
*n
operad, is based on ordered sets and has no symmetric group contribution. (In t*
*his
paper we use the term operad for non operad, where = O for "Ordered.")
Given an operad O, we are interested in the possibility of doing homotopy theory
in the categories of Omodules and Oalgebras, which in practice means putting
a Quillen model structure on these categories of modules and algebras. In this
setting, Oalgebras are left Omodules concentrated at 0 (Section 6.3).
Of course, to get started we need some kind of homotopy theoretic structure on
C itself; this structure should mesh appropriately with the monoidal structure *
*on
C. The basic assumption is this.
Basic Assumption 1.1. From now on in this paper we assume that (C, , k) is a
symmetric monoidal closed category (Section 2) with all small limits and colimi*
*ts,
that C is a cofibrantly generated model category (Definition 11.5) in which the*
* gener
ating cofibrations and acyclic cofibrations have small domains, and that with r*
*espect
to this model structure (C, , k) is a monoidal model category (Definition 11.7*
*).
The main theorem is this.
Theorem 1.2. Assume that C satisfies Basic Assumption 1.1 and in addition
satisfies the monoid axiom (Definition 13.25). Let O be an operad in C. Then
the category of left Omodules and the category of Oalgebras both have natural
model category structures. The weak equivalences and fibrations in these model
structures are inherited in an appropriate sense from the weak equivalences and
fibrations in C.
Remark 1.3. Given any operad O, there is an associated operad O . , such
that algebras over O . are the same as algebras over O (Section 7). It follows
easily from the above theorem that if O0is a operad which is a retract of O .*
* ,
then the category of algebras over O0has a natural model category structure.
The above remark shows how to handle algebras over certain operads. We
can do a lot better if C satisfies a strong cofibrancy condition. In setting up*
* the
machinery for Theorems 1.2 and 1.4, we introduce model category structures on
1
2 JOHN E. HARPER
the category of (symmetric) sequences in C (Definition 3.1) and on the category*
* of
(symmetric) arrays in C (Definition 13.1).
Theorem 1.4. Assume that C satisfies Basic Assumption 1.1 and in addition that
every symmetric array (resp. symmetric sequence) in C is cofibrant in the model
category structure described below (Theorems 13.2 and 12.2). Then for any 
operad O in C, the category of left Omodules (resp. Oalgebras) has a natural *
*model
category structure. The weak equivalences and fibrations in these model structu*
*res
are inherited in an appropriate sense from the weak equivalences and fibrations*
* in
C.
1.1. Some examples of interest. The hypotheses of these theorems may seem
restrictive, but in fact they allow, especially in the case of Theorem 1.2, for*
* many
interesting examples including the case (sSet, x, *) of simplicial sets [5, 9, *
*15, 37],
the case (Chk, , k) of unbounded chain complexes over a commutative ring with
unit [22, 29], and the case (Sp , ^, S) of symmetric spectra [24]. In a related*
* paper
[16], we improve Theorem 1.2 to operads for the case (Sp , ^, S) of symmetric
spectra.
1.2. Relationship to previous work. One of the main theorems of Schwede and
Shipley [39] is that the category of monoids in (C, , k) has a natural model c*
*ategory
structure, provided the monoid axiom (Definition 13.25) is satisfied. Theorem 1*
*.2
improves this result to left modules and algebras over any operad.
One of the main theorems of Hinich [18, 19] is that for unbounded chain com
plexes over a field of characteristic zero, the category of algebras over any *
*operad
has a natural model category structure. Theorem 1.4 improves this result to the
category of left modules, and also provides (Section 14) a simplified conceptual
proof of Hinich's original result. In this rational case our theorem is this.
Theorem 1.5. Let k be a field of characteristic zero and let (Chk, , k) be the
symmetric monoidal closed category of unbounded chain complexes over k. Let O be
any operad or operad. Then the category of left Omodules (resp. Oalgebras)
has a natural model category structure. The weak equivalences are the objectwise
homology isomorphisms (resp. homology isomorphisms) and the fibrations are the
objectwise dimensionwise surjections (resp. dimensionwise surjections).
Another theorem of Hinich [18] is that for unbounded chain complexes over a
commutative ring with unit, the category of algebras over any operad of the f*
*orm
O . for some operad O, has a natural model category structure. Theorem 1.2
improves this result to the category of left modules. Our theorem is this.
Theorem 1.6. Let k be a commutative ring with unit and let (Chk, , k) be the
symmetric monoidal closed category of unbounded chain complexes over k. Let
O be any operad. Then the category of left Omodules (resp. Oalgebras) has a
natural model category structure. The weak equivalences are the objectwise homo*
*logy
isomorphisms (resp. homology isomorphisms) and the fibrations are the objectwise
dimensionwise surjections (resp. dimensionwise surjections).
One of the main theorems of Elmendorf and Mandell [6] is that the category of
simplicial multifunctors from a small multicategory (enriched over simplicial s*
*ets)
to the category of symmetric spectra has a natural simplicial model category st*
*ruc
ture. Their proof involves a filtration in the underlying category of certain p*
*ushouts
HOMOTOPY THEORY OF MODULES OVER OPERADS 3
of algebras. We have benefitted from their paper and our proofs of Theorems 1.2
and 1.4 exploit similar filtrations (Section 13). In the special case of unboun*
*ded
chain complexes, the analysis of certain pushouts reduces to an analysis of cer
tain coproducts and filtration constructions are not required. We have included*
* a
shortened proof for this special case (Section 14).
The framework presented in this paper for doing homotopy theory in the cate
gories of modules and algebras over an operad is largely influenced by Rezk [38*
*].
Acknowledgments. The author would like to thank Bill Dwyer for his constant
encouragement and invaluable help and advice. The author is grateful to Emmanuel
Farjoun for a stimulating and enjoyable visit to Hebrew University of Jerusalem*
* in
spring 2006 and for his invitation which made this possible, and to Paul Goerss*
* and
Mike Mandell for helpful comments and suggestions at a Midwest Topology Semi
nar. Parts of this paper were completed while the author was a visiting researc*
*her
at the Thematic Program on Geometric Applications of Homotopy Theory at the
Fields Institute for Mathematics, Toronto.
2.Preliminaries on group actions
Here, for reference purposes, we collect certain basic properties of group ac*
*tions
and adjunctions involving group actions. Some of the statements we organize into
propositions. Their proofs are left to the reader.
Remark 2.1. The model category assumptions on C stated in Basic Assumption 1.1
will not be needed until Section 12.
2.1. Symmetric monoidal closed categories. By Basic Assumption 1.1, (C, , k)
is a symmetric monoidal closed category with all small limits and colimits.
In particular, C has an initial object ; and a terminal object *. See [30, VI*
*I] for
monoidal categories and [30, VII.7] for symmetric monoidal categories. By closed
we mean there exists a functor
Copx C! C, (Y, Z) 7! Map(Y, Z),
which we call mapping object (or cotensor object), which fits into isomorphisms
(2.2) homC(X Y, Z) ~=hom C(X, Map(Y, Z)),
natural in X, Y, Z.
Remark 2.3. This condition is stronger than only requiring each functor  Y: C*
*! C
to have a specified right adjoint Map(Y, ): C! C.
2.2. Group actions and Gobjects. The symmetric monoidal closed structure
on C induces a corresponding structure on certain diagram categories.
Definition 2.4. Let G be a finite group. CGopis the category with objects the f*
*unc
tors X : Gop! Cand morphisms their natural transformations. CG is the category
with objects the functors X : G! Cand morphisms their natural transformations.
The diagram category CGop(resp. CG) is isomorphic to the category of objects
in C with a specified right action of G (resp. left action of G).
4 JOHN E. HARPER
Proposition 2.5. Let G be a finite group. Then (CGop, , k) has a symmetric
monoidal closed structure induced from the symmetric monoidal closed structure
on (C, , k). In particular, there are isomorphisms
hom CGop(X Y, Z) ~=hom CGop(X, Map(Y, Z))
natural in X, Y, Z.
The proposition remains true when CGop is replaced by CG. We usually leave
such corresponding statements to the reader.
2.3. Copower constructions. If X is a finite set, let's define X to be the nu*
*mber
of elements in X.
Definition 2.6. Let X be a finite set and A 2 C. The copowers A . X 2 C and
X . A 2 C are defined by the same construction:
a a
A . X := A, X . A := A,
X X
the coproduct in C of Xcopies of A.
Sometimes extra structure on the set X induces extra structure on the objects
A . X and X . A. For example, if G is a finite group, then A . G and G . A have*
* op
naturally occurring right and left actions of G, respectively, and hence A.G 2 *
*CG
and G . A 2 CG.
Remark 2.7. In the literature, copower is sometimes indicated by a tensor produ*
*ct
symbol, but because several tensor products already appear in this paper, we are
using the usual dot notation as in [30].
When C = sSetthere are natural isomorphisms A . G ~=A x G, when C = Chk
there are natural isomorphisms A . G ~=A k[G], and when C = Sp there are
natural isomorphisms A . G ~=A ^ G+. Since left Kan extensions may be calcu
lated objectwise in terms of copower, the copower construction appears in sever*
*al
adjunctions below.
2.4. Gorbits and Gfixed points.
Definition 2.8. Let G be a finite group. If Y : Gopx G! Cand Z : G x Gop!C
are functors, then YG 2 C and ZG 2 C are defined by
YG := coendY, ZG := endZ.
The universal properties satisfied by these coends and ends are convenient wh*
*en
working with YG and ZG , but the reader may take the following calculations as
definitions. There are natural isomorphisms,
YG ~=colim( Gop_diag//_Gopx Gop~=Gopx G_Y__//C),
ZG ~=lim( Gop_diag//_Gopx Gop~=G x GopZ__//C).
HOMOTOPY THEORY OF MODULES OVER OPERADS 5
2.5. Adjunctions.
Proposition 2.9. Let G be a finite group, H G a subgroup, and lo:pH! G
the inclusion of groups. Let G1, G2 be finite groups and A2 2 CG2 . There are
adjunctions
_.HG//_ op op____A2___// op
(2.10) C _____//CHoplimoo_CGoo_, CG1 oo________C(G1xG2) ,
l* Map(A2,)G2
with left adjoints on top. In particular, there are isomorphisms
homCGop(A .H G, B)~=homCHop(A, B),
hom CGop(A . (H\G),~B)=homC(A, BH ),
hom CGop(A . G,~B)=homC(A, B),
hom C(G1xG2)op(A1 A2,~X)=homCGop1(A1, Map(A2, X)G2),
natural in A, B and A1, X.
Remark 2.11. The restriction functor l* is sometimes dropped from the notation,
as in the natural isomorphisms in Proposition 2.9.
3.Sequences and symmetric sequences
In preparation for defining operads, we consider sequences and symmetric se
quences of objects in C. We introduce a symmetric monoidal structure on
SymSeq, and a symmetric monoidal structure ^ on Seq. Both of these are rel
atively simple; is a form of the symmetric monoidal structure that is used in
the construction of symmetric spectra [23, 24], while ^ is defined in a way that
is very similar to the definition of the graded tensor product of chain complex*
*es.
These monoidal products possess appropriate adjoints, which can be interpreted
as mapping objects. For instance, there are objects Map (B, C) and Map ^(Y, Z)
which fit into isomorphisms
hom(A B, C)~=hom(A, Map (B, C)),
hom(X ^Y, Z)~=hom(X, Map^(Y, Z)),
natural in the symmetric sequences A, B, C and the sequences X, Y, Z.
3.1. Sequences and symmetric sequences. Let's define the sets n := {1, . .,.n}
for each n 0, where 0 := ; denotes the empty set. When regarded as a totally
ordered set, n is given its natural ordering.
Definition 3.1. Let n 0.
o is the category of finite sets and their bijections. is the category *
*of
totally ordered finite sets and their order preserving bijections.
oA symmetric sequence in C is a functoroAp: op! C. A sequence in C is
a functor X : op! C. SymSeq := C is the category of symmetricosep
quences in C and their natural transformations. Seq:= C is the category
of sequences in C and their natural transformations.
oA (symmetric) sequence A is concentrated at n if A[s] = ; for all s 6= n.
6 JOHN E. HARPER
3.2. Small skeletons and equivalent categories. The indexing categories for
symmetric sequences and sequences are not small, but they have small skeletons,
which will be useful for calculations.
Definition 3.2.
o n is the category with exactly one object n and morphisms the bijections
of sets. n is the category with exactly one object n and morphisms the
identity map.
o 0 is the category with objects the sets n for n 0 and morphisms the
bijections of sets. 0is the category with objects the totally ordered se*
*ts n
for n 0 and morphisms the identity maps.
oA small symmetric sequence in C is a functor A : 0op!C.0oApsmall
sequence in C is a functor X : 0op!C. SymSeq0:= C is the cate
gory of small0symmetricosequencespin C and their natural transformations.
Seq0 := C is the category of small sequences in C and their natural
transformations.
` `
The indexing categories 0op~= n 0 opnand 0op~= n 0 opnare coproducts
of categories, hence giving functors A : 0op!Cand X : 0op!Cis the same as
giving collections {A[n]}n 0 and {X[n]}n 0 of objects in C such that each A[n] *
*is
equipped with a right action of the symmetric group n.
The inclusions i : 0! and i : 0! are equivalences of categories,
__j_//_ __j_//_
(3.3) ooi__ 0, ooi__ 0.
In particular, i has a left adjoint j with corresponding unit j : id!ijand cou*
*nit
" : ji!idisomorphisms.
Remark 3.4. Giving the equivalence of categories j : ! 0is the same as giving
a choice of bijection jT: T !T= ij(T ) (i.e., a choice of ordering) for each*
* finite
set T .
The equivalences of categories (3.3)induce equivalences of categories,
(3.5) SymSeq ____//_SymSeq0,oo_Seq___//Seq0.oo_
3.3. Ordered and unordered tensor products. Given any nonempty totally
ordered finite set N (i.e., a nonempty finite set N equipped with a bijection
j : N !N) and a collection of objects {Xn}n2N in C indexed by N, there is
a naturally occurring ordered tensor product n2N Xn 2 C defined by
n2N Xn := Xj1(1). . .Xj1(N).
Similarly, given any nonempty finite set T and a collection of objects {Xt}t*
*2T
in C indexed by T , we would like to define a naturally occuring unordered tens*
*or
product t2TXt 2 C. One approach is to simply choose an ordering of T (i.e.,
choose a bijection j : T !T) and declare that
t2TXt:= Xj1(1). . .Xj1(T).
This approach is fine, but an isomorphic and more intrinsic definition would be
to replace the choice of ordering with a choice of colimit. The idea is, instea*
*d of
choosing a particular ordering of the unordered set T , take all possible order*
*ings
HOMOTOPY THEORY OF MODULES OVER OPERADS 7
of T , coproduct the corresponding ordered tensor products together, and identi*
*fy
them. The construction is this.
