On the derived category of an algebra over an operad

0.22 MB·English

by  Clemens Berger

ON THE DERIVED CATEGORY OF AN ALGEBRA OVER AN
OPERAD
8
0 CLEMENSBERGERANDIEKEMOERDIJK
0
2
Abstract. We present a general construction of the derived category of an

n algebra over an operad andestablish its invarianceproperties. A central role
a isplayedbytheenvelopingoperadofanalgebraoveranoperad.
J

5
2

Introduction
]
T
It is a classical device in homological algebra to associate to an associative ring
A
R the homotopy category of differential graded R-modules, the so-called derived
h. category D(R) of R. One of the important issues is to know when two rings have
t equivalentderivedcategories;positiveanswerstothis questionmaybe obtainedby
a
m means of the theory of tilting complexes, which is a kind of derived Morita theory,
cf. Rickard [15], Keller [10], Schwede [16], To¨en [19]. In this paper, we provide a
[
solution to the problem of giving a suitable construction of the derived category

2 associated to an algebra over an operad in a non-additive context. Building on
v
earlierworkofours’(cf. [2,3]andtheAppendixtothispaper)weestablishgeneral
4
invariance properties of this derived category under change of algebra, change of
6
6 operad and change of ambient category. In the special case of the operad for
2 differential graded algebras, our construction agrees with the classical one.
. A central role in our proofs is played by the enveloping operad P of A whose
1 A
0 algebrasaretheP-algebrasunder A. Indeed,themonoidofunaryoperationsPA(1)
8 may be identified with the enveloping algebra Env (A) of A. The latter has the
P
0 characteristic property that Env (A)-modules (in the classical sense) correspond
P
:
v to A-modules (in the operadic sense). This indirect construction of the enveloping
i algebraoccurs in specific casesat severalplacesin the literature (cf. Getzler-Jones
X
[6],Ginzburg-Kapranov[7],Fresse[4,5],Spitzweck[18],vanderLaan[20],Basterra-
r
a Mandell [1]). The main point in the use of the enveloping operad PA rather than
the envelopingalgebraP (1) is that the assignment(P,A)7→P extends to a left
A A
adjointfunctorwhichonadmissibleΣ-cofibrantoperadsP andcofibrantP-algebras
A behaves like a left Quillen functor on cofibrant objects. It is precisely this good
homotopicalbehaviourthatallowsthedefinitionofthederivedcategoryD (A)for
P
any P-algebra A.
Acknowledgements : WearegratefultoB.Jahren,K.HessandB.Oliver,theor-
ganizingcommittee ofthe AlgebraicTopologySemester 2006ofthe Mittag-Leffler-
Institut in Stockholm. Most of this work has been carried out during the authors’
visit at the MLI in Spring 2006. We are also grateful to B. Fresse for helpful
comments on an earlier version of this paper.
Date:16January2008.
1991 Mathematics Subject Classification. Primary18D50;Secondary18G55, 55U35.
1
2 CLEMENSBERGERANDIEKEMOERDIJK
1. Enveloping operads and enveloping algebras
LetE be a(bicomplete)closedsymmetricmonoidalcategory. WewriteI forthe
unit, −⊗− for the monoidal structure and Hom (−,−) for the internal hom of E.
E
This section is a recollection of known results on categories of modules over a
P-algebra A, where P is any symmetric operad in E. The main objective of this
section is to fix the notations and definitions we use. The category of A-modules
is a module category in the classicalsense for a suitable monoid in E, the so-called
enveloping algebra of A. We will explain in detail that the enveloping algebra of A
is isomorphic to the monoid of unary operations P (1) of the so-called enveloping
A
operad ofthepair(P,A). TheenvelopingoperadP ischaracterisedbyauniversal
A
propertywhich implies in particular that P -algebrasare the P-algebrasunder A.
