THE BAR CONSTRUCTION OF AN E-INFINITE ALGEBRA
BENOIT FRESSE
Abstract.We consider the classical reduced bar construction of associati*
*ve
algebras B(A). If the product of A is commutative, then B(A) can be equi*
*pped
with the classical shuffle product, so that B(A) is still a commutative *
*algebra.
This assertion can be generalized for algebras which are commutative up *
*to
homotopy. Namely, one observes that the bar construction of an E1 -algeb*
*ra
B(A) can be endowed with the structure of an E1 -algebra.
The purpose of this article is to give an existence and uniqueness th*
*eo-
rem for this claim. We would like to insist on the uniqueness property: *
*our
statement makes the construction of E1 -structures easier and more flexi*
*ble.
Therefore, the proof of our existence theorem differs from other constru*
*ctions
of the literature. In addition, the uniqueness property allows to give e*
*asily a
homotopy interpretation of the bar construction.
Contents
Introduction 1
Article outline 3
0. Conventions 4
1. The existence theorem 7
2. The uniqueness theorem 22
3. The homotopy interpretation of the bar construction 27
4. The relationship with the cochain algebra of a loop space 40
Appendix A. Operads, bar duality and transfer 45
References 58
Introduction
We consider the classical reduced normalized bar complex of augmented asso-
ciative algebras over a fixed ground ring F. More explicitly, for an augmented
associative algebra A, we consider the complex B(A) such that
Bn(A) = ( A~) n ,
where A~denotes the suspension of the augmentation ideal of A, together with t*
*he
classical bar differential @ : B*(A) ! B*-1(A) given by the formula
n-1X
@(a1 . . .an) = a1 . . .ai-1ai . . .an.
i=1
____________
Date: 4 January, 2006. Revised: 9 January, 2006.
2000 Mathematics Subject Classification. Primary: 57T30; Secondary: 55P48, 1*
*8G55, 55P35.
1
2 BENOIT FRESSE
Recall that we have a unital, associative and commutative product
^: B*(A) B*(A) ! B*(A)
defined by the shuffle of tensors. If the product of A is commutative, then the*
* bar
differential is a derivation with respect to the shuffle product so that B(A) i*
*s still
an associative and commutative differential graded algebra.
Unfortunately, in algebraic topology, algebras are usually commutative only u*
*p to
homotopy: a motivating example is provided by the cochain algebra of a topologi*
*cal
space C*(X). In this context, the shuffle product is no longer compatible with *
*the
differential. Thus, the problem is to use commutativity homotopies in order to
add perturbations to the shuffle product so that B(A) can still be equipped with
the structure of a differential graded algebra. In order to state precise resul*
*ts, we
introduce E1 -algebra structures (strongly homotopy associative and commutative
algebras). Recall briefly that an E1 -algebra consists of an algebra over an op*
*erad
E equivalent to the operad of associative and commutative algebras C. Several
authors have observed that B(A) can be equipped with the structure of an E1 -
algebra if A is an E1 -algebra (see [38, 36]). The purpose of this article is t*
*o give
a more precise existence and uniqueness theorem. Explicitly:
Theorem A. Fix a cofibrant E1 -operad E.
a. The bar construction of an E-algebra B(A) can be endowed with the structu*
*re
of an E-algebra, functorially in A, and so that, in the case of a commuta-
tive algebra A, this E-algebra structure reduces to the classical commuta*
*tive
algebra structure of B(A), the one defined by the shuffle product of tens*
*ors.
b. Such structures are homotopically unique. To be more precise, let
ae0A, ae1A: E ! End B(A)
denote operad morphisms which provide the chain complex B(A) with the
structure of an E-algebra as above. The algebras (B(A), ae0A) and (B(A), *
*ae1A)
can be connected by weak-equivalences of E-algebras
(B(A), ae1A) -~ . -~! (B(A), ae0A)
functorially in A.
We would like to insist on the uniqueness property: this statement makes the
construction of E1 -structures easier and more flexible. Therefore, the proof o*
*f our
existence theorem differs from other constructions of the literature. In addit*
*ion,
the uniqueness property allows to give easily a homotopy interpretation of the *
*bar
construction. Namely:
Theorem B. Let FA _~_////_A denote a cofibrant resolution of a given E-algebra *
*A.
Suppose that the bar construction B(A) is equipped with the structure of an E-a*
*lgebra
as in theorem A. Then, we have a weak-equivalence of E-algebras
FA -~! B(A),
where FA denotes the suspension of FA in the closed model category of E-algebr*
*as.
Finally, this work is motivated by the relationship between the bar construc*
*tion
and the cochain complex of loop spaces (see [1] and the historical survey [31]).