Definition 3.6. Let T be a nonempty finite set.
o T is the category with exactly one object T and morphisms the bijections
of sets. Let iT: T! denote the inclusion of categories.
oThe category of orderings of T is the over category Ord(T ) := iT # T; *
*i.e.,
the objects are the bijections j : T !Tand the morphisms ' : j! j0
are the commutative diagrams
'
T ___~=_//_T
j~= ~=j0
fflffl fflffl
T______T.
Let T be a nonempty finite set and {Xt}t2Ta collection of objects in C index*
*ed
by T . There is a functor X : Ord(T)!C defined objectwise by
( X)(j : T !T) := Xj1(1). . .Xj1(T)
which is useful for defining unordered tensor products.
Definition 3.7. Let T be a nonempty finite set and {Xt}t2Ta collection of obje*
*cts
in C indexed by T . The unordered tensor product t2TXt2 C is defined by
t2TXt:= colim( X).
Remark 3.8. Since every object in Ord(T ) is terminal, for each bijection j : T*
* !T
there is a natural isomorphism,
t2TXt~=Xj1(1). . .Xj1(T).
4.Tensor products for (symmetric) sequences
Sequences and symmetric sequences have naturally occuring tensor products.
Definition 4.1. Let A1, . .,.At be symmetric sequences and let X1, . .,.Xt be
sequences. The tensor products A1 . . .At2 SymSeqand X1^ . .^.Xt2 Seqare
the left Kan extensions of objectwise tensor along coproduct of sets,
( op)xt_A1x...xAt//_Cxt__//C ( op)xtX1x...xXt//_Cxt___//C
` `
 
fflffl A1 ...At fflffl X1^...^Xt
op ___left_Kan_extension//_C, op ___left_Kan_extension//_C.
Remark 4.2. The reader may wish to drop the hat in the tensor product notation
^ for sequences. We have included it only to avoid confusion later, since both *
*the
tensor product of sequences and of symmetric sequences appear in this paper, and
sometimes in the same formula (Section 7).
8 JOHN E. HARPER
4.1. Calculations. This gives a conceptual definition of the tensor products, b*
*ut
the reader may take either of the following calculations as a definition.
Proposition 4.3. Let A1, . .,.At be symmetric sequences and S 2 , with s :=
S. Let X1, . .,.Xt be sequences and M 2 , with m := M. There are natural
isomorphisms,
a
(4.4) (A1 . . .At)[S]~= A1[ss1(1)] . . .At[ss1(t)],
ss:S!itn Set
(4.5) ~= a A1[s1] . . .At[st] . s,
s1+...+st=s s1x...x st
a
(4.6) (X1^ . .^.Xt)[M]~= X1[ss1(1)] . . .Xt[ss1(t)],
ss:M!itn OrdSet
(4.7) ~= a X1[m1 ] . . .Xt[mt ],
m1+...+mt=m
Remark 4.8. Giving a map of sets ss : S! tis the same as giving an ordered
partition (I1, . .,.It) of S. Whenever ss is not surjective, at least one Ij wi*
*ll be the
empty set 0.
Proof.Left to the reader.
4.2. Tensor powers. It will be useful to extend the definition of tensor powers
A t and X ^nto situations in which the integers t and n are replaced, respectiv*
*ely,
by a finite set T or a finite ordered set N. The calculations in Proposition 4*
*.3
suggest how to proceed. We introduce here the suggestive bracket notation used
by Rezk [38].
Definition 4.9. Let A be a symmetric sequence and S, T 2 . Let X be a sequence
and M, N 2 . The tensor powers A T 2 SymSeq and X ^N 2 Seqare defined
objectwise by
a
(4.10) (A T )[S] := A[S, T ] := t2TA[ss1(t)], T 6= ; ,
ss:S!iTn Set
a
(A ;)[S] := A[S, ;] := k,
ss:S!i;n Set
a
(X ^N)[M] := X[M, N] := n2N X[ss1(n)], N 6= ;,
ss:M!iNn OrdSet
a
(X ^;)[M] := X[M, ;] := k.
ss:M!i;n OrdSet
We will use the abbreviations A 0 := A ; and X ^0:= X ^;.
Remark 4.11. The reader will notice there is no hat appearing in the bracket no
tation X[M, N] and X[M, ;] for sequences.
Remark 4.12. The tensor products indexed by T (resp. indexed by N) are regarded
as unordered (resp. ordered) (Section 3.3). Set is the category of sets and the*
*ir
maps. OrdSetis the category of totally ordered sets and their order preserving
maps.
HOMOTOPY THEORY OF MODULES OVER OPERADS 9
The above constructions give functors
SymSeqx opx ! C, (A, S, T ) 7! A[S, T ],
Seqx opx ! C, (X, M, N) 7! X[M, N],
SymSeq x SymSeq! SymSeq, (A, B) 7! A B,
Seqx Seq!Seq, (X, Y ) 7! X ^Y.
Observe that the unit for the tensor product on SymSeqand the unit for the
tensor product ^ on Seq, both denoted "1", are given by the same formula
ae
1[S] := k,;for,So=t0,herwise.
It is useful to make some simple calculations.
Proposition 4.13. Let A, B be symmetric sequences. There are natural isomor
phisms,
A 1 ~=A, A 0 ~=A[, 0] ~=1, A B ~=B A
A ; ~=;, A 1 ~=A[, 1] ~=A, (A t)[0] ~=A[0, t] ~=A[0] t, t 0.
Proof.These are verified directly from Definition 4.9.
Similar calculations are true for sequences.
4.3. Mapping objects for (symmetric) sequences. Let B, C be symmetric
sequences and T 2 . Let Y, Z be sequences and N 2 . There are functors
x op! C, (S, S0) 7! Map(B[S], C[T q S0]),
x op! C, (M, M0) 7! Map(Y [M], Z[N q M0]),
which are useful for defining the mapping objects of (SymSeq, , 1) and (Seq, ^*
*, 1).
Definition 4.14. Let B, C be symmetric sequences and T 2 . Let Y, Z be se
quences and N 2 . The mapping objects Map (B, C) 2 SymSeqand Map^(Y, Z) 2
Seqare defined objectwise by the ends
Map (B, C)[T ]:= Map (B, C[T q ]) ,
Map ^(Y, Z)[N]:= Map (Y, Z[N q ]) .
4.4. Universal properties. Hence Map (B, C) satisfies objectwise the universal
property
(4.15) f[S]____________0Map(B[S],0C[T_q_S])___S_____________________*
*_____________________________________________________________________________*
*_____________________________________________________________________________*
*_____________________________________________________________________________*
*____________33__________________@
______________________________________________________________*
*_____________________________________________________________________________*
*_____________________________________________________________________________*
*____________________
_______________________________(id,(idqi)*)______________________*
*____________________________________o[S]_____________________________________*
*_________________________________
________________________________fflffl_______________~f____________*
*______________________
.____________________//_____________Map(B,MC)[Ta]p(B[S],iC[Tq S0])
__________________________________________9!________________________*
*___OO
__________________________________________________o[S0]___________*
*__________________________________________ *
* *
_______________________________________________________________*
*_____________________________________________________________________________*
*________________________(i,id)
___________________________________________________________*
*_____________________________________________________________________________*
*____________________________________________________
f[S0] __________..___________________________________________*
*_____________________________________________________________________________*
*_____________________________________________________________________________*
*_______________**____________Map(B[S0],SC[T0qS0])
10 JOHN E. HARPER
that each wedge f factors uniquely through the terminal wedge o of Map (B, C)[T*
* ].
The mapping objects Map ^(Y, Z) satisfy objectwise a similar universal property.
These constructions give functors
SymSeqopx SymSeq! SymSeq, (B, C) 7! Map (B, C),
Seqopx Seq!Seq, (Y, Z) 7! Map ^(Y, Z).
Proposition 4.16. Let A, B, C be symmetric sequences and let X, Y, Z be se
quences. There are isomorphims
(4.17) hom(A B, C)~=hom(A, Map (B, C)),
(4.18) hom(X ^Y, Z)~=hom(X, Map^(Y, Z)),
natural in A, B, C and X, Y, Z.
Proof.Consider (4.17). Use the calculation (4.4)and the universal property (4.1*
*5)
with the natural correspondence (2.2)to verify that giving a map A B! C is the
same as giving a map A! Map (B, C), and that the resulting correspondence is
natural. Use a similar argument for the case of sequences.
4.5. Monoidal structures.
Proposition 4.19. (SymSeq, , 1) and (Seq, ^, 1) have the structure of symmetric
monoidal closed categories with all small limits and colimits.
Proof.To verify the symmetric monoidal structure, use (4.4)to describe the re
quired natural isomorphisms and to verify the appropriate diagrams commute.
Proposition 4.16 verifies the symmetric monoidal structure is closed. Limits and
colimits are calculated objectwise. Argue similarly for sequences.
4.6. Calculations. Definition 4.14 gives objectwise a conceptual interpretation*
* of
the mapping objects, but the reader may take the following calculations as defi*
*ni
tions.
Proposition 4.20. Let B, C be symmetric sequences and T 2 , with t := T .
Let Y, Z be sequences and N 2 , with n := N. There are natural isomorphisms,
Y
(4.21) Map (B, C)[T~]= Map (B[s], C[t + s]) s,
s 0
Y
(4.22) Map ^(Y, Z)[N]~= Map(Y [m], Z[n + m]).
m 0
Proof.To verify (4.21), use the universal property (4.15)and restrict to a small
skeleton to obtain natural isomorphisms
` Q ____//_Q 0'
Map(B[S], C[T q S])__//_ Map(B[S], C[T q S ])
Map (B, C)[T~]=lim S i:S0!S .
in 0 in 0
This verifies (4.21). Argue similarly for the case of sequences.
HOMOTOPY THEORY OF MODULES OVER OPERADS 11
5. Circle products for (symmetric) sequences
We describe a circle product O on SymSeqand a related circle product ^Oon Seq.
These are monoidal products which are not symmetric monoidal, and they figure
in the definitions of operad and operad respectively (Definition 6.1). Perh*
*aps
surprisingly, these monoidal products possess appropriate adjoints, which can be
interpreted as mapping objects. For instance, there are objects Map O(B, C) and
Map ^O(Y, Z) which fit into isomorphisms
hom(A O B, C)~=hom(A, MapO(B, C)),
hom (X ^OY, Z)~=hom(X, Map^O(Y, Z)),
natural in the symmetric sequences A, B, C and the sequences X, Y, Z.
The material in this section is largely influenced by Rezk [38]. Earlier work
exploiting circle product O for symmetric sequences includes [10, 11, 40]; more
recent work includes [7, 8, 25, 26]. The circle product ^Ois used in [2] for wo*
*rking
with operads and their algebras.
5.1. Circle products (or composition products). Let A, B be symmetric se
quences and S 2 . Let X, Y be sequences and M 2 . There are functors
opx ! C, (T 0, T ) 7! A[T 0] B[S, T ],
opx ! C, (N0, N) 7! X[N0] Y [M, N],
which are useful for defining circle products of sequences and of symmetric se
quences.
Definition 5.1. Let A, B be symmetric sequences and S 2 . Let X, Y be se
quences and M 2 . The circle products (or composition products) AOB 2 SymSeq
and X ^OY 2 Seqare defined objectwise by the coends
(A O B)[S]:= A (B  )[S] = A B[S, ],
(X ^OY )[M]:= X (Y ^)[M] = X Y [M, ].
Remark 5.2. The reader may wish to drop the hat in the circle product notation
^Ofor sequences. We have included it only to avoid confusion later, since both *
*the
circle product of sequences and of symmetric sequences appear in this paper, and
sometimes in the same formula (Section 7).
5.2. Universal properties. Hence AOB satisfies objectwise the universal property
(5.3) T A[T ]OOB[S,_T_]_________________________________f[T]______*
*_____________________________________________________________________________*
*______________________________________________________________________________
 ____________________________________________*
*_____________________________________________________________________________*
*_________________
 ,* [id,id] i[T]_________________________________________*
*________________________________________________
  __OEOE____________________________~~____*
*___________________________________________________________~
, A[T 0] B[S, T ] (A O B)[S]f___//___________.JJ____
 __@@_____9!_____________________________*
*___________________________________
 id [id,,] i[T0]________________________________________*
*____________________________________________
  ___________________________________________*
*__________________________________________________________________
fflffl fflffl_____________________________________________0*
*_____________________________________________________________________________*
*_____________________________________________________________________________*
*_________
T 0 A[T 0] B[S,_T_0]_____f[T_]________________
that each wedge f factors uniquely through the initial wedge i of (A O B)[S]. A
similar universal property is satisfied objectwise by the circle products X ^OY*
* . These
12 JOHN E. HARPER
constructions give functors
SymSeqx SymSeq! SymSeq, (A, B) 7! A O B,
Seqx Seq!Seq, (X, Y ) 7! X ^OY.
5.3. Calculations. Definition 5.1 gives objectwise a conceptual interpretation *
*of
the circle products, but the reader may take the following calculations as defi*
*nitions.
Proposition 5.4. Let A, B be symmetric sequences and S 2 , with s := S. Let
X, Y be sequences and M 2 , with m := M. There are natural isomorphisms,
a a
(5.5) (A O B)[S]~= A[t] t(B t)[s] ~= A[t] tB[s, t],
t 0 t 0
a ^ a
(X ^OY )[M]~= X[n] (Y n)[m] ~= X[n] Y [m, n].
n 0 n 0
Proof.Use the universal property (5.3)and restrict to a small skeleton to obtain
natural isomorphisms
_ ` ____//_` !
A[T 0] B[S,_T_]//_ A[T ] B[S, T ]
(A O B)[S] ~=colim ,:T! T0 T .
in 0 in 0
This verifies (5.5). The other case is similar.
Observe that the unit for the circle product O on SymSeq and the unit for the
circle product ^Oon Seq, both denoted "I", are given by the same formula
ae
I[S] := k,;for,So=t1,herwise.
Definition 5.6. Let A be a symmetric sequence, X a sequence, and Z 2 C. The
corresponding functors A O (): C! Cand X ^O(): C! Care defined objectwise
by,
a
A O (Z):= A[t] tZ t,
t 0
a
X ^O(Z):= X[t] Z t.
t 0
The category C embeds in SymSeq(resp. Seq) as the full subcategory of symmet
ric sequences (resp. sequences) concentrated at 0, via the functor ^: C! SymS*
*eq
(resp. ^: C! Seq) defined objectwise by
ae
(5.7) Z^[S] := Z, for S = 0,
;, otherwise.
It is useful to make some simple calculations.
HOMOTOPY THEORY OF MODULES OVER OPERADS 13
Proposition 5.8. Let A, B be symmetric sequences and Z 2 C. There are natural
isomorphisms,
; O A~=;, I O A ~=A, A O I ~=A,
ae
(A O ;)[S]~= A[0],;for,So=t0,herwise,
ae
(A O ^Z)[S]~=A O (Z),for;S,=o0,therwise,
(A O B)[0]~=A O (B[0]).
Proof.These can be verified directly from Proposition 5.4.
Similar calculations are true for sequences.
5.4. Properties of tensor and circle products. It is useful to understand how
tensor products and circle products interact.