A
Definition 1.1. Let A be a P-algebra in E. An A-module (under P) consists of
an object M of E together with action maps
µ :P(n)⊗A⊗k−1⊗M ⊗A⊗n−k →M, 1≤k ≤n,
n,k
subject to the following three axioms:
(1) (Unit axiom) The operad-unit I →P(1) induces a commutative triangle
∼
=
I ⊗M - M
-
? µ 1,1
P(1)⊗M
(2) (Associativity axiom) Foreach n=n +···+n ≥1, the following diagram
1 s
commutes:
β
P(s)⊗P(n )⊗···⊗P(n )⊗A⊗k−1⊗M ⊗A⊗n−k - P(s)⊗A⊗l−1⊗M ⊗A⊗s−l
1 s
α µ
? ?s,l
P(n)⊗A⊗k−1⊗M ⊗A⊗n−k - M
µ
n,k
where α is induced by the operad structure of P, and β is induced by the
P-algebra structure of A and the A-module structure of M; in particular, l
is the unique natural number such that n +···+n <k ≤n +···+n .
1 l−1 1 l
(3) (Equivariance axiom) For each n≥1, the µ induce a total action
n,k
k=n
µ :P(n)⊗ ( A⊗k−1⊗M ⊗A⊗n−k) - M
n Σn
ka=1
where the symmetric group Σ acts on the coproduct by permuting factors.
n
A morphism f : M → N of A-modules under P is a morphism in E
rendering commutative all diagrams of the form
id⊗id⊗f ⊗id
P(n)⊗A⊗k−1⊗M ⊗A⊗n−k - P(n)⊗A⊗k−1⊗N ⊗A⊗n−k
µM µN
n,k n,k
? ?
-
M N.
f
ON THE DERIVED CATEGORY OF AN ALGEBRA OVER AN OPERAD 3
The category of A-modules under P will be denoted by Mod (A); the forgetful
P
functor (M,µ)7→M will be denoted by U :Mod (A)→E.
A P
Remark 1.2. The pairs (A,M) consisting of a P-algebra A and an A-module M
define a category with morphisms the pairs (φ,ψ):(A,M)→(B,N) consisting of
a map of P-algebras φ : A → B and a map of A-modules ψ : M → φ∗(N). This
categorycanbe identifiedwith the full subcategoryofleft P-modules concentrated
indegrees 0 and1. In orderto make this moreexplicit, recallthat a left P-module
M consistsofacollection(M ) ofΣ -objectstogetherwithamapofcollections
k k≥0 k
P◦M →M satisfying the usualaxiomsofa leftaction. Suchacollection(M )
k k≥0
is concentrated in degrees 0 and 1 precisely when all M for k ≥ 2 are initial
k
objects in E. The left P-module structure restricted to M endows M with a P-
0 0
algebrastructure,whiletheleftP-modulestructurerestrictedto(M ,M )amounts
0 1
precisely to an M -module structure on M under P.
0 1
Thereisyetanotherwaytospecifysuchapair(A,M)ifE isanadditive category.
Recall that a P-algebra structure on the object A of E is equivalent to an operad
map P → End taking values in the endomorphism operad of A. The latter is
A
defined by
End (n)=Hom (A⊗n,A)
A E
where the operadstructure maps aregivenby substitution and permutationof the
factorsinthe domain. ForapairofobjectsAandM inanadditivecategoryE, we
define a linear endomorphism operad End of M relative to A such that operad
M|A
maps P → End correspond to a P-algebra structure on A together with an
M|A
A-module structure on M.
This linear endomorphism operad End is defined as a suboperad of the en-
M|A
domorphism operad End , where −⊕− stands for the direct sum in E. Since
M⊕A
Hom (X ⊕Y,Z⊕W)∼= HomE(X,Z) HomE(Y,Z)
E (cid:18)Hom (X,W) Hom (Y,W)(cid:19)
E E
with the usual matrix rule for composition, it makes sense to define End (n) as
M|A
that subobject of End (n) that takes the summand A⊗n to A, the summands
M⊕A
of the form A⊗k−1 ⊗M ⊗A⊗n−k to M, and all other summands to a null (i.e.
initial and terminal) object of E. It is then readily verified that this subcollection
(End (n)) of (End )(n)) defines a suboperad End of End ,
M|A n≥0 M⊕A n≥0 M|A M⊕A
and that an operad map P → End determines, and is determined by, a P-
M|A
algebra structure on A together with an A-module structure on M.