Namely, one proves classically that the dg-module B(C*(X)) is chain-equivalent
to C*( X) in the situation where the cohomological Eilenberg-Moore spectral
THE BAR CONSTRUCTION OF AN E-INFINITE ALGEBRA 3
*(X)
sequence E2 = TorH* (F, F) ) H*( X) converges. Recall that the cochain
algebra of a space C*(X) can be equipped with the structure an E1 -algebra
(see [5, 21, 30]). One can prove that B(C*(X)) is equivalent to C*( X) as an
E1 -algebra. By induction, we obtain that, for the structure deduced from theo-
rem A, the iterated bar construction Bn(C*(X)) is equivalent to C*( nX). We
would like to point out that such results can easily be obtained by comparing f*
*irst
the cochain algebra C*( nX) with the iterated suspension of C*(X) in the catego*
*ry
E1 -algebras, because the suspension is a categorical construction. To be expli*
*cit,
we have the following theorem:
Theorem C. We let C*(X) denote the cochain algebra of a pointed space X with
coefficients in a field F of characteristic p > 0. We let FX _~_////_C*(X) den*
*ote a
cofibrant resolution of C*(X) in the category of E-algebras. We assume that X
is connected p-complete, nilpotent and of finite p-type (as in [27]). Then, for*
* any
n 0, the natural map
nFX ! C*( nX)
defines a weak-equivalence of E-algebras provided that ssn(X) is a finite p-gro*
*up.
As a corollary, for the iterated bar construction, we obtain H*(BnC*(X)) '
H*( nX) under the assumption of this theorem.
One can introduce the Bousfield-Kan tower {RsX} in order to extend this resu*
*lt.
We obtain the following statement:
Theorem D. We can let F = Fp. We assume that X is a pointed space whose
cohomology modules H*(X, Fp) are degreewise finite. We let RsX denote Bousfield-
Kan' tower of X (for R = Fp). We fix a cofibrant resolution FX of C*(X), as in
theorem C above. We have
^
H0( nFX ) = Fssn(R1pX)p,
the module of maps ff : ssn(R1 X) ! F pwhich are continuous in regard to the
p-profinite topology and
H*( nFX ) = H0( nFX ) colimsH*( n0RsX, Fp),
where n0X denotes the connected component of the base point of X.
As a corollary, for the iterated bar construction, we obtain
^
H0(BnC*(X)) ' Fssn(R1pX)p,
as long as the cohomology H*(X, Fp) is degreewise finite.
Compare these results with [4, 24, 25, 34, 36, 37, 38].
Article outline
Let us outline briefly the plan of this article.
The existence part of theorem A is proved in section 1, in which we introduce
a fundamental tool of the article, namely, the endomorphism operad of the bar
construction. We observe that a good approximation of this operad satisfies a n*
*ice
homotopy invariance property from which we deduce the existence theorem by the
left-lifting property of cofibrant operads. The uniqueness part of theorem A is
proved in section 2. For that purpose, we establish a one-to-one correspondence
between on one hand, weak-equivalences for algebras over an operad, and on the
4 BENOIT FRESSE
other hand, left-homotopies for operad morphisms. The homotopy interpretation
of the bar construction (theorem B) is established in section 3. Briefly, we de*
*fine
by transfer a specific operad action that satisfies the assumptions of the uniq*
*ueness
theorem and which makes the bar complex B(A) equivalent to the suspension FA
by construction. Our theorem follows. Section 4 is devoted to the proof of theo-
rems C and D. These results are obtained by techniques borrowed from [27] and
by classical tower arguments for which we refer to [8, 10]. In appendix A, we r*
*ecall
some fundamental definitions and results on operads. Then, we survey carefully *
*the
bar duality theory for algebras over an operad, from which we deduce the transf*
*er
argument used in section 3.
The sections 2, 3 and 4 are self-contained and independent from each other, o*
*nce
the results and the fundamental constructions of section 1 are established. Ea*
*ch
section contains its own detailed introduction. We refer to the appendix sectio*
*n A.1
for our conventions in operad theory.
0.Conventions
We fix a commutative ground ring F and we work within the category of differ-
ential graded F-modules (dg-modules for short), denoted by dgMod F . We assume
tacitely that any object is projective as an F-module when the ground ring F is*
* not
a field and if this assumption is necessary.
0.1. Differential gradedLmodules. To be precise, a dg-module denotes a lower Z-
graded F-module V = *2ZV* equipped with a differential ffiV : V* ! V*-1 that
decreases degrees by 1. The notation |v| = d indicates the degree of a homogene*
*ous
element v 2 Vd. In general, we do not specify the module V in the notation of t*
*he
differential, so that the differential of V is usually denoted by ffi = ffiV .
Symmetrically, we do not specify the differential in the notation of a dg-mo*
*dule.
Nevertheless, we can equip a dg-module V with a non-canonical differential, usu*
*ally
defined by a homogeneous map @ : V* ! V*-1 of degree -1 which is added to the
internal differential of V . In this case, the resulting dg-module, formed by *
*V*
equipped with the differential ffi + @ : V* ! V*-1, is denoted by the pair (V, *
*@). Let
us recall that the sum ffi + @ defines a differential if and only if we have th*
*e identity
ffi(@) + @2 = 0, where ffi(@) = ffi@ - @ffi represents the differential of the*
* map @ in
the internal hom-set of dg-modules (we recall this definition in paragraph 0.2).