Proposition 5.9. Let A, B, C be symmetric sequences, X, Y, Z be sequences, and
t 0. There are natural isomorphisms
(A B) O C~=(A O C) (B O C),
(X ^Y ) ^OZ~=(X ^OZ) ^(Y ^OZ),
(5.10) (B t) O C~=(B O C) t,
(Y ^t) ^OZ~=(Y ^OZ)^t.
Proof.Using (4.5)and (5.5), there are natural isomorphisms
a
(A B) O C~= (A B)[s] sC s
s 0
~=a a A[s1] B[s2] s1x s2C (s1+s2)
si0 s1+s2=s j i j
~= a A[s1] s1C s1 a B[s2] s2C s2
s1 0 s2 0
~=(A O C) (B O C).
The argument for sequences is similar.
Proposition 5.11. Let A, B be symmetric sequences and let X, Y be sequences.
Suppose Z 2 C and t 0. There are natural isomorphisms
(A B) O (Z)~=(A O (Z)) (B O (Z)),
(X ^Y ) ^O(Z)~=(X ^O(Z)) (Y ^O(Z)),
(B t) O (Z)~=(B O (Z)) t,
(Y ^t) ^O(Z)~=(Y ^O(Z)) t.
Proof.Argue as in the proof of Proposition 5.9, or use the embedding (5.7)to
deduce it as a special case.
14 JOHN E. HARPER
Proposition 5.12. Let A, B, C be symmetric sequences and let X, Y, Z be se
quences. There are natural isomorphisms
(A O B) O C~=A O (B O C),
(X ^OY ) ^OZ~=X ^O(Y ^OZ).
Proof.Using (5.5)and (5.10), there are natural isomorphisms
a
A O (B O C)~= A[t] t(B O C) t
t 0
~=a A[t] t(B t) O C
t 0
~=a a A[t] t(B t)[s] sC s
s 0t 0
~=(A O B) O C.
Argue similarly for the case of sequences.
Proposition 5.13. Let A, B be symmetric sequences and let X, Y be sequences.
Suppose Z 2 C. There are natural isomorphisms
(A O B) O (Z) ~=A O B O (Z) ,
(X ^OY ) ^O(Z) ~=X ^OY ^O(Z) .
Proof.Argue as in the proof of Proposition 5.12, or use the embedding (5.7)to
deduce it as a special case.
5.5. Mapping sequences. Let B, C be symmetric sequences and T 2 . Let Y, Z
be sequences and N 2 . There are functors
x op! C, (S, S0) 7! Map(B[S, T ], C[S0]),
x op! C, (M, M0) 7! Map(Y [M, N], Z[M0]),
which are useful for defining mapping objects of (SymSeq, O, I) and (Seq, ^O, I*
*).
Definition 5.14. Let B, C be symmetric sequences and T 2 . Let Y, Z be se
quences and N 2 . The mapping sequences MapO(B, C) 2 SymSeqand Map^O(Y, Z) 2
Seqare defined objectwise by the ends
Map O(B, C)[T:]= Map ((B T )[], C) = Map (B[, T ], C) ,
Map ^O(Y, Z)[N]:= Map ((Y ^N)[], Z) = Map (Y [, N], Z) .
5.6. Universal properties. Hence MapO(B, C) satisfies objectwise the universal
property
(5.15) f[S]____________Map0(B[S,0T_],_C[S])_S______________________*
*_____________________________________________________________________________*
*_____________________________________________________________________________*
*_____________________________________________________________________________*
*______________________________33@
_____________________________________________________________*
*_____________________________________________________________________________*
*_____________________________________________________________________________*
*____
______________________________([id,id],i*)______________________*
*_____________________________________o[S]____________________________________*
*_________________________________
_______________________________fflffl_______________~f____________*
*_________________________
.____________________//_____________MapO(B,MC)[Ta]p(B[S,iT], C[S0])
__________________________________________9!_______________________*
*_______OO
___________________________________________________o[S0]_________*
*_____________________________________________
_______________________________________________________________*
*_____________________________________________________________________________*
*_________________([i,id],id)
__________________________________________________________*
*_____________________________________________________________________________*
*_____________________________________________________________
f[S0] _________..___________________________________________*
*_____________________________________________________________________________*
*_____________________________________________________________________________*
*__________________**______________Map(B[S0,ST0], C[S0])
HOMOTOPY THEORY OF MODULES OVER OPERADS 15
that each wedge f factors uniquely through the terminal wedge o of MapO(B, C)[T*
* ].
A similar universal property is satisfied objectwise by the mapping objects Map*
*^O(Y, Z).
These constructions give functors
SymSeqopx SymSeq! SymSeq, (B, C) 7! MapO(B, C),
Seqopx Seq!Seq, (Y, Z) 7! Map^O(Y, Z).
Proposition 5.16. Let A, B, C be symmetric sequences and let X, Y, Z be se
quences. There are isomorphims
(5.17) hom(A O B, C)~=hom(A, MapO(B, C)),
(5.18) hom (X ^OY, Z)~=hom(X, Map^O(Y, Z)),
natural in A, B, C and X, Y, Z.
Proof.Use the universal properties (5.3)and (5.15)with the natural correspon
dence (2.2)to verify that giving a map A O B! C is the same as giving a map
A! Map O(B, C), and that the resulting correspondence is natural. Argue simi
larly for the case of sequences.
5.7. Monoidal structures.
Proposition 5.19. (SymSeq, O, I) and (Seq, ^O, I) have the structure of monoidal
closed categories with all small limits and colimits. Circle product is not sym*
*metric.
Proof.To verify the monoidal structure, use (5.5)along with properties of from
Proposition 4.19 to describe the required natural isomorphisms and to verify the
appropriate diagrams commute. Proposition 5.16 verifies the monoidal structure *
*is
closed. Limits and colimits are calculated objectwise. Argue similarly for the *
*case
of sequences.
5.8. Calculations. Definition 5.14 gives objectwise a conceptual interpretation*
* of
the mapping sequences, but the reader may take the following calculations as de*
*fi
nitions.
Proposition 5.20. Let B, C be symmetric sequences and T 2 , with t := T .
Let Y, Z be sequences and N 2 , with n := N. There are natural isomorphisms,
(5.21) Y Y
Map O(B, C)[T~]= Map((B t)[s], C[s]) s ~= Map(B[s, t], C[s]) s,
s 0 s 0
Y ^ Y
Map ^O(Y, Z)[N]~= Map ((Y n)[m], Z[m]) ~= Map(Y [m, n], Z[m]).
m 0 m 0
Proof.Use the universal property (5.15)for MapO(B, C)[T ] and restrict to a sma*
*ll
skeleton to obtain natural isormorphisms
` Q ____//_Q ' 0
Map (B[S, T ], C[S])//_ Map (B[S, T ], C[S ])
Map O(B, C)[T~]=lim S i:S0!S .
in 0 in 0
This verifies (5.21). The case for sequences is similar.
It is useful to make some simple calculations.
16 JOHN E. HARPER
Proposition 5.22. Let B, C be symmetric sequences, T 2 , and Z 2 C. There
are natural isomorphisms,
MapO(B, *)~=*,
ae
MapO(;, C)[T~]= C[0],*for,Tot=h0,erwise,
Map O(B, C)[0]~=C[0],
Y
Map O(B, C)[1]~= Map(B[s], C[s]) s,
s 0
Map O(Z^, C)[T~]=Map(Z T, C[0]).
Proof.These can be verified directly from Proposition 5.20.
Similar calculations are true for sequences.
5.9. Circle products as Kan extensions. Circle products can also be under
stood as Kan extensions. Let F be the category of finite sets and their maps an*
*d let
OrdF be the category of totally ordered finite sets and their order preserving *
*maps.
Then circle products are left Kan extensions of  *  along projection onto sou*
*rce,
op A*B .!. op X*Y
Iso(F.!.) __________//C Iso(OrdF ) __________//C
proj proj
fflfflAOB fflfflX ^OY
op left_Kan_extension//_C, op _left_Kan_extension//_C.
The functors  *  are defined objectwise by
(A * B)(ss : S !:T=)A[T ] t2TB[ss1(t)],
(X * Y )(ss : M ! N):= X[N] n2NY [ss1(n)].
Remark 5.23. The tensor products indexed by T (resp. indexed by N) are regarded
as unordered (resp. ordered) (Section 3.3).
6. Operads, modules, and algebras
In this section we define operads and the objects they act on.
6.1. Operads.
Definition 6.1.
oA operad is a monoid object in (SymSeq, O, I) and a morphism of 
operads is a morphism of monoid objects in (SymSeq, O, I).
oAn operad is a monoid object in (Seq, ^O, I) and a morphism of operads
is a morphism of monoid objects in (Seq, ^O, I).
These two types of operads were originally defined by May [32]; the operad
has symmetric groups and the operad is based on ordered sets and is called a
non operad [27, 32].
HOMOTOPY THEORY OF MODULES OVER OPERADS 17
Example 6.2. More explicitly, for instance, a operad is a symmetric sequence
O together with maps m : O O O!O and j : I! Oin SymSeq which make the
diagrams
jOid idOj
O O O O OmOid//_O O O I O O____//_O OoOo__O O I
idOm m ~= m ~=
fflfflm fflffl fflffl fflffl fflffl
O O O________//O O _________O________O_
commute. If O and O0 are operads, then a morphism of operads is a map
f : O! O0in SymSeqwhich makes the diagrams
j
O O O__m__//_O O oo___I
fOf f f 
fflfflm fflfflfflfflj
O0O O0 ____//_O0 O0 oo___I
commute.
6.2. Modules over operads. Similar to the case of any monoid object, we intro
duce operads because we are interested in the objects they act on. Compare the
following definition with [29, Chapter VII].
Definition 6.3. Let Q, R, S be operads (resp. operads).
oA left Rmodule is an object in (SymSeq, O, I) (resp. an object in (Seq, *
*^O, I))
with a left action of R and a morphism of left Rmodules is a map which
respects the left Rmodule structure.
oA right Smodule is an object in (SymSeq, O, I) (resp. an object in (Seq,*
* ^O, I))
with a right action of S and a morphism of right Smodules is a map which
respects the right Smodule structure.
oAn (R, S)bimodule is an object in (SymSeq, O, I) (resp. an object in (Se*
*q, ^O, I))
with compatible left Rmodule and right Smodule structures and a mor
phism of (R, S)bimodules is a map which respects the (R, S)bimodule
structure.
Example 6.4. More explicitly, for instance, if R is a operad, then a left Rm*
*odule
is a symmetric sequence B together with a map m : R O B!B in SymSeq which
makes the diagrams
jOid
(6.5) R O R O BmOid//_R O B I O B____//R O B
idOm m ~= m
fflfflm fflffl fflffl fflffl
R O B_______//_B B _________B
commute. If B and B0are left Rmodules, then a morphism of left Rmodules is a
map f : B! B0in SymSeqwhich makes the lefthand diagram
(6.6) R O B _m__//_B R O B O SmOid//_B O S
idOf f idOm m
fflfflm fflffl fflfflm fflffl
R O B0____//_B0 R O B_______//_B
18 JOHN E. HARPER
commute. If R and S are operads, then an (R, S)bimodule is an object in
(SymSeq, O, I) with left Rmodule and right Smodule structures which make the
righthand diagram in (6.6)commute.
6.3. Algebras over operads. An algebra over an operad O is a left Omodule
which is concentrated at 0.
Definition 6.7. Let O be an operad.
oAn Oalgebra is an object X 2 C with a left Omodule structure on ^X.
oLet X and X0be Oalgebras. A morphism of Oalgebras is a map f : X! X0
in C such that ^f: ^X!X^0is a morphism of left Omodules.
Giving a symmetric sequence Y a left Omodule structure is the same as giving
a morphism of operads
m : O! Map O(Y, Y.)
Similarly, giving an object X 2 C an Oalgebra structure is the same as giving a
morphism of operads
m : O! Map O(X^,.^X)
This is the original definition given by May [32] of an Oalgebra structure on *
*X,
where MapO(X^, ^X) is called the endomorphism operad of X (see Proposition 5.22*
*),
and motivates the suggestion in [27, 32] that O[t] should be thought of as obje*
*cts
of parameters for tary operations.
Algebras over an operad can also be described as objects in C with a left ac
tion of a particular monad (or triple). Each operad O determines a functor
O() : C! Cdefined objectwise by
a
O(X) := O O (X) = O[t] tX t,
t 0
along with natural transformations m : OO ! Oand j : id!Owhich make ap
propriate associativity and unit diagrams commute. Giving an Oalgebra structure
on X is the same as giving a left action of O() on X; i.e., a map m : O(X)! X
in C which makes the appropriate associativity and unit diagrams commute. From
this perspective, operads correspond to special functors in such a way that cir*
*cle
product corresponds to composition, but because these functors have such simple
descriptions in terms of symmetric sequences, operads are easier to work with t*
*han
arbitrary functors.
7.Freely adding actions
Proposition 7.1. There is an adjunction
_._//_
Seq oo___SymSeq
*
with left adjoint on top and : ! the forgetful functor.
Proof.There is a chain of adjunctions
_._//_ ____//_
Seq_____//Seq0oo_SymSeq0oo_ooSymSeq_,
with left adjoints on top; here, (A. )[s] := A[s]. s. Composition gives the des*
*ired
adjunction, with left adjoint also denoted  . .
HOMOTOPY THEORY OF MODULES OVER OPERADS 19
7.1. Properties of the left adjoint. It is useful to understand how tensor prod
ucts and circle products behave with respect to  . .
Proposition 7.2. Let A, B be sequences and t 0. There are natural isomor
phisms,
(7.3) (A . ) (B .~=)(A ^B) . ,
(7.4) (B . ) t~=(B ^t) . ,
(7.5) (A . ) O (B .~=)(A ^OB) . ,
in SymSeq.
Proof.To verify (7.3), let s 0 and consider the natural isomorphisms,
a
(A . ) (B . )~[s]= (A[s1] . s1) (B[s2] . s2) . s1x s2 s
s1+s2=s
~= (A ^B) . [s].
To verify (7.5), consider the natural isomorphisms,
a ^
(A . ) O (B . ) ~= (A . )[t] t(B t) . ~=(A ^OB) . .
t 0
7.2. Naturally occurring maps.
Proposition 7.6. Let B 2 Seqand C 2 SymSeq. There is a map
i : B ^OC!(B . ) O C
in Seq, natural in B, C.
Proof.Each M 2 may be regarded as an unordered set; use this together with
(4.4)and (4.6)to look for naturally occurring maps
iM : (B ^OC)[M]!(B . ) O C [M],
for each M 2 .
7.3. Adjunctions.
Proposition 7.7. Let O be an operad. There are adjunctions
_._//_ _____//
LtOoo___LtO. AlgOoo___AlgO.
U U
with left adjoints on top and U the forgetful functor. The righthand adjunctio*
*n is
an isomorphism of categories.
Proof.Use Propositions 7.2 and 7.6.
8.Limits and colimits for modules and algebras
We have introduced modules and algebras over an operad, but to work with
them, we need to understand how to build basic constructions such as limits and
colimits. The material in this section is largely influenced by Rezk [38].