It follows from the preceding considerations that for each A-module M in an
additive categoryE, the directsum M⊕Acarriesa canonicalP-algebrastructure,
induced by the composite operad map P → End → End ; the resulting
M|A M⊕A
P-algebra is often denoted by M ⋊ A, cf. [8]. Projection on the second factor
defines a map of P-algebras M ⋊A→A, hence an object of the category Alg /A
P
of P-algebras over A. This assignment extends to a functor ρ : Mod → Alg /A.
A P
The following lemma is due to Quillen [14]; it is the starting point of the definition
of the cotangent complex of the P-algebra A.
Lemma1.3. LetAbeanalgebraoveranoperad P inanadditive,closedsymmetric
monoidal category E. Under the above construction, the category of A-modules is
isomorphic to the category of abelian group objects of Alg /A.
P
4 CLEMENSBERGERANDIEKEMOERDIJK
Proof. SinceE isadditive,thecategoryofA-modulesisadditiveandanyA-module
carries a canonical abelian group structure in Mod . By inspection, the functor
A
ρ:Mod →Alg /A preserves finite products, thus abelian group objects, so that
A P
for any A-module M, the image ρ(M) carries a canonical abelian group structure
inAlg /A. Thezeroelementofthisabeliangroupstructureisgivenbythesection
P
A→M⊕A;inparticular,thefunctorρisfullandfaithful,providedρisconsidered
as taking values in category of abelian group objects of Alg /A.
P
It remains to be shownthat any abelian groupobject of Alg /A arises as ρ(M)
P
for a uniquely determined A-module M. Indeed, an abelian group object N → A
hasasectionA→N bythezeroelementsothatN splitscanonicallyasN =M⊕A.
TheP-algebrastructureonN =M⊕ArestrictstothegivenP-algebrastructureon
A. Theabeliangroupstructure(α,id ):(M⊕M)⊕A=N× N −→N =M⊕A
A A
commutes with the P-algebra structure of N; thus, the square
id ⊗α⊗α
P(2)⊗(M ⊕M)⊗(M ⊕M) P(2) - P(2)⊗M ⊗M
µ ⊕µ µ
2 2 2
? ?
M ⊕M - M
α
is commutative which implies that µ is zero. This shows that the operad action
2
P →End factors through End and we are done. (cid:3)
M⊕A M|A
Lemma1.4. Let P bean operad. The category ofP(0)-modules underP is canon-
ically isomorphic to the module category of the monoid P(1).
Proof. ForaP(0)-moduleM underP,theactionmapµ :P(1)⊗M →M defines
1,1
an action on M by the monoid P(1). Conversely,an action on M by P(1) extends
uniquely to action maps µ :P(n)⊗P(0)⊗k−1⊗M ⊗P(0)⊗n−k →M where we
n,k
use the symmetry of the monoidal structure as well as the operad structure maps
P(n)⊗P(0)⊗k−1⊗I⊗P(0)⊗n−k →P(n)⊗P(0)⊗k−1⊗P(1)⊗P(0)⊗n−k →P(1).
(cid:3)
Definition 1.5. Let P be an operad and A be a P-algebra.
The envelopingoperadP oftheP-algebraAisdefinedbytheuniversalproperty
A
that operad maps P → Q correspond precisely to pairs (φ,ψ) consisting of an
A
operad map φ : P → Q and a P-algebra map ψ : A → φ∗Q(0), and that this
correspondence is natural in Q.
Alternatively,wecanconsiderthecategoryPairs(E)ofpairs(P,A)consistingof
anoperadP andaP-algebraA,withmorphismsthepairs(φ,ψ):(P,A)→(Q,B)
consistingofanoperadmapφ:P →QandaP-algebramapψ :A→φ∗(B). There
isacanonicalembeddingofthe categoryOper(E)ofoperadsinE intothe category
Pairs(E) given by P 7→(P,P(0)). The universalproperty of the envelopingoperad
then expresses (provided it exists for all P and A) that Oper(E) is a reflective
subcategory of Pairs(E) and that the left adjoint of the embedding is precisely the
enveloping operad construction Pairs(E) →Oper(E) :(P,A) 7→P . In particular,
A
if this left adjoint exists, it preserves all colimits.