The homology of a dg-module is denoted by H*(V ). Recall that a quasi-isomor-
phism f : U -~! V denotes a morphism of dg-modules which induces an isomor-
phism in homology f* : H*(U) -'!H*(V ). The category of dg-modules is equipped
with the structure of a cofibrantly generated closed model category in which a *
*mor-
phism f : U ! V is a weak-equivalence (-~!), respectively a fibration (___////_*
*), if f is
a quasi-isomorphism, respectively a surjective morphism. The cofibrations ( //_*
*//)
are characterized by the left-lifting property as usual. If the ground ring is *
*a field,
then all dg-modules are cofibrant, otherwise, we assume tacitely that a dg-modu*
*le
is cofibrant if this assumption is necessary. We follow the classical conventio*
*ns of
model categories (we refer to Quillen's original monograph [32] or to [22, 23])*
*. In
particular, a map which is both a weak-equivalence and a fibration, respectivel*
*y a
cofibration, is called an acyclic fibration, respectively an acyclic cofibratio*
*n. Given
an object X in a closed model category, a cofibrant resolution of X denotes a
cofibrant object Q endowed of an acyclic fibration Q _~////_X.
THE BAR CONSTRUCTION OF AN E-INFINITE ALGEBRA 5
0.2. Tensor product and maps of dg-modules. We equip the category dgMod with
the classical tensor product of dg-modules together with the symmetry isomorphi*
*sm
o : V W ! W V that follows the usual sign convention. Let us mention that
we do not make explicit the sign which arises from a permutation of homogeneous
tensors since this sign is determined by the rules of differential graded calcu*
*lus.
We let Hom__(V, W ) denote the internal hom of (dg Mod, ), characterized by the
adjunction relation
Hom dgMod(U V, W ) = Hom dgMod(U, Hom__(V, W )).
Recall that the module Hom__d(V, W ) consists of linear maps f : V ! W such
that f(V*) W*+d. We say also that f : V ! W is a homogeneous map of lower
degree |f| = d. The differential of f in Hom__d(V, W ) is given by the classica*
*l formula
ffi(f) = ffiW f - (-1)dfffiV . In particular, a morphism of dg-modules f : V !*
* W is
equivalent to a map f 2 Hom__0(V, W ) such that ffi(f) = 0.
For any reasonable category C together with functors F, G : C ! dg Mod we
let Hom__X2C(F (X), G(X)) denote the dg-module formed by collections of homoge-
neous maps `X : F (X) ! G(X) which define a natural transformation in X 2 C.
Equivalently, the dg-module Hom__X2C(F (X), G(X)) is defined by the end formula
Z X2C
Hom__X2C(F (X), G(X)) = Hom__(F (X), G(X)).
(In this formula, the notation for ends and coends is converse to the usual one,
nevertheless we shall adopt this notation in our articles, because it extends t*
*he
classical conventions for invariants and coinvariants.)
0.3. Suspensions. The suspension of a dg-module, denoted by V , is defined by *
*the
tensor product V = Fe1 V , where deg(e1) = 1. Hence, we have ( V )* ' V*-1
and ffi V (e1 v) = -e1 ffiV (v). By an abuse of notation, we omit the tensor *
*e1 in
our notation, so that we identify an element of degree d in V with an element *
*of
degree d - 1 in V .
The suspension of a dg-operad P denotes an operad P such that the suspen-
sion functor A 7! A defines an isomorphism from the category of P-algebras to
the category of P-algebras. This dg-operad can be characterized by the relation
between free objects P( V ) = P(V ) (see [15, x1.3]).
0.4. Operads. We consider symmetric operads in the category of dg-modules. We
assume in addition that an operad P satisfies the connectedness condition P(0) *
*= 0
and P(1) = F1. The symmetric group on r letters is denoted by r.
We give a recall of our conventions on operads in appendix A.1 and we refer
to the literature and more particularly to our article [13], from which we take*
* our
conventions, for more background. To be precise, we adopt the notation of [13],
except that the free algebra over an operad P is denoted by P(V ) instead of S(*
*P, V ).
Anyway, let us recall briefly that an operad P is *-projective, respectively
*-cofibrant, if the underlying collection P(r), r 2 N, defines a projective ob*
*ject,
respectively a cofibrant object, in the category of dg- *-modules, the category
formed by sequences of dg- r-modules M(r), r 2 N. If an operad P is cofibrant
(in the category of operads), then P is necessarily *-cofibrant and *-project*
*ive,
but the converse implication does not hold (see paragraphs A.1.3-A.1.5).
6 BENOIT FRESSE
0.5. The associative operad and A1 -operads. The associative operad, associated
to the category of associative F-algebras, is denoted by the letter A . Recall *
*that
A (r) = F[ r], the regular representation of the symmetric group r. To be more
precise, as mentioned above, we assume A(0) = 0, so that we consider the operad
of non-unital associative algebras. Let us recall that the category of non-uni*
*tal
algebra is equivalent to the category of augmented unital algebras, since a non-
unital algebra A is the augmentation ideal of the unital algebra A+ such that
A+ = F 1 A, and, conversely, any augmented algebra A satisfies A ' F 1 ~A,
where ~Adenotes the augmentation ideal of A.