20 JOHN E. HARPER
8.1. Reflexive coequalizers and filtered colimits.
od0o_
Definition 8.1. A pair of maps of the form X0 oo__X1_in C is called a reflexive
d1
pair if there exists s0: X0! X1in C such that d0s0 = idand d1s0 = id. A reflex*
*ive
coequalizer is the coequalizer of a reflexive pair.
Reflexive coequalizers will be useful for building colimits in the categories*
* of
modules and algebras over an operad.
Proposition 8.2.
(a)Suppose X1 oo__X0_oo__X1_oo_and Y1oo___Y0oo__Y1_oo_are reflexive co
equalizer diagrams in C. Then their objectwise tensor product
X1 Y1oo___X0 Y0 oo___X1oY1o_
is a reflexive coequalizer diagram in C.
(b)Suppose X, Y: D! Care filtered diagrams. Then objectwise tensor product
of their colimiting cones is a colimiting cone. In particular, there are *
*natural
isomorphisms
colimd2D(Xd Yd) ~=(colimd2DXd) (colimd2DYd).
in C.
Proof.Consider (a) and the diagram
X0 Y1 oo___X1oY1o_
 
 
fflfflfflfflfflfflfflffl
X0 Y0 oo___X1oY0.o_
Look for a naturally occurring cone into X1 Y1, and verify this cone is init*
*ial
with respect to all cones. Use relations satisfied by the reflexive pairs and n*
*ote that
tensoring with any X 2 C preserves colimiting cones. Verification of (b) is sim*
*ilar to
(a), except instead of properties satisfied by reflexive pairs, use properties *
*satisfied
by filtered diagrams.
Hence objectwise tensor product of diagrams in (C, , k) respects certain col*
*im
iting cones. Objectwise circle product of diagrams in (SymSeq, O, I) and (Seq, *
*^O, I)
behave similarly.
Proposition 8.3.
(a)Suppose A1 oo__A0_oo__A1oo_and B1 oo__B0_oo__B1oo_are reflexive co
equalizer diagrams in SymSeq. Then their objectwise circle product
(8.4) A1 O B1oo___A0O B0oo___A1OoB1o_
is a reflexive coequalizer diagram in SymSeq.
(b)Suppose A, B: D! SymSeqare filtered diagrams. Then objectwise circle
product of their colimiting cones is a colimiting cone. In particular, th*
*ere
are natural isomorphisms
colimd2D(AdO Bd) ~=(colimd2DAd) O (colimd2DBd).
in SymSeq.
HOMOTOPY THEORY OF MODULES OVER OPERADS 21
(c)For sequences, the corresponding statements in (a) and (b) remain true;
i.e., when (SymSeq, O, I) is replaced by (Seq, ^O, I).
Proof.Consider (a). We want to verify that (8.4)is a colimiting cone, hence suf*
*fi
cient to verify it is initial with respect to all cones. Use the universal prop*
*erty (5.3)
together with (4.10)and Proposition 8.2(a). Verification of (b) is similar, exc*
*ept
use Proposition 8.2(b).
Proposition 8.5.
(a)Suppose A1oo___A0oo___A1oo_is a reflexive coequalizer diagram in SymSeq
and Z1 oo__Z0_oo__Z1oo_is a reflexive coequalizer diagram in C. Then
their objectwise evaluation
A1 O (Z1)oo__A0O (Z0)oo___A1Oo(Z1)o_
is a reflexive coequalizer diagram in C.
(b)Suppose A : D! SymSeqand Z : D! Care filtered diagrams. Then ob
jectwise evaluation of their colimiting cones is a colimiting cone. In pa*
*rtic
ular, there are natural isomorphisms
colimd2DAdO (Zd) ~=(colimd2DAd) O (colimd2DZd).
in C.
(c)For sequences, the corresponding statements in (a) and (b) remain true;
i.e., when (SymSeq, O, I) is replaced by (Seq, ^O, I).
Proof.Argue as a special case of Proposition 8.3 using the embedding in (5.7).
8.2. Freeforgetful adjunctions.
Definition 8.6. Let O, R, S be operads.
oLtO is the category of left Omodules and their morphisms.
oRtO is the category of right Omodules and their morphisms.
oBi(R,S)is the category of (R, S)bimodules and their morphisms.
oAlgO is the category of Oalgebras and their morphisms.
It will be useful to establish the following freeforgetful adjunctions.
Proposition 8.7.
(a)Let O, R, S be operads. There are adjunctions
OO_//_ _OO_// ROOS//_
SymSeq oo___LtO, SymSeqoo___RtO, SymSeqoo___Bi(R,S),
U U U
with left adjoints on top and U the forgetful functor.
(b)Let O, R, S be operads. There are adjunctions
_O_^O//_ O^O//_ R_^OO^S//_
Seqoo___LtO, Seq oo___RtO, Seqoo___Bi(R,S),
U U U
with left adjoints on top and U the forgetful functor.
Proof.To verify the first adjunction of (a), it is enough to show there are iso*
*mor
phisms
homLtO(O O B, C) ~=hom SymSeq(B, UC)
22 JOHN E. HARPER
natural in B, C. For this, it is enough to verify the universal_property that g*
*iven
any map f : B! UCin SymSeq, there exists a unique map f: O O B!C in LtO
such that f factors through the map
B ~=idO BjOid//_O O B = U(O O B)
__
via the map Uf . The other cases are similar.
There are similar freeforgetful adjunctions for algebras over an operad.
Proposition 8.8. Let O be a operad and O0be an operad. There are adjunc
tions
OO()//_ O0^O()//_
C oUo__AlgO, CooU__AlgO0,
with left adjoints on top and U the forgetful functor.
Proof.Argue as in the proof of Proposition 8.7.
8.3. Construction of colimits.
Proposition 8.9. Let O, R, S be operads. Reflexive coequalizers and filtered co*
*l
imits exist in LtO, RtO, Bi(R,S), and AlgO, and are preserved (and created) by *
*the
forgetful functors in Propositions 8.7 and 8.8.
Proof.Let O be a operad and consider the case of left Omodules. Suppose
A0 oo__A1_oo_is a reflexive pair in LtOand consider the solid commutative diagr*
*am
O O O_O_A1oo__O O O O A0oo__OoOoO_O A1
________  
d0_______d1_____mOididOm mOididOm
fflffl___fflffl___fflfflfflfflfflfflfflffl
O OCA1_oo_____OCOCA0oo______OoOoA1_C_CC_
______ ______________________________________________*
*________________
s0__m_________jOidm____________m__________________jOid________*
*______________________
__fflffl___ __fflffl_______fflffl_________________________*
*____
A1 oo_________A0 oo_________A1oo_
in SymSeq, with bottom row the reflexive coequalizer diagram of the underlying
reflexive pair in SymSeq. By Proposition 8.3, the rows are reflexive coequaliz*
*er
diagrams and hence there exist unique dotted arrows m, s0, d0, d1 in SymSeqwhich
make the diagram commute. By uniqueness, s0 = jOid, d0 = mOid, and d1 = idOm.
Verify that m gives A1 the structure of a left Omodule and also verify that t*
*he
bottom row is a reflexive coequalizer diagram in LtO. First check the diagram l*
*ives
in LtO, then check the colimiting cone is initial with respect to all cones in *
*LtO.
The case for filtered colimits is similar. The cases for RtO, Bi(R,S), and AlgO*
*can
be argued similarly. Argue similarly for operads.
Proposition 8.10. Let O, R, S be operads. All small colimits exist in LtO, RtO,
Bi(R,S), and AlgO.
Proof.Let O be a operad and consider the case of left Omodules. Suppose
A : D! LtOis a small diagram. Want to show that colimA exists. This colimit
HOMOTOPY THEORY OF MODULES OVER OPERADS 23
may be calculated by a reflexive coequalizer in LtOof the form,
i (mOid)*oo_ j
colimA ~=colim colimd2D(O O(Ad)colimd2D(OiOdOOOmAd))*oo_,
provided the indicated colimits appearing in this reflexive pair exist in LtO. *
*The
underlying category SymSeq has all small colimits, and left adjoints preserve c*
*ol
imiting cones, hence there is a commutative diagram
(mOid)*
colimd2D(O OoAd)o_colimd2D(OoOoO_O Ad)
(idOm)*
~= ~=
fflffl fflffl
O O colimd2DAdoo_OoOocolimd2D(O_O Ad)
in LtO. The colimits in the bottom row exist since they are in the underlying
category SymSeq(we have dropped the notation for the forgetful functor U), hence
the colimits in the top row exist in LtO. Therefore colimA exists. The cases for
RtO, Bi(R,S), and AlgOcan be argued similarly. Argue similarly for operads.
Example 8.11. For instance, if O is a operad and A, B 2 LtO, then the coprodu*
*ct
A q B in LtOmay be calculated by a reflexive coequalizer of the form
i oo___ j
A q B ~=colim O O (A q B)oo__O O (O O A) q (O O B)
in the underlying category SymSeq. The coproducts appearing inside the parenthe
ses are in the underlying category SymSeq.
8.4. Colimits for right modules. Colimits in right modules over an operad are
particularly simple.
Proposition 8.12. Let O be an operad. The forgetful functors from right O
modules in Proposition 8.7 preserve (and create) all small colimits.
Proof.Let O be a operad and suppose A : D! RtOis a small diagram. By the
proof of Proposition 8.10, colimA may be calculated in the underlying category
by the colimit in SymSeq of the reflexive pair in the top row of the commutative
diagram
oo___
colimd2DAd OoOo_ colimd2D(AdO O) O O
~= ~=
fflffl fflffl
colimd2D(AdOoO)o__colimd2D(AdOoOoO_O),
and since the functor  O O: SymSeq! SymSeqpreserves colimiting cones, this
is the same as calculating the colimit of the bottom row in SymSeq, which is the
colimit of the underlying diagram of A in SymSeq. We have dropped the notation
for the forgetful functor U. Argue similarly for operads.
24 JOHN E. HARPER
8.5. Construction of limits.
Proposition 8.13. Let O, R, S be operads. All small limits exist in LtO, RtO,
Bi(R,S), and AlgO, and are preserved (and created) by the forgetful functors in
Propositions 8.7 and 8.8.
Proof.This can be argued similar to the proof of Proposition 8.9.
9.Basic constructions for modules
In this section we present some basic constructions for modules. The material
in this section is largely influenced by Rezk [38]. Compare the following defin*
*ition
with [29, Chapter VI.5].
9.1. Circle products (mapping sequences) over an operad.
Definition 9.1. Let R be a operad (resp. operad), A a right Rmodule, and
B a left Rmodule. Define A OR B 2 SymSeq(resp. A ^ORB 2 Seq) by the reflexive
coequalizer
i d0 j
A OR B:= colim A O Boo__A_OoRoO_B ,
d1
i i d0 jj
resp. A ^ORB:= colim A ^OBoo__Ao^ORo^OB_,
d1
with d0 induced by m : A O R!A and d1 induced by m : R O B!B (resp. d0
induced by m : A ^OR!Aand d1 induced by m : R ^OB!B).
Definition 9.2. Let S be a operad (resp. operad) and let B and C be right S
modules. Define MapOS(B, C) 2 SymSeq(resp. Map ^OS(B, C) 2 Seq) by the equalizer
i d0 j
MapOS(B, C):= lim MapO(B, C)____//_//_MapO(B O S,,C)
d1
i i d0 jj
resp. Map^OS(B, C):= lim Map^O(B, C)___//_//_Map^O(B ^OS,,C)
d1
with d0 induced by m : B O S!B and d1 induced by m : C O S!C (resp. d0
induced by m : B ^OS!Band d1 induced by m : C ^OS!C).
9.2. Adjunctions. Compare the following adjunctions with [29, Chapter VI.8].
Proposition 9.3. Let Q, R, S be operads. There are isomorphisms,
(9.4) homRtS(A O B, C)~=hom(A, MapOS(B, C)),
(9.5) hom(A OR B, C)~=homRtR(A, MapO(B, C)),
(9.6) hom (Q,S)(A OR B,~C)=hom(Q,R)(A, MapOS(B, C)),
natural in A, B, C.
Remark 9.7. In (9.4), A is a symmetric sequence, and both B and C have right
Smodule structures. In (9.5), A has a right Rmodule structure, B has a left R
module structure, and C is a symmetric sequence. In (9.6), A has a (Q, R)bimod*
*ule
structure, B has a (R, S)bimodule structure, and C has a (Q, S)bimodule struc
ture.
HOMOTOPY THEORY OF MODULES OVER OPERADS 25
Proof.The natural correspondence (9.6)implies both (9.4)and (9.5), and its proof
can be built up from those for (9.4)and (9.5). Use the natural correspondences
(5.17)together with the commutative diagrams satisfied by each object as defined
in Section 6.
There is a corresponding statement for operads.
Proposition 9.8. Let Q, R, S be operads. There are isomorphisms,
homRtS(A ^OB, C)~=hom(A, Map^OS(B, C)),
hom (A ^ORB, C)~=homRtR(A, Map^O(B, C)),
hom (Q,S)(A ^ORB,~C)=hom(Q,R)(A, Map^OS(B, C)),
natural in A, B, C.
Proof.Argue as in the proof of Proposition 9.3.
9.3. Cancellation and associativity properties.
Proposition 9.9. Let R be a operad (resp. operad), A a right Rmodule, and
B a left Rmodule. There are natural isomorphisms
A OR R ~=A R OR B ~=B,
resp. A ^ORR ~=A R ^ORB ~=B .
Proof.Suppose R is a operad. We want to verify A is naturally isomorphic to a
particular coequalizer. Look for a naturally occurring cone into A and verify i*
*t is
initial with respect to all cones. The other cases are similar.
Proposition 9.10. Let R, S be operads (resp. operads), A a right Rmodule,
B an (R, S)bimodule, and C a left Smodule. There are natural isomorphisms
(A OR B) OS C~=A OR (B OS C),
resp. (A ^ORB) ^OSC~=A ^OR(B ^OSC) .
Proof.Use Proposition 8.3.
9.4. Change of operads adjunction.
Proposition 9.11. Let f : R! Sbe a morphism of operads (resp. operads).
There is an adjunction
__f*_//
LtRoo___LtS,
f*
with left adjoint f* := S OR  (resp. f* := S ^OR) and f* the forgetful functo*
*r. In
particular, there are isomorphisms
hom LtS(S OR A, B)~=homLtR(A, f*(B)),
*
resp.hom LtS(S ^ORA, B)~=homLtR(A, f (B)) ,
natural in A, B.
Proof.Look for natural transformations j : id!f*f*and " : f*f*!idand verify
* f*"
that f*_f*j//_f*f*f*"f*//_f*and f*_jf_//_f*f*f*__//f*each factor the identity.
26 JOHN E. HARPER
10.Basic constructions for algebras
Here we present some corresponding constructions for algebras.