Proposition 1.6. The enveloping operad P exists for any P-algebra A.
A
ON THE DERIVED CATEGORY OF AN ALGEBRA OVER AN OPERAD 5
Proof. For a free P-algebraA=F (X), where X is anobject ofE, the enveloping
P
operad of F (X) is given by
P
P (n)= P(n+k)⊗ X⊗k,
FP(X) Σk
ka≥0
seefor instanceGetzler andJones[6]. AgeneralP-algebraA is partofa canonical
coequalizer
F F (A)⇉F (A)→A,
P P P
whence the corresponding coequalizer of operads
(1) P ⇉P →P
FPFP(A) FP(A) A
has the required universal property of the enveloping operad of A. (cid:3)
The identity P →P correspondsby the universalpropertyto anoperadmap
A A
η : P →P together with a map of P-algebras η¯ : A →η∗P (0). We will now
A A A A A
show that the latter map is an isomorphism.
Lemma 1.7. For any P-algebra A, the category of P -algebras is canonically iso-
A
morphic to the category of P-algebras under A, and P (0) is isomorphic to A.
A
Proof. The pair(η ,η¯ )induces a functorfromthe categoryofP -algebrasto the
A A A
category of P-algebras under A which is compatible with the forgetful functors.
This functor is an isomorphism of categories since a P -algebra structure on B is
A
givenequivalentlybyanoperadmapP →End orbyanoperadmapP →End
A B B
(i.e. a P-algebra structure on B) together with a map of P-algebrasA→B. (cid:3)
Lemma 1.8. Let P be an operad and α : A → B be a map of P-algebras. Write
B for the P -algebra defined by α. The enveloping operad of the P-algebra B is
α A
isomorphic to the enveloping operad of the P -algebra B .
A α
Proof. An operad map (P ) → Q gives rise to a pair (φ,ψ) consisting of an
A Bα
operad map φ : P → Q and a P -algebra map ψ : B → φ∗Q(0). According
A A α
to Lemma 1.7, the latter yields a P-algebra map ψ′ : B → η∗φ∗Q(0) (under
A
A) for the canonical operad map η : P → P . Conversely, the pair (φη ,ψ′)
A A A
uniquely determines the operad map (P ) → Q we started from. Therefore,
A Bα
the enveloping operad (P ) has the same universal property as the enveloping
A Bα
operad P so that both operads are isomorphic. (cid:3)
B
Thefollowingpropositionispreparatoryfortherelationshipbetweenenveloping
operadandenvelopingalgebra. TheresultisimplicitlyusedbyGoerssandHopkins,
compare [8, Lemma 1.13].
Proposition 1.9. Let T be a monad on a closed symmetric monoidal category E.
The category of T-algebras is a module category for a monoid M in E if and
T
only if the tensor-cotensor adjunction of E lifts to the category of T-algebras along
the forgetful functor U : Alg → E. If the latter is the case, the monad T is
T T
isomorphic to T(I)⊗(−), i.e. the monoid M is given by T(I).
T
Proof. Assume first that the monad T is given by tensoring with a monoid M .
T
Then the tensor-cotensor adjunction of E lifts as follows. For any M -modules
T
M,N and object X of E, the tensor M ⊗X inherits an M -module structure by
T
ǫ ⊗X
M ⊗M ⊗X M - M ⊗X;
T
6 CLEMENSBERGERANDIEKEMOERDIJK
the cotensor Hom (X,N) inherits an M -module structure by the adjoint of
E T
M ⊗ev ǫ
M ⊗Hom (X,N)⊗X T -X M ⊗N N- N.
T E T
It follows that the adjunction E(M ⊗X,N) ∼= E(M,Hom (X,N)) lifts to an ad-
E
junction Alg (M ⊗X,N)∼=Alg (M,Hom (X,N)).