An A1 -operad K denotes a *-cofibrant operad in the category of dg-modules
equipped with a fixed acyclic fibration K _~_////_A. An A1 -algebra is by defin*
*ition
an algebra over some fixed A1 -operad K. We do not assume necessarily that K is
a cofibrant operad, though this assumption is often necessary. In fact, if Q de*
*notes
a cofibrant A1 -operad, then, by the left-lifting-property, we have automatical*
*ly a
weak-equivalence Q -~! K such that the diagram
"K??
~"" |~|
" fflfflfflffl|
Q __~_////_A
commutes. Consequently, any algebra over K defines an algebra over Q by restric-
tion of structure. A classical instance of a cofibrant A1 -operad is provided b*
*y the
cell complex of Stasheff's associahedra. Another cofibrant A1 -operad is define*
*d by
the operadic cobar-bar construction Bc(B(A )). In fact, this operad can be iden*
*ti-
fied with the cell complex of Boardmann-Vogt' W -construction for the operad of
associative monoids, which, in turn, can be identified with a cubical subdivisi*
*on of
Stasheff's operad.
0.6. The commutative operad and E1 -operads. The commutative operad, associ-
ated to the category of associative and commutative F-algebras, is denoted by t*
*he
letter C. Recall that C(r) = F, the trivial representation of the symmetric gro*
*up
r. As for associative algebras, we assume C(0) = 0, so that we consider non-un*
*ital
associative and commutative algebras, which are equivalent to the augmentation
ideal of augmented unital associative and commutative algebras.
An E1 -operad E denotes a *-cofibrant operad in the category of dg-modules
equipped with a fixed acyclic fibration E _~////_C. An E1 -algebra is by defin*
*ition
an algebra over some fixed E1 -operad E. Any E1 -operad E is endowed with a
morphism K ! E, for some A1 -operad K. For instance, if we fix a cofibrant A1 -
operad for K, then such a morphism can be deduced from the left-lifting-property
in the diagram
K ` ` `//E.
~|| ~||
fflfflfflffl|fflfflfflffl|
A _____//C
Consequently, any E1 -algebra is equipped with the structure of an algebra over*
* K
and admits a bar construction (see below).
THE BAR CONSTRUCTION OF AN E-INFINITE ALGEBRA 7
0.7. The bar construction. Let us recall that the bar construction B(A) can be
extended to algebras over any A1 -operad K. The modules Bn(A) are unchanged,
except that we set now Bn(A) = ( A) n and B0(A) = 0 since we deal with non-
unital algebras, but the bar differential contains perturbative terms. We have *
*more
precisely:
Xn n-r+1X
@(a1 . . .an) = a1 . . .~r(ak, . .,.ak+r-1) . . .an,
r=2 k=1
for a fixed sequence of operations ~r 2 K(r) such that |~r| = r-2. (These eleme*
*nts
are determined by the image of the generators of Stasheff's chain operad Q under
an operad equivalence Q -~! K.)
1. The existence theorem
1.1. Introduction. In this section, we prove the existence part of theorem A. To
be precise, we would like to state a slightly more general result. Namely:
Theorem 1.A. Fix an E1 -operad E and a cofibrant E1 -operad Q . The bar con-
struction of an E-algebra B(A) can be endowed with the structure of a Q -algebr*
*a,
functorially in A, and so that, in the case of a commutative algebra A, this Q -
algebra structure reduces to the classical commutative algebra structure of B(A*
*),
the one defined by the shuffle product of tensors.
For that purpose, we introduce the endomorphism prop of the bar construction
End PBassociated to an operad P equipped with a morphism K ! P, where K
is some A1 -operad. This object is the structure defined by the collection of d*
*g-
modules EndPB(r, s), r, s 2 N, formed by the natural transformations
`A : B(A) r ! B(A) s,
where A ranges over the category of P-algebras.
In the first subsection, we prove that any natural transformation `A : A m !
A r is the composite of a tensor permutation
*
A m w--!A m ,
with a tensor product of P-algebra operations
A m = A m1 . . .A mr p1-...-pr----!A . . .A = A r,
where p1 2 P(m1), . .,.pr 2 P(mr), provided that the ground ring F is an infini*
*te
field or the operad P is *-projective.
In the second subsection, we consider the dg-modules Op PB(r, s) End PB(r,*
* s)
formed by the transformations above. The module Op PB(r, s) is in general small*
*er
and behaves better than EndPB(r, s), so that we can prove that the functor P 7!
Op PB(r, s) maps weak-equivalences of operads to quasi-isomorphisms.
The endomorphism operad of the bar construction, also denoted by End PB, is
defined by the sequence of dg-modules EndPB(r) = EndPB(r, 1), and represents the
universal operad operating functorially on the bar construction of P-algebras B*
*(A).
Hence, the classical commutative algebra structure given by the shuffle product
is equivalent to an operad morphism r : C ! End CB. In fact, the dg-modules
Op CB(r) = Op CB(r, 1) form a suboperad of End CBand we observe that r factors
8 BENOIT FRESSE
through OpCB. Consequently, the existence assertion of theorem 1.A can be deduc*
*ed
from the lifting problem
pOp7EB_____//EndEB7
p
9?pp |~| ||
p p fflfflfflffl|fflffl|
Q p_~_////_Cr_//OpCB____//EndCB
which has automatically a solution as long as Q is a cofibrant operad and has t*
*he
left-lifting property with respect to acyclic fibrations. In the announcement *
*of a
preliminary version of this work [12], a lifting is made explicit for certain (*
*non-
cofibrant) E1 -operads, namely, the Barratt-Eccles operad for Q and the surject*
*ion
operad for E.