10.1. Circle products over an operad.
Definition 10.1. Let R be a operad (resp. operad), A a right Rmodule,
and Z an Ralgebra. Define A OR (Z) 2 C (resp. A ^OR(Z) 2 C) by the reflexive
coequalizer
i d0 j
A OR (Z):= colim A O (Z)oo__(AoOoR)_O (Z),
d1
i i d0 jj
resp.A ^OR(Z):= colim A ^O(Z)oo__(Ao^OR)o^O(Z)_,
d1
with d0 induced by m : A O R!Aand d1 induced by m : R O (Z)!Z(resp. d0
induced by m : A ^OR!Aand d1 induced by m : R ^O(Z)!Z).
10.2. Cancellation and associativity properties.
Proposition 10.2. Let R be a operad (resp. operad) and Z an Ralgebra.
There are natural isomorphisms
R OR (Z) ~=Z, resp. R ^OR(Z) ~=Z .
Proof.Argue as a special case of Proposition 9.9 by taking B := ^Z.
Proposition 10.3. Let R, S be operads (resp. operads), A a right Rmodule,
B an (R, S)bimodule, and Z an Salgebra. There are natural isomorphisms
(A OR B) OS (Z)~=A OR B OS (Z) ,
i j
resp.(A ^ORB) ^OS(Z)~=A ^ORB ^OS(Z) .
Proof.Argue as a special case of Proposition 9.10 by taking C := ^Z.
10.3. Change of operads adjunction.
Proposition 10.4. Let f : R! Sbe a morphism of operads (resp. operads).
There is an adjunction
f* //
AlgR_____AlgS,oo_
f*
with left adjoint f* := S OR () (resp. f* := S ^OR()) and f* the forgetful fu*
*nctor.
In particular, there are isomorphisms
homAlgS(S OR (A),~B)=homAlgR(A, f*(B)),
*
resp. homAlgS(S ^OR(A),~B)=homAlgR(A, f (B)) ,
natural in A, B.
Proof.Argue as in the proof of Proposition 9.11.
HOMOTOPY THEORY OF MODULES OVER OPERADS 27
11. Model categories and definitions
In this section we establish some notation; the definitions appearing below a*
*re
only intended to make precise Basic Assumption 1.1. We assume the reader is
familiar with model categories. A useful introduction is given in [4], from whi*
*ch we
have taken the model category axioms listed below. See also the original articl*
*es
by Quillen [36, 37], and the more recent [3, 15, 21, 22].
In this paper, our primary method of constructing model categories from exist*
*ing
ones involves the additional structure of a cofibrantly generated model category
together with (possibly transfinite) small object arguments. Schwede and Shipley
provide an account of these techniques in [39, Section 2] which will be suffici*
*ent for
our purposes. The reader unfamiliar with the small object argument may consult
[4, Section 7.12] for a useful introduction; after which the (possibly transfin*
*ite)
versions in [21, 22, 39] appear quite natural.
11.1. Model categories.
Definition 11.1. In C, a map i : A! Bhas the left lifting property (LLP) with
respect to a map p : X! Yif every solid commutative diagram of the form
(11.2) A _____//X>>___
____
i,____p____
fflfflfflffl____
B _____//Y
has a lift ,. In C, a map p : X! Yhas the right lifting property (RLP) with re*
*spect
to a map i : A! Bif every solid commutative diagram of the form (11.2)has a
lift ,.
Definition 11.3. A model category is a category C with three distinguished sub
categories of C:
oW the subcategory of weak equivalences
oFibthe subcategory of fibrations
oCof the subcategory of cofibrations
each of which contains all objects of C. A map which is both a fibration (resp.
cofibration) and a weak equivalence is called an acyclic fibration (resp. acyc*
*lic
cofibration). An object X 2 C is called cofibrant (resp. fibrant) if ;! X is*
* a
cofibration (resp. X! * is a fibration). We require the following axioms:
(MC1) Finite limits and colimits exist in C.
(MC2) If f and g are maps in C such that gf is defined and if two of the three
maps f, g, gf are weak equivalences, then so is the third.
(MC3) If f is a retract of g and g is a fibration, cofibration, or weak equival*
*ence,
then so is f.
(MC4) Cofibrations have the LLP with respect to acyclic fibrations. Acyclic cof*
*i
brations have the LLP with respect to fibrations.
(MC5) Any map f can be factored in two ways: (i) f = pi, where i is a cofibrati*
*on
and p is an acyclic fibration, and (ii) f = pi, where i is an acyclic cof*
*ibration
and p is a fibration.
Remark 11.4. The definition above describes what was originally called a "close*
*d"
model category [37]; following [4] we have dropped the term "closed" in this pa*
*per.
28 JOHN E. HARPER
11.2. Cofibrantly generated model categories. To construct model category
structures on (symmetric) sequences and on modules and algebras over an operad,
we will require some extra conditions on the model category structure of C.
Definition 11.5. A model category C is cofibrantly generated if it has all small
limits and colimits and there exists a set I of cofibrations and a set J of acy*
*clic
cofibrations satisfying certain properties [39, Definition 2.2]; these properti*
*es will
imply that I and J completely determine the model category structure.
Remark 11.6. The maps in I (resp. J) are called generating cofibrations (resp.
generating acyclic cofibrations). The properties referred to in Definition 11.5*
* are
useful for creating a model category structure on the target D of a left adjoin*
*t, from
an existing cofibrantly generated model structure on the source C. Sometimes the
single left adjoint is replaced by a set of left adjoints with the same target.
11.3. Monoidal model categories.
Definition 11.7. A monoidal model category is a model category C with a sym
metric monoidal closed structure (C, , k) such that the following axiom is sat*
*isfied:
(ENR) If j : A! Bis a cofibration and p : X! Yis a fibration, then the pullba*
*ck
corner map
Map (B, X)____//_Map(A, X) xMap(A,YM)ap(B, Y )
is a fibration that is an acyclic fibration if either j or p is a weak eq*
*uivalence.
Remark 11.8. A model category with a symmetric monoidal closed structure (C, ,*
* k)
satisifies the axiom (ENR) if and only if it satisfies the axiom:
(ENR')If i : K! Land j : A! Bare cofibrations, then the pushout corner map
`
L A K A K B ____//_L B
is a cofibration that is an acyclic cofibration if either i or j is a weak
equivalence.
In particular, Definition 11.7 is equivalent to the definition by Schwede and S*
*hip
ley [39, Section 3] of a monoidal model category. Lewis and Mandell [28] refer *
*to
(ENR) as the enrichment axiom, and also include an additional condition involvi*
*ng
the unit of which we will not require in this paper.
12.Model categories for Gobjects and (symmetric) sequences
So far in this paper, except in Section 1, we have only used the property that
(C, , k) is a symmetric monoidal closed category with all small limits and col*
*imits.
In this section, we begin to make use of the model category assumptions on (C, *
* , k)
described in Basic Assumption 1.1.
In this section, we establish natural model catgory structures on Gobjects a*
*nd
(symmetric) sequences, and investigate how well tensor product and circle produ*
*ct
mesh with this model category structure.
12.1. Model category structure for Gobjects.
Theorem 12.1. Let G be a finite group. Then the category CGopof right G objects
has a natural model category structure. The weak equivalences are the objectwise
weak equivalences and the fibrations are the objectwise fibrations. The model s*
*truc
ture is cofibrantly generated.
HOMOTOPY THEORY OF MODULES OVER OPERADS 29
Proof.The model category structure on CGopis created by the adjunction
_.G//_op
C ooU__CG
with left adjoint on top and U the forgetful functor. Define a map f in CGopto *
*be
a weak equivalence (resp. fibration)oifpUf is a weak equivalence (resp. fibrati*
*on)
in C. Define a map f in CG toobepa cofibration if it has the LLP with respect *
*to
all acyclic fibrations in CG . To verify this gives a model category structure*
*, argue
as in the proof of Theorem 12.2. By construction, the model category structure *
*is
cofibrantly generated.
12.2. Model category structure for (symmetric) sequences.
Theorem 12.2. The symmetric monoidal closed categories (SymSeq, , 1) of sym
metric sequences and (Seq, ^, 1) of sequences have natural model category struc
tures. The weak equivalences are the objectwise weak equivalences and the fibr*
*a
tions are the objectwise fibrations. The model structures are cofibrantly gener*
*ated
and give (SymSeq, , 1) and (Seq, ^, 1) the structure of monoidal model categor*
*ies.
Proof.The model category structure on SymSeqis created by the set of adjunctions
._s//_op____//_ ____//_
C oUo__C s oo___SymSeq0oo___SymSeq, s 0,
in*s
with left adjoints on top, U the forgetful functor, and inclusion ins: ops! 0*
*op.
The righthand adjunction is the equivalence of categories (3.5). Define a map *
*f in
SymSeq to be a weak equivalence (resp. fibration) if U(in*s(f)) is a weak equiv*
*alence
(resp. fibration) in C for every s 0. Define a map f in SymSeqto be a cofibra*
*tion
if it has the LLP with respect to all acyclic fibrations in SymSeq.
We want to verify the model category axioms (MC1)(MC5). We already know
(MC1) is satisfied, and verifying (MC2) and (MC3) is clear. The arguments in the
proof of [39, Lemma 2.3] use (possibly transfinite) small object arguments to v*
*erify
(MC5). The first part of (MC4) is satisfied by definition, and the second part *
*of
(MC4) follows from the usual lifting and retract argument, as described in the *
*proof
of [39, Lemma 2.3]. This verifies the model category axioms. By construction, t*
*he
model category is cofibrantly generated. By Theorem 12.4, this gives (SymSeq, *
*, 1)
the structure of a monoidal model category.
The model category structure on Seqis created by the set of adjunctions
C ____//_CUopsoo_//_Seq0oo_//_Seqoo_, s 0,
in*s
with left adjoints on top, U the forgetful functor, and inclusion ins: ops! 0*
*op.
Argue as in the SymSeqcase.
Remark 12.3. Since the right adjoints in this proof all commute with filtered c*
*ol
imits, the smallness conditions needed for the (possibly transfinite) small obj*
*ect
arguments in [39, Lemma 2.3] are satisfied. Also, condition (1) of [39, Lemma
2.3] is easily verified since colimits in SymSeq and Seq are computed objectwise
in the underlying category, and since acyclic cofibrations in C are preserved u*
*nder
coproducts, pushouts, and transfinite compositions.
30 JOHN E. HARPER
12.3. Pushout corner map for tensor products. Here we verify that tensor
products of symmetric sequences mesh nicely with the model structure.
Theorem 12.4.
(a)In symmetric sequences (resp. sequences), if i : K! Land j : A! Bare
cofibrations, then the pushout corner map
`
L A K A K B ____//_L B,
`
resp. L ^A K ^AK ^B ____//_L ^B,
is a cofibration that is an acyclic cofibration if either i or j is a wea*
*k equiv
alence.
(b)In symmetric sequences (resp. sequences), if j : A! Bis a cofibration and
p : X! Yis a fibration, then the pullback corner map
Map (B, X)____//_Map(A, X) xMap (A,YM)ap (B, Y,)
^ ^ ^
resp. Map (B, X)____//_Map(A, X) xMap^(A,YM)ap (B, Y,)
is a fibration that is an acyclic fibration if either j or p is a weak eq*
*uivalence.
We prove this theorem in section 12.8.
12.4. Pushout corner map for circle products. These model structures also
mesh nicely with circle product, provided an additional cofibrancy condition is
satisfied. A version of the following theorem is given by Rezk [38] for symmetr*
*ic
sequences of simplicial sets, using a model category structure with fewer weak
equivalences.
Theorem 12.5. Let A be a cofibrant symmetric sequence (resp. cofibrant se
quence).
(a)In symmetric sequences (resp. sequences), if i : K! Land j : A! Bare
cofibrations, then the pushout corner map
`
L O A KOA K O B___//_L O,B
`
resp. L ^OA K ^OAK ^OB___//L ^OB,
is a cofibration that is an acyclic cofibration if either i or j is a wea*
*k equiv
alence.
(b)In symmetric sequences (resp. sequences), if j : A! Bis a cofibration and
p : X! Yis a fibration, then the pullback corner map
Map O(B, X)____//MapO(A, X) xMapO(A,YM)apO(B,,Y )
^O ^O
resp. Map ^O(B, X)___//Map(A, X) xMap^O(A,YM)ap(B, Y,)
is a fibration that is an acyclic fibration if either j or p is a weak eq*
*uivalence.
We prove this theorem in section 12.8.
HOMOTOPY THEORY OF MODULES OVER OPERADS 31
12.5. Fixed points and the pullback corner map.
Proposition 12.6. Let G be a finite group and H G a subgroup. In CGop,
suppose j : A! Bis a cofibration and p : X! Yis a fibration. Then in C the
pullback corner map
Map(B, X)H_____//Map(A, X)H xMap(A,Y )HMap(B, Y )H
is a fibration that is an acyclic fibration if either j or p is a weak equivale*
*nce.
Proof.Suppose j : A! Bis a cofibration and p : X! Yis an acyclic fibration.
Let i : C! Dbe a cofibration in C. We want to verify the pullback corner map
satisfies the right lifting property with respect to i.
(12.7) C ______________//_Map(B, X)H
 ____55________
 ____________
 ___________ 
fflffl_____________ fflffl
D ____//_Map(A, X)H xMap(A,Y )HMap(B, Y )H .
The solid commutativeodiagramp(12.7)in C has a lift if and only if the solid di*
*agram
(12.8)in CG has a lift,
(12.8) C . (H\G)_____________//_Map(B,4X)4__
_______
 _____________
 _____________ 
fflffl_______ fflffl
D . (H\G)____//_Map(A, X) xMap(A,YM)ap(B, Y ).
if and only if the solid diagram (12.9)in CGophas a lift.
(12.9) A ___________________//_Map(D . (H\G), X)
 _____33__________
 _______________
 ________________ 
fflffl__________________ fflffl
B ____//_Map(C . (H\G), X) xMap(C.(H\G),YM)ap(D . (H\G), Y ).
Hence it is sufficient to verify that theorighthandpvertical map in (12.9)is an
acyclic fibration in C, and hence in CG . The map i . id: C . (H\G)!D . (H\G)
is isomorphic in C to a coproduct of cofibrations in C, hence is itself a cofib*
*ration
in C, and the (ENR) axiom finishes the argument for this case. The other cases *
*are
similar.
12.6. Calculations for mapping sequences.
Proposition 12.10. Let B and X be symmetric sequences and t 1. Then for
each s 0 there is a natural isomorphism in C,
Y
Map (B[s, t], X[s]) s ~= Map (B[s1] . . .B[st], X[s]) s1x...x st.
s1+...+st=s
Proof.Use the calculation in Proposition 4.3.
32 JOHN E. HARPER
12.7. Tensor products and (acyclic) cofibrations for Gobjects.
Proposition 12.11. Let G1, . .,.Gn be finite groups.
(a)Suppose for ko=p1, . .,.n that jk: Ak! Bkis a cofibration between cofibr*
*ant
objects in CGk . Then the induced map
j1 . . .jn: A1 . . .An!B1 . . .Bn
is a cofibration in C(G1x...xGn)opthat is an acyclic cofibration if each *
*jk is
a weak equivalence. op
(b)Suppose for k = 1, . .,.n that Ak is a cofibrant objectoinpCGk . Then
A1 . . .An is a cofibrant object in C(G1x...xGn).