T T E
Assumeconverselythatsuchaliftedtensor-cotensoradjunctionexistsforagiven
monad T on E. By adjointness we get for any objects X,Y binatural isomorphims
of T-algebras
FT(X)⊗Y ∼=FT(X ⊗Y).
Since by assumption U preserves tensors this implies (setting X = I) that the
T
monadT =U F isisomorphictoT(I)⊗(−);inparticular,T(I)carriesacanonical
T T
monoid structure. (cid:3)
Theorem 1.10. For any algebra A over an operad P in a closed symmetric
monoidal category E, the category of A-modules under P is canonically isomor-
phic to the module category of the monoid P (1).
A
Proof. First of all, it follows immediately from Definition 1.1 that the forgetful
functor U : Mod → E creates colimits (hence permits an application of Beck’s
A A
tripleability theorem) and allows a lifting to Mod of the tensors and cotensors
A
by objects of E. In order to apply Proposition 1.9, and to compare the resulting
monoidtoP (1),wegiveanexplicitdescriptionoftheleftadjointF :E →Mod
A A A
of U , again following Goerss and Hopkins, compare [8, Proposition 1.14]. For
A
objects M and A of E, denote by Ψ(A,M) the positive collection in E given by
n
Ψ(A,M)(n)= A⊗k−1⊗M ⊗A⊗n−k, n≥1,
ka=1
the symmetric group Σ acting by permutation of the factors. Moreover, define
n
the object P(A,M) = P(n)⊗ Ψ(A,M). The axioms of an A-module M
n≥1 Σn
amountthentotheexist`enceofanactionmapµM :P(A,M)→M whichisunitary
and associative in a natural sense. For instance, the associativity constraint uses a
canonical isomorphism (P ◦P)(A,M) ∼= P(F (A),P(A,M)) where F (A) is the
P P
free P-algebra on A. It follows that for a free P-algebra A = F (X) the free A-
P
module on M is given by P(X,M), the A-module structure being induced by the
isomorphim just cited. A general P-algebra A is part of a reflexive coequalizer
F F (A)⇉F (A)→A
P P P
whichispreservedundertheforgetfulfunctorAlg →E. Therefore,theunderlying
P
object of the free A-module F (M) on M is part of a reflexive coequalizer in E
A
(2) P(F (A),M)⇉P(A,M)→U F (M).
P A A
Proposition1.9impliesthatthecategoryMod isamodulecategoryforthemonoid
A
MA ∼= UAFA(I). Putting M = I in (2) we end up with the following reflexive
coequalizer diagram in E
(3) P(F (A),I)⇉P(A,I)→M .
P A
Forthesecondstepoftheproofobservefirstthatthecoequalizer(1)inOper(E)is
preservedunderthe forgetfulfunctor fromoperadstocollections,sinceoperadsare
ON THE DERIVED CATEGORY OF AN ALGEBRA OVER AN OPERAD 7
monoids in collections with respect to the circle product, and since the coequalizer
is reflexive. Therefore we get the following reflexive coequalizer diagram in E
(4) P (1)⇉P (1)→P (1).
FPFP(A) FP(A) A
It follows from the definitions that (3) and (4) are isomorphic diagrams in E. It
remainstobeshownthatthemonoidstructuresofM andofP (1)coincideunder
A A
this isomorphism. Lemmas 1.4 and 1.7 imply that the category of P (1)-modules
A
is isomorphic to the category of A-modules under P ; the canonical operad map
A
η :P →P induces thus a functor (over E) from the category of P (1)-modules
A A A
to the category of A-modules under P, and therefore (by Proposition 1.9) a map
of monoids from P (1) to M ; this map of monoids may be identified with the
A A
isomorphism between the coequalizers of (4) and (3). (cid:3)
Definition 1.11. For any algebra A over an operad P in a closed symmetric
monoidalcategoryE,the envelopingalgebraEnv (A)isthemonoidP (1)ofunary
P A
operations of the enveloping operad of A.