In fact, a prop is a structure P which consists of a collection of dg-modules
P(r, s), r, s 2 N , which parametrize operations with r inputs and s outputs p :
A r ! A s. Accordingly, the notion of a prop generalizes the notion of an operad
since an operad P contains only operations with 1 output p : A r ! A. The en-
domorphism prop of the bar construction represents the universal prop operating
functorially on the bar construction of P-algebras B(A). Consequently, we have a
morphism B^ ! End CB, where B^ denotes a prop associated to connected com-
mutative bialgebras, since the deconcatenation coproduct and the shuffle product
provides the bar construction of commutative algebras with this structure. We
prove the following theorem in the third subsection:
Theorem 1.B. The classical bialgebra structure on the bar construction of com-
mutative algebras B(A) is associated to a composite morphism of props
B^ -r! OpCB,! End CB.
Furthermore, the morphism r which occurs in this construction defines a weak-
equivalence from the prop of connected commutative bialgebras B^ to the prop of
bar operations Op CB.
As a remark, let us mention that our arguments can be carried out for EndAB,
the endomorphism prop of the bar construction for associative algebras. In this
case, we obtain a weak-equivalence
K ^ ~-!Op AB,
where K^ denotes the prop of connected coalgebras. Accordingly, the bar constru*
*c-
tion of an associative algebra does not carry any natural (non-trivial) multipl*
*icative
structure.
To conclude this introduction, let us mention that props were precisely intr*
*o-
duced by Adams and Mac Lane in order to model the algebraico-homotopic struc-
ture of the bar construction. In particular, our theorems 1.A and 1.B should be
compared with theorem 25.1 in [26].
1.2. Natural operations for algebras over an operad.
1.2.1. On *-modules. In this section, we consider structures, called *-module*
*s,
formed by sequences of dg-modules M(r), r 2 N, equipped with an action of the
symmetric groups r, like the underlying sequence of a symmetric operad P(r),
r 2 N. To be precise, we shall use the relationship between *-modules M and
THE BAR CONSTRUCTION OF AN E-INFINITE ALGEBRA 9
associated functors V 7! M(V ) which generalizes the free algebra functor V 7!
P(V ) associated to an operad. We recall this relationship and refer to our art*
*icle [13,
x1.2] for more details.
Explicitly, the functor V 7! M(V ) is defined by the formula
1M
M(V ) = (M(r) V r) r, forV 2 dgMod .
r=0
As for free algebras, we let x(v1, . .,.vr) denotes the element of M(V ) repres*
*ented
by the tensor x v1 . . .vr 2 M(r) V r. Observe that an element x 2 M(r)
gives rise to a natural transformation
x* : V r ! M(V ), forV 2 dgMod :
we set simply x*(v1 . . .vr) = x(v1, . .,.vr), for all v1 . . .vr 2 V r. C*
*onse-
quently, we have a natural morphism of dg- r-modules
: M(r) ! Hom__V 2FMod(V r, M(V )).
This morphism is clearly split injective and one proves classically that it is *
*often
an isomorphism. More precisely, we have the following statement:
1.2.2. Fact (see x1.2 in [13]). The natural morphism
: M(r) ! Hom__V 2FMod(V r, M(V ))
is an isomorphism of dg-modules, provided that the ground ring F is an infinite*
* field
or M is a projective *-module.
Let us mention that a *-module M is projective if and only if M(r) is a cha*
*in
complex of projective r-modules, for all r 2 N.
1.2.3. Strictly polynomial transformations. In general, the morphism identifi*
*es
M(r) with a submodule of Hom__V 2dgMod(V r, M(V )) formed by natural transfor-
mations `V : V r ! M(V ) which are in some sense strictly polynomial.
We make this idea more precise for functors on F-modules since this setting *
*is
used in section 1.4. We refer to [14] and to the discussion of [13, x1.2] for *
*more
details. One considers the category (F Mod ) formed by free F-modules V = Fm
as objects together with the divided power algebras (Hom (Fm , Fn)) as morphism
sets. A strictly polynomial functor on F-modules denotes a functor on this cat-
egory F : (F Mod ) ! dg Mod . Observe that a natural morphism of categories
F Mod ! (F Mod ) is provided by the total divided power of morphisms, so that
any strictly polynomial functors defines a functor in the usual sense. By defin*
*ition,
a homogeneous transformation `V : F (V ) ! G(V ), where F, G are strictly polyn*
*o-
mials, belongs to Hom__V 2 (FMod)(F (V ), G(V )) if `V commutes with all morphi*
*sms
of (F Mod ).
The functors V 7! V r and V 7! M(V ) are strictly polynomial and the con-
struction of paragraph 1.2.1 gives an isomorphism of dg-modules
: M(r) -'!Hom__V 2 (FMod)(V r, M(V )).