Remark 12.12. By the righthand adjunction in (2.10),  Ak preserves initial
objects. In particular, if A1, . .,.An in the statementoofp(a) are all initial *
*objects,
then A1 . . .An is an initial object in C(G1x...xGn).
Proof.For each n, statement (b) is a special case of statement (a), hence it is
sufficient to verify (a). By induction on n, it is enough to verify the case n *
*= 2.
Supposeoforpk = 1, 2 that jk: Ak! Bkis a cofibration between cofibrant objects
in CGk . The induced map j1 j2: A1 A2 !B1 B2 factors as
j1 id id j2
A1 A2 ____//_B1 A2___//_B1 B2,
hence it is sufficient to verify each of theseoispa cofibration in C(G1xG2)op. *
*Consider
any acyclic fibration p : X! Yin C(G1xG2) . We want to show that j1 id has
the LLP with respect to p.
A1 A2 ____//_X;;___A1___//Map(A2,9X)G29_
 _________  ___________
 ______  _______ 
fflffl__fflffl__fflffl___fflffl______
B1 A2 ____//_Y B1_____//Map(A2, Y )G2
The lefthand solid commutative diagramoinpC(G1xG2)ophas a lift if and only ifop
the righthand solid diagram in CG1 has a lift. We know A2 is cofibrant in CG2 ,
hence by Proposition 12.6 the righthand solid diagramohaspa lift, finishing the
argument thatoj1pidis a cofibration in C(G1xG2) . Similarly, id j2 is a cofibra*
*tion
in C(G1xG2) . The case for acyclic cofibrations is similar.
The following proposition is also useful.
Proposition 12.13. Let G1 and G2 beofinitepgroups. Suppose for k = 1, 2 that
jk: Ak! Bkis a cofibration in CGk . Then the pushout corner map
(12.14) B1 A2qA1 A2A1 B2 ____//_B1 B2
is a cofibration in C(G1xG2)opthat is an acyclic cofibration if either j1 or j2*
* is a
weak equivalence.
op
Proof.Suppose for k = 1, 2 that jk: Ak! Bkisoapcofibration in CGk . Consider
any acyclic fibration p : X! Yin C(G1xG2) . We want to show that the pushout
HOMOTOPY THEORY OF MODULES OVER OPERADS 33
corner map (12.14)has the LLP with respect to p. The solid commutative diagram
B1 A2qA1 A2A1 B2 ____//_X66_______
_____
 _________
 __________ 
fflffl_____ fflffl
B1 B2 __________//_Y
in C(G1xG2)ophas a lift if and only if the solid diagram
A1________________//Map(B2,4X)G24__
________
 ______________
 _____________ 
fflffl______________ fflffl
B1_____//Map(A2, X)G2 xMap(A2,Y )G2Map(B2, Y )G2
op Gop
in CG1 has a lift. We know A2! B2is a cofibration in C 2 , henceoPropositionp1*
*2.6
finishes the argument that (12.14)is a cofibration in C(G1xG2) . The other cases
are similar.
12.8. Proofs for the pushout corner map theorems.
Proof of Theorem 12.4.Statements (a) and (b) are equivalent. This can be verifi*
*ed
using the natural correspondence (4.17)together with the various lifting charac*
*ter
izations [4, Proposition 3.13] satisfied by any closed model category. Hence it*
* is
sufficient to verify statement (b).
Suppose j : A! Bis an acyclic cofibration and p : X! Yis a fibration. We
want to verify each pullback corner map
Map (B, X)[t]___//_Map(A, X)[t] xMap (A,Y )[t]Map(B, Y )[t],
is an acyclic fibration in C. By Proposition 4.20 it is sufficient to show each*
* map
Map(B[s], X[t + s]) s


fflffl
Map(A[s], X[t + s]) sxMap(A[s],Y [t+Ms])asp(B[s], Y [t + s]) s
is an acyclic fibration in C. Proposition 12.6 completes the proof for this cas*
*e. The
other cases are similar.
Proof of Theorem 12.5.Statements (a) and (b) are equivalent. This can be verifi*
*ed
using the natural correspondence (5.17)together with the various lifting charac*
*ter
izations [4, Proposition 3.13] satisfied by any closed model category. Hence it*
* is
sufficient to verify statement (b).
Suppose j : A! Bis an acyclic cofibration between cofibrant objects and p : *
*X! Y
is a fibration. We want to verify each pullback corner map
Map O(B, X)[t]__//_MapO(A, X)[t] xMapO(A,Y )[t]MapO(B, Y )[t],
34 JOHN E. HARPER
is an acyclic fibration in C. If t = 0, this map is an isomorphism by a calcula*
*tion
in Proposition 5.22. If t 1, by Proposition 5.20 it is sufficient to show eac*
*h map
Map(B[s, t], X[s]) s


fflffl
Map(A[s, t], X[s]) sxMap(A[s,t],Y [s])Msap(B[s, t], Y [s]) s
is an acyclic fibration in C. By Propositions 12.10 and 12.6, it is enough to v*
*erify
each map
A[s1] . . .A[st]_//_B[s1] . . .B[st],
is an acyclic cofibration in C( s1x...x st)op. Proposition 12.11 completes the *
*proof
for this case. The other cases are similar.
12.9. A special case.
Remark 12.15. Consider Theorem 12.5. It is useful to note that L O ; and K O ;
may not be isomorphic, and similarly Map O(;, X) and Map O(;, Y ) may not be
isomorphic. On the other hand, Theorem 12.5 reduces the proof of the following
proposition to a trivial inspection at the emptyset 0.
Proposition 12.16. Let B be a cofibrant symmetric sequence (resp. cofibrant
sequence).
(a)In symmetric sequences (resp. sequences), if i : K! Lis a cofibration,
then the induced map
K O B! L O B , resp. K ^OB! L ^OB ,
is a cofibration that is an acyclic cofibration if i is a weak equivalenc*
*e.
(b)In symmetric sequences (resp. sequences), if p : X! Yis a fibration, then
the induced map
^O ^O
Map O(B, X)! Map O(B, Y ) , resp. Map (B, X)! Map (B, Y ) ,
is a fibration that is an acyclic fibration if p is a weak equivalence.
Proof.Statements (a) and (b) are equivalent. This can be verified using the nat*
*ural
correspondence (5.17)together with the various lifting characterizations satisf*
*ied
by any closed model category. Hence it is sufficient to verify (b). Suppose B is
cofibrant and p : X! Yis an acyclic fibration. We want to verify each induced
map
MapO(B, X)[t]! Map O(B, Y )[t]
is an acyclic fibration in C. Theorem 12.5(b) implies this for t 1. For t =*
* 0,
it is enough to note that X[0]! Y [0] is an acyclic fibration. The other case*
* is
similar.
13.Proofs
In this section we give proofs of Theorems 1.2 and 1.4. When working with left
modules over an operad, we are led naturally to replace (C, , k) with (SymSeq,*
* , 1)
as the underlying monoidal model category, and hence to working with symmetric
arrays.
HOMOTOPY THEORY OF MODULES OVER OPERADS 35
13.1. Arrays and symmetric arrays.
Definition 13.1.
oA symmetric array in C is a symmetric sequence in SymSeq; i.e. a functor
A : op! SymSeq. An array in C is a sequence in Seq; i.e. a functor
A : op! Seq. op op op
oSymArray := SymSeq ~=C x is the category of symmetricoarrayspopop
in C and their natural transformations. Array:= Seq ~=C x is the
category of arrays in C and their natural transformations.
All of the statements and constructions which were previously described in te*
*rms
of (C, , k) are equally true for (SymSeq, , 1) and (Seq, ^, 1), and we usuall*
*y cite
and use the appropriate statements and constructions without further comment.
Theorem 13.2. The categories SymArrayof symmetric arrays and Arrayof arrays
have natural model category structures. The weak equivalences are the objectwise
weak equivalences and the fibrations are the objectwise fibrations. The model s*
*truc
tures are cofibrantly generated.
Proof.This is a special case of Theorem 12.2 with (C, , k) replaced by (SymSeq*
*, , 1)
and (Seq, ^, 1).
13.2. Model category structures in operad case.
Proof of Theorem 1.4.The model category structure on LtO(resp. AlgO) is created
by the adjunction
OO_//_ i OO()//_ j
SymSeq oo__LtO_ resp. Coo___AlgO
U U
with left adjoint on top and U the forgetful functor. Define a map f in LtOto be
a weak equivalence (resp. fibration) if U(f) is a weak equivalence (resp. fibra*
*tion)
in SymSeq. Similarly, define a map f in AlgO to be a weak equivalence (resp.
fibration) if U(f) is a weak equivalence (resp. fibration) in C. Define a map f*
* in
LtO (resp. AlgO) to be a cofibration if it has the LLP with respect to all acyc*
*lic
fibrations in LtO(resp. AlgO).
Consider the case of LtO. We want to verify the model category axioms (MC1)
(MC5). We already know (MC1) is satisfied, and verifying (MC2) and (MC3) is
clear. The arguments in the proof of [39, Lemma 2.3] use (possibly transfinite)
small object arguments to reduce (MC5) to verifying Proposition 13.4 below. The
first part of (MC4) is satisfied by definition, and the second part of (MC4) fo*
*llows
from the usual lifting and retract argument, as described in the proof of [39, *
*Lemma
2.3]. This verifies the model category axioms. By construction, the model categ*
*ory
is cofibrantly generated.
Consider the case of AlgO. Argue similar to the case of LtO, except use Propo
sition 13.4 together with Remark 13.7. By construction, the model category is
cofibrantly generated.
Remark 13.3. Since the forgetful functors in this proof commute with filtered c*
*ol
imits, the smallness conditions needed for the (possibly transfinite) small obj*
*ect
arguments in [39, Lemma 2.3] are satisfied.
36 JOHN E. HARPER
13.3. Analysis of pushouts for operad case.
Proposition 13.4. Let O be a operad and A 2 LtO. Assume every object in
SymArrayis cofibrant. Then every (possibly transfinite) composition of pushouts*
* in
LtO of the form
(13.5) O O X_________//_A
idOi j
fflffl fflffl
O O Y____//_A qOOX O O Y,
such that i : X! Yan acyclic cofibration in SymSeq, is a weak equivalence in t*
*he
underlying category SymSeq.
Remark 13.6. The proof of this proposition verifies the stronger statement that*
* the
(possibly transfinite) composition of pushouts in LtOis an acyclic cofibration *
*in the
underlying category SymSeq.
Remark 13.7. If X, Y, A are concentrated at 0, then the pushout diagram (13.5)
is concentrated at 0. To verify this, use Proposition 5.8 and the construction *
*of
colimits described in the proof of Proposition 8.10.
This subsection is devoted to proving Proposition 13.4, which we cited above *
*in
the proof of Theorem 1.4. A first step in analyzing the pushouts in (13.5)is an
analysis of certain coproducts. The following proposition is motivated by a sim*
*ilar
construction given in [14, Section 2.3] and [31, Section 13] in the context of *
*algebras
over an operad.
Proposition 13.8. Let O be a operad, A 2 LtO, and Y 2 SymSeq. Consider
any coproduct in LtOof the form
(13.9) A q (O O Y ).
There exists a symmetric array OA and natural isomorphisms
a
A q (O O Y ) ~=OA O (Y ) = OA[q] qY q
q 0
in the underlying category SymSeq. If Q 2 and q := Q, then OA[Q] is natural*
*ly
isomorphic to a colimit of the form
` ` pod0o_` 'p
OA[Q] ~=colim p 0O[p + q] pA ood1_p 0O[p + q] p(O O A) ,
in SymSeq, with d0 induced by operad multiplication and d1 induced by m : O O A*
*!A.
Proof.The coproduct in (13.9)may be calculated by a reflexive coequalizer in LtO
of the form,
i d0 j
A q (O O Y ) ~=colim (O O A) q (O O Yo)o(O_OoOoO_A) q (O O Y,)
d1
HOMOTOPY THEORY OF MODULES OVER OPERADS 37
The maps d0 and d1 are induced by maps m : O O O!O and m : O O A!A,
respectively. There are natural isomorphisms in the underlying category SymSeq,
(O O A) q (O O Y~)=O O (A q Y )
~=a O[t] t(A q Y ) t
t 0i j
~=a a O[p + q] pA p qY q,
q 0p 0
and similarly,
a ia j
(O O O O A) q (O O Y ) ~= O[p + q] p(O O A) p qY q.
q 0 p 0
The maps d0 and d1 similarly factor in the underlying category SymSeq.
Remark 13.10. We have used the natural isomorphisms
a
(A q Y ) t ~= p+q. px qA p Y q,
p+q=t
in the proof of Proposition 13.8.
Definition 13.11. Let i : X! Ybe a morphism in SymSeq and t 1. Define
Qt0:= X tand Qtt:= Y t. For 0 < q < t define Qtqinductively by the pushout
diagrams
pr* t
t. tqx qX (tq)Qqq1___//_Qq1
i* i*
fflffl fflffl
t. tqx qX (tq)Y q____//_Qtq
in SymSeq t.
Remark 13.12. The construction Qt1tcan be thought of as a tequivariant versi*
*on
of the colimit of a punctured tcube (Proposition 13.23). If the category C is *
*pointed,
there is a natural isomorphism Y t=Qtt1~=(Y=X) t.
The following proposition provides a useful description of certain pushouts of
left modules, and is motivated by a similar construction given in [6, section 1*
*2] in
the context of simplicial multifunctors of symmetric spectra.
Proposition 13.13. Let O be a operad, A 2 LtO, and i : X! Yin SymSeq.
Consider any pushout diagram in LtOof the form,
f
(13.14) O O X __________//_A
idOi 
fflffl fflffl
O O Y_____//A q(OOX)(O O Y ).
The pushout in (13.14)is naturally isomorphic to a filtered colimit of the form
i j1 j2 j3 j
(13.15) A q(OOX)(O O Y ) ~=colim A0 ____//_A1___//A2___//_. . .