Theenvelopingalgebraconstructionisafunctorthattakesmapsofpairs(φ,ψ):
(P,A) → (Q,B) to maps of monoids Env (A) → Env (B) in E. Theorem 1.10
P Q
shows that the category of A-modules under P is canonically isomorphic to the
module category of the enveloping algebra Env (A).
P
The purposeofthe remainingpartofthis sectionis togiveasufficientcondition
for the enveloping algebra Env (A) to be a bialgebra, i.e. to have a compatible
P
comonoid structure; this amounts to the existence of a monoidal structure on the
category of A-modules under P.
Recall that a Hopf operad P in E is by definition an operad in the symmet-
ric monoidal category Comon(E) of comonoids in E; for such a Hopf operad, a
P-bialgebra is defined to be a P-algebra in Comon(E). Alternatively, a “Hopf
structure” on an operad P amounts to a monoidal structure on the category of P-
algebrassuch that the forgetful functor is stronglymonoidal, cf. [12]; P-bialgebras
are then precisely comonoids in this monoidal category of P-algebras.
For any two operads P and Q, the tensor product P ⊗Q denotes the operad
defined by (P ⊗Q)(n)=P(n)⊗Q(n).
Proposition 1.12. For any Hopf operad P and P-bialgebra A, the enveloping
operad P is again a Hopf operad. In particular, the enveloping algebra of A is a
A
bialgebra in E.
Proof. AnyHopfoperadP hasadiagonalP →P⊗P. Therefore,foranyP-algebras
A and B, there is a canonical operad map P →P ⊗P ηA−⊗→ηB P ⊗P , and hence
A B
by the universal property of P , a canonical operad map P →P ⊗P .
A⊗B A⊗B A B
IfAisaP-bialgebra,thereisadiagonalA→A⊗AinthecategoryofP-algebras
and hence a diagonal P → P → P ⊗P in the category of operads. This
A A⊗A A A
shows that P is a Hopf operad, and that P (1) is a bialgebra in E. (cid:3)
A A
2. The derived category of an algebra over an operad
In this section, we study the derived category of an algebra over an operad.
For any P-algebra A, the derived category D (A) is defined to be the homotopy
P
category of the category of cA-modules, where cA is a cofibrant resolution of A
in the category of P-algebras. Thanks to Theorem 1.10, invariance properties
8 CLEMENSBERGERANDIEKEMOERDIJK
of D (A) correspond to invariance properties of the enveloping algebra Env (A).
P P
Sincetheenvelopingalgebramaybeidentifiedwiththemonoidofunaryoperations
oftheenvelopingoperadP ,themethodsof[2]apply(seetheAppendixforasmall
A
correctionto[2]),andwegetquite preciseinformationontheinvarianceproperties
of the derived category D (A).
P
Fromnowon,E denotesamonoidal modelcategory. Recall(cf. Hovey[9])thata
monoidal model category is simultaneously a closed symmetric monoidal category
andaQuillenmodelcategorysuchthattwocompatibilityaxiomshold: thepushout-
product axiom and the unit axiom. The unit axiom requires the existence of a
cofibrantresolution of the unit cI −∼→I such that tensoring with cofibrant objects
X induces weak equivalences X ⊗cI −∼→ X. The latter is of course automatic if
the unit of E is already cofibrant. We assume throughout that E is cocomplete and
cofibrantly generated as a model category.
For any unitary associative ring R, the category of simplicial R-modules is an
additive monoidal model category with weak equivalences (resp. fibrations) those
maps of simplicial R-modules whose underlying map is a weak equivalence (resp.
fibration)ofsimplicialsets. Similarly,thecategoryofdifferential graded R-modules
is an additive monoidal model category with weak equivalences (resp. fibrations)
the quasi-isomorphisms (resp. epimorphisms) of differential graded R-modules.
These two examples generalise to any Grothendieck abelian category A equipped
with a set of generators.