Let us mention that we identify any non-graded module to a dg-module concen-
trated in degree 0. Equivalently, the dg-module Hom__V 2 (FMod)(V r, M(V )) is
equipped with the grading, respectively the differential, defined by the intern*
*al
grading of M, respectively the internal differential of M.
10 BENOIT FRESSE
Let us recall that the functor V 7! V rdefines a projective object in the ca*
*te-
gory of strictly polynomial functors. Consequently, any strictly polynomial tra*
*ns-
formation ` : F ! G such that `V : F (V ) ! G(V ) is a quasi-isomorphism of
dg-modules, for any V 2 FMod , induces a quasi-isomorphism
`* : Hom V 2 (FMod)(V r, F (V )) -~!Hom V 2 (FMod)(V r, G(V )).
1.2.4. The tensor product of *-modules. Let M and N be *-modules. Recall that
the tensor product M N denotes the *-module characterized by the relation
M(V ) N(V ) ' (M N)(V ), for V 2 dgMod
and defined by the formula
M
(M N)(r) = Ind rsx tM(s) N(t), for r 2 N.
s+t=r
Thus, the module (M N)(r) is spanned by tensors w . x y where x 2 M(s),
y 2 N(t) and w 2 r.
The map M, N 7! M N defines an associative, unital and symmetric bifunctor.
In the following paragraphs, we consider the tensor powers of an operad P s, wh*
*ich
are *-modules such that P s(V ) = P(V ) s, for V 2 dgMod . We have clearly
M
(P s)(m) = Ind mm1x...x msP(m1) . . .P(ms), for m 2 N,
m1+...+ms=m
and an element of (P s)(m) is represented by a sum of tensors w . p1 . . .ps
where p1 2 P(m1), . .,.ps 2 P(ms) and w 2 m .
1.2.5. Operations and natural transformations. Let P be an operad. Recall that
Hom__A2P Alg(A m , A s) denote the dg-module formed by the collections of homog*
*e-
neous maps `A : A m ! A s which define a natural transformation in A 2 P Alg.
Let us observe that a natural transformation `A is associated to any tensor
w.p1 . . .ps 2 P s(m). Explicitly, the map `A is the composite of the permutati*
*on
of tensors on the source *
A m w--!A m
with the s-fold P-algebra operation
A m = A m1 . . .A ms p1-...-ps----!A . . .A = A s.
Hence, we have a morphism of dg-modules : P s(m) ! Hom__A2P Alg(A m , A s)
and this map is clearly functorial in P. Explicitly, for any operad morphism O*
*E :
P ! Q , the restriction functor OE! : Q Alg ! P Alggives rise to a morphism of
dg-modules
Hom__A2P Alg(A m , A s) -OE*!Hom_A2Q Alg(A m , A s)
and we have a commutative diagram
P s(m) ____//_Hom_A2P Alg(A m , A.s)
| |
| |
fflffl| fflffl|
Q s(m) ____//_Hom_A2Q Alg(A m , A s)
THE BAR CONSTRUCTION OF AN E-INFINITE ALGEBRA 11
1.2.6. Lemma. The morphism
: P s(m) ! Hom__A2P Alg(A m , A s)
is an isomorphism of dg-modules provided that either P is a *-projective operad
or the ground ring F is an infinite field.
This statement is an immediate corollary of lemma 1.2.7 below and of the
fact 1.2.2. To be more precise, we can check readily that the natural transfor-
mation `A defined in paragraph 1.2.5 is the element of Hom__A2P Alg(A m , A s)
associated to w . p1 . . .ps 2 P s(m) by the isomorphisms of lemma 1.2.7 and
by the morphism of the fact 1.2.2.
1.2.7. Lemma. Let P be any operad. Let QF ree(P Alg), respectively Free(P Alg),
denote the full subcategory of P Alggenerated by quasi-free P-algebras, respect*
*ively
by free P-algebras. We have natural isomorphisms
Hom__A2P Alg(A m , A s)' Hom__A2QF ree(P Alg)(A m , A s)
' Hom__A2Free(P Alg)(A m , A s)
' Hom__V 2dgMod(V m, P(V ) s)
given by the obvious restriction process and by the postcomposition of natural *
*trans-
formations
`P(V ) s
P(V ) m ----! P(V )
with tensor powers of the universal morphism jV : V ! P(V ) of free algebras.
Proof.We prove that the first module is isomorphic to the last one, since the o*
*ther
cases can be deduced from our construction. Consider the morphism
: Hom__A2P Alg(A m , A s) ! Hom__V 2dgMod(V m, P(V ) s)
specified in the lemma. We define a map
: Hom__V 2dgMod(V m, P(V ) s) ! Hom__A2P Alg(A m , A s)
such that = Idand = Id.
Let ` denote a homogeneous natural transformation
`V : V m ! P(V ) s, where V 2 dgMod .