38 JOHN E. HARPER
in the underlying category SymSeq, with A0 := OA[0] ~=A and At defined induc
tively by pushout diagrams in SymSeq of the form
(13.16) OA[t] tQtt1f*//_At1.
idti* jt
fflffl ,t fflffl
OA[t] tY t____//_At
Proof.The pushout in (13.14)may be calculated by a reflexive coequalizer in LtO
of the form
i _i j
A q(OOX)(O O Y ) ~=colim A q (O O Yo)o__A_qo(OoO_X) q (O O Y )
f
_ __
The maps i and f are induced by maps idO i* and idO f* which fit into the
commutative diagram
od0o_
(13.17) OA O (X q Yo)o__O O (A q X q Yo)o__O O (O O A) q X q Y
d1
_i_  
f idOi*idOf* idOi*idOf*
fflfflfflffl fflfflfflffld0 fflfflfflffl
OA O (Yo)o______O O (A q Yo)o______OoOo(O_O A) q Y )
d1
in LtO, with rows reflexive coequalizer diagrams, and maps i* and f* in SymSeqi*
*n
duced by i : X! Yand f : X! A. By Proposition 13.8, the coproduct in (13.14)
may be calculated by the colimit of the left hand column of (13.17)in the under
lying category SymSeq. We want to reconstruct this colimit via a suitable filte*
*red
colimit. __
Using (13.17), there exist maps fq,pwhich make the diagrams
` ` ijinq,ip j
OA O (X q Y ) ~= oo___ OA[p + q] x X p Y q
q 0p 0 _p q
__
_f ______
 9_fq,p___
fflffl ij fflffl____
` inq i j
OA O (Y ) ~= oo____________ OA[q] Y q
t 0 q
_
in SymSeqcommute. Similarly, there exist maps iq,pwhich make the diagrams
` ` ijinq,ip j
OA O (X q Y ) ~= oo___ OA[p + q] x X p Y q
q 0p 0 _p q
__
_i ______
 9_iq,p___
fflffl ij fflffl____
` inp+q i j
OA O (Y ) ~= oo________ OA[p + q] Y (p+q)
t 0 p+q
HOMOTOPY THEORY OF MODULES OVER OPERADS 39
in SymSeq commute._Let_'_: OA O (Y)!.be a morphism in SymSeq and define
'q := 'inq. Then 'i= 'f if and only if the diagrams
_f
OA[p + q] px qX p Y _qq,p//_OA[q] qY q
_iq,p 
 'q
fflffl 'p+q fflffl
OA[p + q] p+qY (p+q)_________//.
_ __
commute for every p, q 0. Since iq,0= idand fq,0= id, it is sufficient to con*
*sider
q 0 and p > 0.
To motivate the construction (13.16), it is useful to describe a preliminary *
*con
struction which also calculates the pushout in (13.14). Define A0 := OA[0] ~=A
and for each t 1 define Atby the pushout diagram
`
OA[p + q] px qX p Y q f*
(13.18) p+q=t __________//At1
q 0,p>0

i* jt
fflffl ,t fflffl
OA[t] tY _t__________________//At
__ _
in SymSeq. The maps f* and i* are induced by the appropriate maps fq,pand iq,p.
Verify that (13.15)is satisfied.__
The collection of maps fq,pand iq,psatisfy many compatibility relations. This
suggests we replace the coproduct in (13.18), which is isomorphic to
h i
OA[t] t(X q Y ) t Y t,
with an appropriate (possibly smaller) pushout construction. Here, (XqY ) tY t
means the coproduct of all factors in (X q Y ) t except Y t.
Define A0 := OA[0] ~=A and for each t 1 define At by the pushout diagram_
(13.16)in_SymSeq. The maps f* and i* are induced by the appropriate maps fq,p
and iq,p. Verify that (13.15)is satisfied.
Proposition 13.19. Let O be a operad, A 2 LtO, and i : X! Ya cofibration
(resp. acyclic cofibration) in SymSeq. Assume every object in SymArrayis cofibr*
*ant.
Consider any pushout diagram in LtOof the form,
(13.20) O O X_________//_A
idOi j
fflffl fflffl
O O Y____//_A qOOX O O Y.
Then each map jt: At1!Atin the filtration (13.15)is a cofibration (resp. acyc*
*lic
cofibration) in SymSeq. In particular, j is a cofibration (resp. acyclic cofibr*
*ation)
in the underlying category SymSeq.
Proof.Suppose i : X! Yis an acyclic cofibration in SymSeq. We want to show
each jt: At1!Atis an acyclic cofibration in SymSeq. By the construction of jt
in Proposition 13.13, it is sufficient to verify each id ti* in (13.16)is an a*
*cyclic
40 JOHN E. HARPER
cofibration. Suppose p : C! Dis a fibration in SymSeq. We want to verify id t*
*i*
has the LLP with respect to p.
OA[t] tQtt1__//_C99__OA[t]__Qtt1___//C_99__
____ _______
 ________  _______
 _______   _______ 
fflffl____ fflffl fflffl____fflffl
OA[t] tY t___//_D OA[t] Y t___//_D,
The lefthand solid commutative diagramoinpSymSeq has a lift if and only if the
righthand solid diagram in SymSeq t has a lift. Hence it is sufficient to veri*
*fy
that the solid diagram
; __________________//_Map(Y t, C)
 ____44_________
 _____________(*)
 _____________ 
fflffl______________ fflffl
OA[t]____//Map(Qtt1, C) xMap (Qtt1,D)Map(Y t, D)
op op
in SymSeq t has a lift. We know OA[t] is cofibrant in SymSeq t , hence suffi
cient to verify (*) is an acyclic fibration. By Theorem 12.4 it is enough to ve*
*rify
i*: Qtt1!Y tis an acyclic cofibration in the underlying category SymSeq, and
Proposition 13.21 finishes the proof. The other case is similar.
13.4. Punctured cube.
Proposition 13.21. Let i : X! Ybe a cofibration (resp. acyclic cofibration) in
SymSeq. Then the induced map i*: Qtt1!Y tis a cofibration (resp. acyclic
cofibration) in the underlying category SymSeq.
Before proving this proposition, we establish some notation.
Definition 13.22. Let t 2.
oCubet is the category with objects the vertices (v1, . .,.vt) 2 {0, 1}t o*
*f the
unit tcube; there is at most one morphism between any two objects, and
there is a morphism
(v1, . .,.vt)! (v01, . .,.v0t)
if and only if vi v0ifor every 1 i t. In particular, Cubetis the cat*
*egory
associated to a partial order on the set {0, 1}t.
oThe punctured cube pCubetis the full subcategory of Cubetwith all objects
except the terminal object (1, . .,.1) of Cubet.
Let i : X! Ybe a morphism in SymSeq. It will be useful to introduce an
associated functor w : pCubet!SymSeqdefined objectwise by
ae
w(v1, . .,.vt) := c1 . . .ct with ci:= X,Y,forfvi=o0,r v
i= 1,
HOMOTOPY THEORY OF MODULES OVER OPERADS 41
and with morphisms induced by i : X! Y. In particular, for t = 3 the diagram w
looks like,
X XOOY ____//_X OYO Y
ppp  
pppp  
wwpppp  
Y XOOY X X X ____//_X Y X
 ppp pppp
 pppp pppp
 wwpppp wwppp
Y X X ____//_Y Y X.
Proposition 13.23. Let i : X! Ybe a morphism in SymSeq and t 2. There
are natural isomorphisms
Qtt1~=colimw : pCubet!SymSeq
in the underlying category SymSeq.
Proof.Use the t maps in SymSeqobtained from the map
t. 1x t1X Y (t1)!Qtt1
in Definition 13.11 to define a cone into Qtt1and verify this cone is initial *
*with
respect to all cones.
Proof of Proposition 13.21.Suppose i : X! Yis an acyclic cofibration in SymSeq.
The colimit of the diagram w : pCubet!SymSeqmay be computed inductively
using pushout corner maps, and hence by Proposition 13.23 there are natural iso
morphisms
Q21~=Y X qX X X Y,
Q32~=Y Y X q Y XqX XX Y X Y X qX X X Y Y, . . .
in the underlying category SymSeq. The same argument provides an inductive con
struction of the induced map i*: Qtt1!Y itn the underlying category SymSeq;
using the natural isomorphisms in Proposition 13.23, for each t 2 the Qtt1fit
into pushout squares
Qt1t2Xid_i_//Qt1t2Y__
_________________________________________*
*__________
i*id  _______________________________________*
*____
fflffl fflffli*_id_______________________________*
*_______
(t1) ______//Qt ____________________
Y __X_________t1_ ________________________
______________________________________________________*
*__________________________________i_______
____________________________________________________*
*__________________________________________________________________*_________
__idi__________''_______________________________*
*_____________________________________________________________________________*
*____________________________________________9!$$______
_______//_________________________________*
*_____________________________________________________________________________*
*________________________Y t
in the underlying category SymSeqwith induced map i*: Qtt1!Y tthe indicated
pushout corner map. By iterated applications of Theorem 12.4, i* is an acyclic
cofibration in SymSeq. The case for cofibrations is similar.
Remark 13.24. This construction of i* by iterated pushout corner maps is used in
the proof of the main theorem in [39].
42 JOHN E. HARPER
Proof of Proposition 13.4.By Proposition 13.19, each map j is an acyclic cofibr*
*a
tion in the underlying category SymSeq. Noting that (possibly transfinite) comp*
*o
sitions of acyclic cofibrations are acyclic cofibrations, completes the proof.
13.5. Model category structures in operad case. In the following subsec
tions we prove Theorem 1.2. The argument and required constructions are related
to those of the previous subsections, but different enough to require some expo*
*si
tion. The strong cofibrancy condition exploited in Theorem 1.4 is replaced here
by the weaker monoid axiom [39], but at the cost of dropping all actions; i.e*
*.,
working with operads instead of operads.
Definition 13.25. A monoidal model category satisfies the monoid axiom if every
map which is a (possibly transfinite) composition of pushouts of maps in
(13.26) {acyclic cofibrations} C
is a weak equivalence.
Remark 13.27. In this definition, (13.26)is notation for the collection of maps*
* of
the form
f id: K B! L B
such that f : K! Lis an acyclic cofibration and B 2 C.
Proof of Theorem 1.2.The model category structure on LtO(resp. AlgO) is created
by the adjunction
O_^O//_ i O_^O()//_j
Seqoo___LtO resp. C oo___AlgO
U U
with left adjoint on top and U the forgetful functor. Define a map f in LtOto b*
*e a
weak equivalence (resp. fibration) if U(f) is a weak equivalence (resp. fibrati*
*on) in
Seq. Similarly, define a map f in AlgOto be a weak equivalence (resp. fibration)
if U(f) is a weak equivalence (resp. fibration) in C. Define a map f in LtO(res*
*p.
AlgO) to be a cofibration if it has the LLP with respect to all acyclic fibrati*
*ons in
LtO (resp. AlgO).
Consider the case of LtO. We want to verify the model category axioms (MC1)
(MC5). We already know (MC1) is satisfied, and verifying (MC2) and (MC3) is
clear. The arguments in the proof of [39, Lemma 2.3] use (possibly transfinite)
small object arguments to reduce (MC5) to verifying Proposition 13.29 below. The
first part of (MC4) is satisfied by definition, and the second part of (MC4) fo*
*llows
from the usual lifting and retract argument, as described in the proof of [39, *
*Lemma
2.3]. This verifies the model category axioms. By construction, the model categ*
*ory
is cofibrantly generated.
Consider the case of AlgO. Argue similar to the case of LtO, except use Propo
sition 13.29 together with Remark 13.31. By construction, the model category is
cofibrantly generated.
Remark 13.28. Since the forgetful functors in this proof commute with filtered
colimits, the smallness conditions needed for the (possibly transfinite) small *
*object
arguments in [39, Lemma 2.3] are satisfied.
HOMOTOPY THEORY OF MODULES OVER OPERADS 43
13.6. Analysis of pushouts for operad case.
Proposition 13.29. Let O be an operad and A 2 LtO. Assume (C, , k) satisfies
the monoid axiom. Then every (possibly transfinite) composition of pushouts in *
*LtO
of the form
(13.30) O ^OX_________//_A
id^Oi j
fflffl fflffl
O ^OY____//_A qO ^OXO ^OY,
such that i : X! Yis an acyclic cofibration in Seq, is a weak equivalence in t*
*he
underlying category Seq.
Remark 13.31. If X, Y, A are concentrated at 0, then the pushout diagram (13.30)
is concentrated at 0. To verify this, argue as in Remark 13.7.
This subsection is devoted to proving Proposition 13.29, which we cited above
in the proof of Theorem 1.2. Similar to the previous subsections, a first step*
* in
analyzing the pushouts in (13.30)is an analysis of certain coproducts. The foll*
*owing
is an operad version of Proposition 13.8.
Proposition 13.32. Let O be an operad, A 2 LtO, and Y 2 Seq. Consider any
coproduct in LtOof the form
(13.33) A q (O ^OY ).
There exists an array OA and natural isomorphisms
a ^
A q (O ^OY ) ~=OA ^O(Y ) = OA[q] ^Y q
q 0
in the underlying category Seq. If Q 2 and q := Q, then OA[Q] is naturally
isomorphic to a colimit of the form
` ` h ^di0o`o_ h ' ^ i
colim p 0O[p + q] _p+q_.pxAqpd1opo0O[p_+ q] _p+q_.px(qO ^OA),p
in Seq, with d0 induced by operad multiplication and d1 induced by m : O ^OA!A.
Proof.Verify the coproduct in (13.33)may be calculated by a reflexive coequaliz*
*er
in LtOof the form,
i d0 j
A q (O ^OY ) ~=colim (O ^OA) q (O ^OYo)o(O_^OOo^OA)oq_(O ^OY.)
d1
The maps d0 and d1 are induced by maps m : O ^OO!O and m : O ^OA!A,
respectively. There are natural isomorphisms in the underlying category Seq,
(O ^OA) q (O ^OY~)=O ^O(A q Y )
~= a O[t] (A q Y )^t
t 0
ia h ij
~= a O[p + q] __p+q__. A^p ^ Y ^q,
q 0p 0 px q
44 JOHN E. HARPER
and similarly,
a ia h p+q ^ ij ^
(O ^OO ^OA) q (O ^OY ) ~= O[p + q] _______. (O ^OA) p ^ Y q.
q 0 p 0 px q
The maps d0 and d1 similarly factor in the underlying category Seq. It is impor*
*tant
to note that the ordering of all tensor power factors is respected, and that we*
* are
simply using the symmetric groups in the isomorphisms
a ^ ^
(A q Y )^t ~= p+q. px qA p^ Y q
p+q=t
to build convenient indexing sets for the tensor powers.
Definition 13.34. Let i : X! Ybe a morphism in Seqand t 1. Define Qt0:=
X ^tand Qtt:= Y ^t. For 0 < q < t define Qtqinductively by the pushout diagrams
pr* t
t. tqx qX ^(tq)^Qqq1__//_Qq1
i* i*
fflffl fflffl
t. tqx qX ^(tq)^Y ^q___//_Qtq
in Seq t.
The following is an operad version of Proposition 13.13, and provides a use*
*ful
description of certain pushouts of left modules.
Proposition 13.35. Let O be an operad, A 2 LtO, and i : X! Yin Seq.
Consider any pushout diagram in LtOof the form,
f
(13.36) O ^OX___________//_A
id^Oi 
fflffl fflffl
O ^OY_____//A q(O ^OX)(O ^OY ).
The pushout in (13.36)is naturally isomorphic to a filtered colimit of the form
i j1 j2 j3 j
(13.37) A q(O ^OX)(O ^OY ) ~=colim A0 ____//_A1___//A2___//_. . .
in the underlying category Seq, with A0 := OA[0] ~=A and At defined inductively
by pushout diagrams in Seqof the form
f*
(13.38) OA[t] ^Qtt1___//At1
id^i* jt
fflffl,t fflffl
OA[t] ^Y ^t____//_At.