Recall from [3] that a map of operads is called a Σ-cofibration if the underlying
map is a cofibration of collections and an operad P is called Σ-cofibrant if the
unique map from the initial operad to P is a Σ-cofibration. In particular, for a
Σ-cofibrant operad P, the unit I → P(1) is a cofibration in E. This terminology
differs slightly from [2] where an operad with the latter property has been called
well-pointed.
An operad P is called admissible if the model structure on E transfers to the
category Alg of P-algebras along the free-forgetful adjunction
P
F :E ⇆Alg :U ,
P P P
i.e. if Alg carries a model structure whose weak equivalences (resp. fibrations)
P
are those maps f : A → B of P-algebras for which the underlying map U (f) :
P
U (A)→U (B)isweakequivalence(resp. fibration)inE. See[2,Proposition4.1]
P P
for conditions on P which imply admissibility.
Lemma 2.1. For any admissible operad P and P-algebra A, the enveloping operad
P is again admissible.
A
Proof. This follows immediately from Lemma 1.7. (cid:3)
A (trivial) cellular extension of P-algebrasis any sequential colimit of pushouts
A→A[u] of the form
ǫ
F U (A) A- A
P P
F (u)
P
? ?
F (X) - A[u]
P
ON THE DERIVED CATEGORY OF AN ALGEBRA OVER AN OPERAD 9
where ǫ denotes the counit of the free-forgetful adjunction and u:U (A)→X is
A P
agenerating(trivial)cofibrationinE. Acellular P-algebra Aisacellularextension
of the initial P-algebra P(0).
Lemma 2.2. Let P be a Σ-cofibrant operad and A be a P-algebra. If the unique
map of P-algebras P(0) → A is a (trivial) cellular extension of P-algebras, then
the induced map P →P is a (trivial) Σ-cofibration of operads.
A
Proof. The case of a cellular extension is [2, Proposition 5.4]. Exactly the same
proof applies to a trivial cellular extension as well. (cid:3)
A monoid M in E will be called well-pointed if the unit map I → M is a
cofibration in E. In particular, if I is cofibrant then a well-pointed monoid M has
a cofibrant underlying object.
Proposition 2.3. Let P be an admissible Σ-cofibrant operad and A be a cofibrant
P-algebra. Then the enveloping operad P is an admissible Σ-cofibrant operad. In
A
particular, the enveloping algebra Env (A) is well-pointed.
P
Proof. Admissibility was dealt with in Lemma 2.1. Any cofibrant P-algebra is
retract of a cellular P-algebra, whence P is retract of Σ-cofibrant operad and
A
therefore Σ-cofibrant. The second statement follows from the identification of the
enveloping algebra Env (A) with P (1). (cid:3)
P A
Corollary 2.4. Let P be an admissible Σ-cofibrant operad and α : A → B be a
weak equivalence of cofibrant P-algebras. The induced map P : P → P is a
α A B
weak equivalence of admissible Σ-cofibrant operads. In particular, the induced map
of enveloping algebras Env (A) → Env (B) is a weak equivalence of well-pointed
P P
monoids.
Proof. By K. Brown’s Lemma, it suffices to consider the case of a trivial cellular
extension α. The statement then follows from Lemmas 1.8, 2.1 and 2.2, since P
B
may be identified with (P ) , and P is an admissible Σ-cofibrant operad. (cid:3)
A Bα A
Remark 2.5. The homotopical properties of the enveloping operad construction
(P,A)7→P ,asexpressedbyLemma2.2andCorollary2.4,arethemaintechnical
A
ingredients in establishing the homotopy invariance of the derived category of an
algebraover an operad. BenoˆıtFresse pointed out to us that in recentwork [5], he
independentlyobtainedsimilarhomotopicalpropertiesoftheassignment(R,A)7→
R◦ A where R is a Σ-cofibrant right P-module and A is a cofibrant P-algebra.
P
Since the enveloping operad P may be identified with R◦ A for a certain right
A P
P-module R, see [5, Section 10], Lemma 2.2 and Corollary 2.4 may be recovered
from[5,Lemma13.1.B]and[5,Theorem13.A.2]. Themoregeneralcontextofright
P-moduleshowevermakesthe proofsofthesestatements moreinvolvedthanthose
of our results which are immediate consequences of [2, Section 5].