The associated map
(`)A : A m ! A s,
for A a P-algebra, is defined as follows. Let a1 . . .am 2 A m . We can as-
sume that a1, . .,.am are homogeneous elements of A of degree d1, . .,.dm respe*
*c-
tively. Let us consider the dg-module X generated by elements x1, . .,.xm of de*
*gree
d1, . .,.dm , and by elements y1, . .,.ym of degree d1 - 1, . .,.dm - 1 togethe*
*r with
the differential ffi : X ! X such that ffi(xi) = yi for i = 1, . .,.m. Let e : *
*X ! A
denote the morphism of dg-modules such that e(xi) = ai for i = 1, . .,.m, and
consider the induced P-algebra morphism "e: P(X) ! A. We set
(`)A (a1 . . .am ) := "e s. `X (x1 . . .xm ).
One checks readily that this definition gives a well-defined linear map (`)A on
A m , since the maps `V : V m ! P(V ) s are assumed to be natural in V 2
dg Mod. Furthermore, the maps (`)A define a natural transformation in A 2
P Alg. In fact, if f : A ! A0 is a morphism of P-algebras, then the composite
12 BENOIT FRESSE
fe": P(X) ! A0can be identified with the morphism of P-algebras "e0: P(X) ! A0
induced by the map e0: X ! A0such that e0(xi) = f(ai), for i = 1, . .,.m. Hence,
by definition of (`)A0, we have
(`)A0(f m (a1 . . .am ))= (`)A0(f(a1) . . .f(am ))
= (fe") s`X (x1 . . .xm )
= f se" s`X (x1 . . .xm )
= f s (`)A (a1 . . .am ).
We check that the composite
jVm m (`)P(V ) s
V m ---! P(V ) ------! P(V )
can be identified with `V , so that = Id. In fact, given a tensor v1 . . .*
*vm 2
V m, we let e : X ! V denote the map such that e(xi) = vi, for i = 1, . .,.m,
as in the definition of (`), and we consider the induced morphism of P-algebras
"e: P(X) ! P(V ). We have by definition
(`)P(V )(jVm (v1 . . .vm=)) (`)P(V )(jV (v1) . . .jV (vm ))
= "e s`X (x1 . . .xm ),
We have then
"e s`X (x1 . . .xm ) = `V (e(x1) . . .e(xm )),
because "e: P(X) ! P(V ) is induced by a map of dg-modules e : X ! V , and
because the transformation ` is natural with respect to such maps. Therefore, we
obtain
(`)P(V )(jVm (v1 . . .vm )) = `V (v1 . . .vm )
and our claim follows: = Id.
Conversely, suppose given a natural transformation !A : A m ! A s, for
A 2 P Alg. Fix a1 . . .am 2 A m and let e : X ! A be as in the definition of
(`)A , where ` = (!). We have a commutative diagram
(!)X
_____________________AEo_
GF_ jXm m!P(X) fflffl| s
X m _____//JP(X) ____//_P(X)
JJJ | |
JJJ |"e m |"e s
e m JJ%%Jfflffl|! fflffl|
______A_// s
A m A
from which we deduce the identity
!A (a1 . . .am=)!A (e m (x1 . . .xm ))
= "e s. (!)X (x1 . . .xm )
= ( (!))A (a1 . . .am ).
Hence, we obtain = Id.
1.3. The prop of bar operations and the proof of our existence result.
THE BAR CONSTRUCTION OF AN E-INFINITE ALGEBRA 13
1.3.1. Props. Let us recall the definition of a prop in the category of dg-modu*
*les
(a pact in the terminology of [26]). We refer to Mac Lane's original article [2*
*6], to
Adams's survey [2, Chapter 2] to Boardmann-Vogt' monograph [7, Chapter 2], and
to Vallette's recent thesis [41] for a solid introduction of this notion.
In general, a prop consists of a symmetric monoidal category (B , ), enrich*
*ed
over dg Mod, whose objects are the non-negative integers n 2 N and such that
m n = m + n for objects. Hence, the structure of a prop is determined by a
collection of morphism sets B(r, s), r, s 2 N, together with tensor product ope*
*rations
: B(r1, s1) B(r2, s2) ! B(r1 + r2, s1 + s2)
and unital and associative composition products
O : B(t, s) B(r, t) ! B(r, s).
Furthermore, the morphism sets B(r, s) are equipped with a right r-action since
the tensor product is symmetric by assumption and r = 1 r. Similarly, the mor-
phism set B(r, s) is equipped with a left s-action and the products above have*
* to
be equivariant.
There is a natural category of representations (also called models) associat*
*ed
to a prop B . Namely, a representation of B is defined by a monoidal functor
R : B ! dgMod . But, since R (r) = R (1 r) = R (1) r, a representation is
determined by a dg-module = R(1) together with evaluation morphisms
B(r, s) ! Hom__( r, s).
As for operads, we consider the endomorphism prop End of a dg-module defined
by the collection of dg-modules End (r, s) = Hom__( r, s), so that a repres*
*en-
tation of B is equivalent to a dg-module together with a morphism of props
B ! End .
For any operad P, the modules B(r, s) = P s(r) are equipped with the structu*
*re
of a prop such that the morphism of paragraph 1.2.5
: P s(r) ! Hom__A2P Alg(A r, A s)
makes any P-algebra equivalent to a representation of B. Equivalently, the mod-
ules B(r, s) = P s(r) define a sub-prop of Hom A2P Alg(A r, A s). In fact, any
representation of this prop B(r, s) = P s(r) is associated to a P-algebra.