Proof.Verify the pushout in (13.36)may be calculated by a reflexive coequalizer
in LtOof the form
i _i j
A q(O ^OX)(O ^OY ) ~=colim A q (O ^OYo)o__A_qo(Oo^OX)_q (O ^OY )
f
HOMOTOPY THEORY OF MODULES OVER OPERADS 45
_ __
The maps iand f are induced by maps id^Oi* and id^Of* which fit into the com
mutative diagram
od0o_
(13.39) OA ^O(X q Yo)o__O ^O(A q X q Yo)o__O ^O(O ^OA) q X q Y
d1
_i_  
f id^Oi*id^Of* id^Oi*id^Of*
fflfflfflffl fflfflfflffld0 fflfflfflffl
OA ^O(Yo)o______O ^O(A q Yo)o______Oo^O(Oo^OA)_q Y )
d1
in LtO, with rows reflexive coequalizer diagrams, and maps i* and f* in Seqindu*
*ced
by i : X! Yand f : X! A. By Proposition 13.32, the coproduct in (13.36)may
be calculated by the colimit of the left hand column of (13.39)in the underlying
category Seq. We want to reconstruct_this colimit via a suitable filtered colim*
*it.
Using (13.39), there exist maps fq,pwhich make the diagrams
` ` ijinqi,p j
OA ^O(X q Y ) ~= oo___ OA[p + q] ^ p+q. x X ^p^Y ^q
q 0p 0 _ p q
__
_f ______
 9_fq,p___
fflffl ij fflffl____
` inq i j
OA ^O(Y ) ~= oo________________ OA[q] ^Y ^q
t 0
_
in Seqcommute. Similarly, there exist maps iq,pwhich make the diagrams
` ` ijinqi,p j
OA ^O(X q Y ) ~= oo___ OA[p + q] ^ p+q. x X ^p^Y ^q
q 0p 0 _ p q
__
_i ______
 9_iq,p___
fflffl ij fflffl____
` inp+q i j
OA ^O(Y ) ~= oo_____________ OA[p + q] ^Y ^(p+q)
t 0
in Seqcommute._Let_' : OA ^O(Y)!.be a morphism in Seqand define 'q := 'inq.
Then 'i= 'f if and only if the diagrams
^ ^ _fq,p ^
OA[p + q] ^ p+q. px qX p^Y q_____//OA[q] ^Y q
_iq,p '
  q
fflffl 'p+q fflffl
OA[p + q] ^Y ^(p+q)_____________//_.
_ __
commute for every p, q 0. Since iq,0= idand fq,0= id, it is sufficient to con*
*sider
q 0 and p > 0.
To motivate the construction (13.38), it is useful to describe a preliminary *
*con
struction which also calculates the pushout in (13.36). Define A0 := OA[0] ~=A
46 JOHN E. HARPER
and for each t 1 define Atby the pushout diagram
` ^ ^
OA[p + q] ^ p+q. px qX p^Y q f* //
(13.40) p+q=t __________ At1
q 0,p>0 
i* jt
fflffl ,t fflffl
OA[t] ^Y ^t_______________________//At
__ _
in Seq. The maps f* and i* are induced by the appropriate maps fq,pand iq,p.
Verify that (13.37)is satisfied.__
The collection of maps fq,pand iq,psatisfy many compatibility relations. This
suggests we replace the coproduct in (13.40), which is isomorphic to
h i
OA[t] ^ (X q Y )^t Y ^t,
with an appropriate pushout construction. Here, (X q Y )^t  Y ^tmeans the
coproduct of all factors in (X q Y )^t except Y ^t.
Define A0 := OA[0] ~=A and for each t 1 define At by the pushout diagram_
(13.38)in_Seq. The maps f* and i* are induced by the appropriate maps fq,pand
iq,p. Verify that (13.37)is satisfied.
Proposition 13.41. Let i : X! Ybe a cofibration (resp. acyclic cofibration)
in Seq. Then the induced map i*: Qtt1!Y ^tis a cofibration (resp. acyclic
cofibration) in the underlying category Seq.
Proof.Argue as in the proof of Proposition 13.21, replacing (SymSeq, , 1) with
(Seq, ^, 1).
Proposition 13.42. Assume that C satisfies Basic Assumption 1.1 and in addition
satisfies the monoid axiom. Then (Seq, ^, 1) satisfies the monoid axiom.
Proof.Since colimits in Seqare calculated objectwise, use (4.7)together with an
argument that the pushout of a coproduct qfffffof a finite set of maps can be
written as a finite composition of pushouts of the maps fff.
Proof of Proposition 13.29.By Proposition 13.35, j is a (possibly transfinite) *
*com
position of pushouts of maps of the form id^i*, and Propositions 13.42 and 13.41
finish the argument.
14. Shortened proof for chain complexes
In this section, we include a shortened proof of Theorem 1.4, for the special*
* case
(Chk, , k) of unbounded chain complexes over a field of characteristic zero.
Consider the proof of Theorem 1.4 given in Section 13.2, for the case of LtO;*
* for
the special case of AlgOsee Remark 13.7. The (possibly transfinite) small object
arguments only require the pushouts in Proposition 13.4 to be constructed from a
set of generating acyclic cofibrations. In the special case of chain complexes *
*[22,
Section 2.3], a set of generating acyclic cofibrations for SymSeqmay be chosen *
*such
that each has the form i : ;!.DSince acyclic cofibrations are preserved under
(possibly transfinite) compositions, in this special case, the proof of Theorem*
* 1.4
reduces to the following proposition.
HOMOTOPY THEORY OF MODULES OVER OPERADS 47
Proposition 14.1. Let O be a operad and D 2 SymSeq such that i : ;! Dis
an acyclic cofibration. Assume every object in SymArrayis cofibrant. Consider a*
*ny
pushout diagram in LtOof the form,
O O ;________//_A
idOi j
fflffl fflffl
O O D ____//_A q (O O D).
Then j is an acyclic cofibration in the underlying category SymSeq.
Remark 14.2. Suppose G is any finite group. Since k is a field of characterist*
*ic
zero, every k[G]module is projective. It follows that every symmetric array in*
* Chk
is cofibrant.
Proof.By Proposition 13.8, it is enough to verify
OA O (i): OA O (;)!OA O (D)
is an acyclic cofibration in SymSeq. Consider any fibration p : X! Yin SymSeq.
We want to show that OA O (i) has the LLP with respect to p.
OA O (;)____//_X;;____OA O_^;_//_^X<<___
___
 _________  ______
 _______p  _______^p
fflffl___fflffl_fflffl___fflffl_
OA O (D)____//_Y OA O ^D___//_^Y
The lefthand solid commutative diagram in SymSeq has a lift if and only if the
righthand solid diagram in SymArrayhas a lift. Hence it is sufficient to verif*
*y that
the solid diagram
; ______________//_MapO(D^, ^X)
 ____55_________
 __________ (*)
 ___________ 
fflffl___________ fflffl
OA ____//_MapO(^;, ^X) xMapO(^;,^Y)MapO(D^, ^Y)
in SymArrayhas a lift. We know OA is cofibrant in SymArray, hence it is suffici*
*ent
to verify (*) is an acyclic fibration. By Proposition 5.22, it is enough to sho*
*w each
map
Map (D t, X)____//Map(; t, X) xMap (; t,YM)ap(D t, Y )
is an acyclic fibration in SymSeq. Theorem 12.4 together with Propositions 12.6
and 12.11 finish the proof.
References
[1]Maria Basterra and Michael A. Mandell. Homology and cohomology of E1 ring s*
*pectra.
Math. Z., 249(4):903944, 2005.
[2]Mikhail A. Batanin. Homotopy coherent category theory and A1 structures in*
* monoidal
categories. J. Pure Appl. Algebra, 123(13):67103, 1998.
[3]Wojciech Chach'olski and J'er^ome Scherer. Homotopy theory of diagrams. Mem*
*. Amer. Math.
Soc., 155(736):x+90, 2002.
[4]W. G. Dwyer and J. Spali'nski. Homotopy theories and model categories. In H*
*andbook of
algebraic topology, pages 73126. NorthHolland, Amsterdam, 1995.
48 JOHN E. HARPER
[5]William G. Dwyer and HansWerner Henn. Homotopy theoretic methods in group *
*cohomology.
Advanced Courses in Mathematics. CRM Barcelona. Birkh"auser Verlag, Basel, 2*
*001.
[6]A. D. Elmendorf and M. A. Mandell. Rings, modules, and algebras in infinite*
* loop space
theory. Adv. Math., 205(1):163228, 2006.
[7]Benoit Fresse. Lie theory of formal groups over an operad. J. Algebra, 202(*
*2):455511, 1998.
[8]Benoit Fresse. Koszul duality of operads and homology of partition posets. *
*In Homotopy
theory: relations with algebraic geometry, group cohomology, and algebraic K*
*theory, volume
346 of Contemp. Math., pages 115215. Amer. Math. Soc., Providence, RI, 2004.
[9]P. Gabriel and M. Zisman. Calculus of fractions and homotopy theory. Ergebn*
*isse der Math
ematik und ihrer Grenzgebiete, Band 35. SpringerVerlag New York, Inc., New *
*York, 1967.
[10]Ezra Getzler and J. D. S. Jones. Operads, homotopy algebra and iterated int*
*egrals for double
loop spaces. arXiv:hepth/9403055v1, 1994.
[11]Victor Ginzburg and Mikhail Kapranov. Koszul duality for operads. Duke Math*
*. J.,
76(1):203272, 1994.
[12]P. G. Goerss and M. J. Hopkins. Moduli spaces of commutative ring spectra. *
*In Structured
ring spectra, volume 315 of London Math. Soc. Lecture Note Ser., pages 1512*
*00. Cambridge
Univ. Press, Cambridge, 2004.
[13]Paul G. Goerss and Michael J. Hopkins. Andr'eQuillen (co)homology for sim*
*plicial algebras
over simplicial operads. In Une d'egustation topologique [Topological morsel*
*s]: homotopy the
ory in the Swiss Alps (Arolla, 1999), volume 265 of Contemp. Math., pages 41*
*85. Amer.
Math. Soc., Providence, RI, 2000.
[14]Paul G. Goerss and Michael J. Hopkins. Moduli problems for structured ring *
*spectra. Avail
able at http://hopf.math.purdue.edu, 2005.
[15]Paul G. Goerss and John F. Jardine. Simplicial homotopy theory, volume 174 *
*of Progress in
Mathematics. Birkh"auser Verlag, Basel, 1999.
[16]John E. Harper. Homotopy theory of modules over operads in symmetric spectr*
*a. In prepa
ration, 2007.
[17]V. A. Hinich and V. V. Schechtman. On homotopy limit of homotopy algebras. *
*In Ktheory,
arithmetic and geometry (Moscow, 19841986), volume 1289 of Lecture Notes in*
* Math.,
pages 240264. Springer, Berlin, 1987.
[18]Vladimir Hinich. Homological algebra of homotopy algebras. Comm. Algebra, 2*
*5(10):3291
3323, 1997.
[19]Vladimir Hinich. Erratum to "homological algebra of homotopy algebr*
*as".
arXiv:math/0309453v3 [math.QA], 2003.
[20]Vladimir Hinich and Vadim Schechtman. Homotopy Lie algebras. In I. M. Gel0f*
*and Seminar,
volume 16 of Adv. Soviet Math., pages 128. Amer. Math. Soc., Providence, RI*
*, 1993.
[21]Philip S. Hirschhorn. Model categories and their localizations, volume 99 o*
*f Mathematical
Surveys and Monographs. American Mathematical Society, Providence, RI, 2003.
[22]Mark Hovey. Model categories, volume 63 of Mathematical Surveys and Monogra*
*phs. Amer
ican Mathematical Society, Providence, RI, 1999.
[23]Mark Hovey. Spectra and symmetric spectra in general model categories. J. P*
*ure Appl. Al
gebra, 165(1):63127, 2001.
[24]Mark Hovey, Brooke Shipley, and Jeff Smith. Symmetric spectra. J. Amer. Mat*
*h. Soc.,
13(1):149208, 2000.
[25]M. Kapranov and Yu. Manin. Modules and Morita theorem for operads. Amer. J.*
* Math.,
123(5):811838, 2001.
[26]G. M. Kelly. On the operads of J. P. May. Repr. Theory Appl. Categ., (13):1*
*13 (electronic),
2005.
[27]Igor K~r'i~z and J. P. May. Operads, algebras, modules and motives. Ast'eri*
*sque, (233):iv+145pp,
1995.
[28]L. Gaunce Lewis, Jr. and Michael A. Mandell. Modules in monoidal model cate*
*gories. J. Pure
Appl. Algebra, 210(2):395421, 2007.
[29]Saunders Mac Lane. Homology. Classics in Mathematics. SpringerVerlag, Berl*
*in, 1995.
Reprint of the 1975 edition.
[30]Saunders Mac Lane. Categories for the working mathematician, volume 5 of Gr*
*aduate Texts
in Mathematics. SpringerVerlag, New York, second edition, 1998.
[31]Michael A. Mandell. E1 algebras and padic homotopy theory. Topology, 40(1)*
*:4394, 2001.
HOMOTOPY THEORY OF MODULES OVER OPERADS 49
[32]J. P. May. The geometry of iterated loop spaces. SpringerVerlag, Berlin, 1*
*972. Lectures Notes
in Mathematics, Vol. 271.
[33]James E. McClure and Jeffrey H. Smith. A solution of Deligne's Hochschild c*
*ohomology
conjecture. In Recent progress in homotopy theory (Baltimore, MD, 2000), vol*
*ume 293 of
Contemp. Math., pages 153193. Amer. Math. Soc., Providence, RI, 2002.
[34]James E. McClure and Jeffrey H. Smith. Multivariable cochain operations and*
* little ncubes.
J. Amer. Math. Soc., 16(3):681704 (electronic), 2003.
[35]James E. McClure and Jeffrey H. Smith. Operads and cosimplicial objects: an*
* introduction.
In Axiomatic, enriched and motivic homotopy theory, volume 131 of NATO Sci. *
*Ser. II Math.
Phys. Chem., pages 133171. Kluwer Acad. Publ., Dordrecht, 2004.
[36]Daniel Quillen. Rational homotopy theory. Ann. of Math. (2), 90:205295, 19*
*69.
[37]Daniel G. Quillen. Homotopical algebra. Lecture Notes in Mathematics, No. 4*
*3. Springer
Verlag, Berlin, 1967.
[38]Charles Rezk. Spaces of Algebra Structures and Cohomology of Operads. PhD t*
*hesis, MIT,
1996. Available at http://www.math.uiuc.edu/~rezk.
[39]Stefan Schwede and Brooke E. Shipley. Algebras and modules in monoidal mode*
*l categories.
Proc. London Math. Soc. (3), 80(2):491511, 2000.
[40]V. A. Smirnov. Homotopy theory of coalgebras. Izv. Akad. Nauk SSSR Ser. Mat*
*., 49(6):1302
1321, 1343, 1985.
Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, USA
Email address: jharper1@nd.edu