Theorem2.6. LetP bean admissible Σ-cofibrantoperad inacofibrantlygenerated
monoidal model category. For any cofibrant P-algebra A, the category Mod (A) of
P
A-modules carries a transferred model structure.
Any map of cofibrant P-algebras f : A → B induces a Quillen adjunction f :
!
Mod (A) → Mod (B) : f∗. If f is a weak equivalence, then (f,f∗) is a Quillen
P P !
equivalence.
10 CLEMENSBERGERANDIEKEMOERDIJK
Proof. By Proposition 2.3, the enveloping algebra of a cofibrant P-algebra is well-
pointed. Theorem1.10,Corollary2.4andProposition2.7thenyieldtheconclusion.
(cid:3)
Proposition 2.7. Let E be a cofibrantly generated monoidal model category.
(a) For any well-pointed monoid M, the category Mod of M-modules carries
M
a transferred model structure; if in addition M has a compatible cocommu-
tative comonoid structure, then Mod is a monoidal model category;
M
(b) Eachmapofwell-pointedmonoidsf :M →N inducesaQuillenadjunction
f : Mod ⇆ Mod : f∗; if f is a weak equivalence, then (f,f∗) is a
! M N !
Quillen equivalence.
Proof. Thefirstpartof(a)followsbyatransferargument(cf. [2,Section2.5])from
thefactthattensoringwithM preservescolimits,aswellascofibrationsandtrivial
cofibrations; the preservation of cofibrations and trivial cofibrations follows from
the pushout-product axiom and the well-pointedness of M. If in addition M has
a compatible comonoid structure, then Mod carries a closed monoidal structure
M
which is strictly preservedby the forgetful functor Mod →E. Since the forgetful
M
functor preserves and reflects limits as well as fibrations and weak equivalences,
this implies that Mod satisfies the axioms of a monoidal model category.
M
The first statement of (b) follows,since f∗ preservesfibrations and trivialfibra-
tions by definition of the transferred model structures on Mod resp. Mod . For
M N
thesecondstatementof(b),weusethat(f,f∗)isaQuillenequivalenceifandonly
!
if the unit η : X → f∗f(X) is a weak equivalence at each cofibrant M-module
X !
X. Assume that f is a weak equivalence; since any cofibrant M-module is retract
of a “cellular extension” of the initial M-module, Reedy’s patching and telescope
lemmas (cf. [3, Section 2.3]) imply that it is sufficient to consider M-modules of
the form M ⊗C, where C is a cofibrant object of E. In this case, the unit η
M⊗C
may be identified with f ⊗id :M ⊗C →N ⊗C; the latter is a weak equivalence
C
by an application of the pushout-product axiom and of K. Brown’s Lemma to the
functor (−)⊗C :I/E →C/E. (cid:3)
Remark 2.8. It follows from Proposition 2.7 (b) that for each weak equivalence
f : M → N of well-pointed monoids, the unit η : X → f∗f(X) is a weak
X !
equivalence at cofibrant M-modules X. For later use, we observe that this holds
alsoifX isonlycofibrantasanobjectofE providedN iscofibrantasanM-module.
Remark 2.9. If E satisfies the monoid axiom of Schwede and Shipley [17], the
categoryof M-modules carriesa transferredmodel structure for any monoid M in
E. The monoid axiom holds in many interesting situations, in particular if either
all objects of E are cofibrant, or all objects of E are fibrant. However, even if
the monoid axiom holds, Proposition 2.7 (b) does not carry over to a base-change
alongarbitrarymonoids;indeed, the unitofthe base-changeadjunctionbehavesin
general badly at cofibrant M-modules if M is not supposed to be well-pointed.
In general it is more restrictive for f :A→B to be a weak equivalence than to
induce a Quillen equivalence. A complete characterisationof those f which induce
a Quillen equivalence on module categories would require a homotopical Morita
theory. We give here, in a particular case, a precise criterion for when a Quillen
equivalence between module categories comes from a weak equivalence.