As mentioned in the introduction, the notion of a prop is more general than *
*the
notion of an operad, since a prop can contain operations p 2 B(r, s) with more
than one input and more than one output which are indecomposable in regard
to the tensor product operation : B(r1, s1) B(r2, s2) ! B (r1 + r2, s1 + s2*
*).
In particular, the prop B associated to the category of commutative bialgebras
considered in the next section is not associated to an operad.
1.3.2. The endomorphism prop and the endomorphism operad of the bar construc-
tion. Fix an A1 -operad K. Let P denote an operad equipped with an operad mor-
phism K ! P so that we can extend the bar construction B(A) to the category
of P-algebras. As mentioned in the introduction of this section, the endomorphi*
*sm
prop of the functor B : P Alg ! dg Mod denotes the structure EndPB defined by
the dg-modules of natural transformations
End PB(r, s) = Hom__A2P Alg(B(A) r, B(A) s),
14 BENOIT FRESSE
and the endomorphism operad of B denotes the operad EndPBsuch that
EndPB(r) = EndPB(r, 1) = Hom__A2P Alg(B(A) r, B(A)).
Let us observe that the map P ! End PBdefines a functor from the category of
operads over K to the category of props, respectively to the category of operad*
*s.
1.3.3. The prop of bar operations. We use the constructions of section 1.2 in o*
*rder
to define a good subprop of EndPB. Explicitly, we consider the dg-modules
Y
OpPB(r, s) = P n1+...+ns(m1 + . .+.mr),
m*,n*
where m* = (m1, . .,.mr) and n* = (n1, . .,.ns) range over collections of integ*
*ers
mi 1 and nj 1. For m, n 2 N, we let also Op PB(r, s)mn denote the product of
components of OpPB(r, s) indexed by collections m*, n* such that m1+. .+.mr = m
and n1+. .+.ns = n. The construction of paragraph 1.2.1 (see also paragraph 0.3)
supplies a split injective morphism of dg-modules
: P n1+...+ns(m1 + . .+.mr) ,! Hom__A02 P Alg(A0 m1+...+mr, A0 n1+...+ns)
' Hom__A2P Alg(( A) m1+...+mr, ( A) n1+...+ns)
= Hom__A2P Alg(Bm1(A) . . .Bmr(A), Bn1(A) . . .Bns(A)),
which is an isomorphism under the assumptions of lemma 1.2.6. In this definitio*
*n,
the dg-module Hom__A2P Alg(B(A) r, B(A) s) is only equipped with an internal
differential ffi, induced by the differential of A, but the bar construction yi*
*elds
additional differentials
@hi: Bmi(A) ! Bmi-*(A), for i = 1, . .,.r,
and @vj: Bnj(A) ! Bnj-*(A), for j = 1, . .,.s,
so that Hom__A2P Alg(B(A) r, B(A) s) forms a multiple dg-module.
1.3.4. Claim. The dg-module OpPB(r, s) can be equipped with additional differen*
*tials
@hiand @vjthat correspond to the differentials above under the embedding
: P n1+...+ns(m1 + . .+.mr)
,! Hom__A2P Alg(Bm1(A) . . .Bmr(A), Bn1(A) . . .Bns(A)),
so that OpPB(r, s) forms a sub-multiple-dg-module of Hom__A2P Alg(B(A) r, B(A) *
*s).
Proof.For future reference, we make explicit maps
@hi: P n (m1 + . .+.mi+ . .+.mr)
! P n (m1 + . .+.mi- * + . .+.mr), for i = 1, . .,.r,
and @vj: P n1+...+nj+...+ns(m)
! P n1+...+nj-*+...+ns(m), for j = 1, . .,.r,
such that @hi= @hi and @vj= @vj , but we leave the straightforward verifica-
tion of these relations to the reader. Furthermore, we do not specify signs in *
*our
formulas.
Recall that the partial composite p Ok q 2 P(m + n - 1), where p 2 P(m) and
q 2 P(n) denotes the operation such that p Ok q = p(1, . .,.q, . .,.1), where q*
* is
THE BAR CONSTRUCTION OF AN E-INFINITE ALGEBRA 15
substituted to the kth entry of p. This construction can be extended to a tensor
power P s(m). Explicitly, for a tensor p1 . . .ps 2 P(m1) . . .P(ms), we set
(p1 . . .ps) Ok q = p1 . . .(pjOk0q) . . .ps
for k = m1+ . .+.mj-1+ k0, where k0= 1, . .,.mj, and we extend the maps Ok to
P s(m) by m -equivariance.
For w . p1 . . .pn 2 P n1+...+ns(m1 + . .+.mr), with the conventions of
paragraph 0.5, we set
X
@hi(w . p1 . . .pn) = (w . p1 . . .pn) Ok ~d,
d,k
where the sum ranges over the intervals d = 2, . .,.mi and k = m1 + . .+.mi-1+
1, . .,.m1 + . .+.mi-1+ mi- d + 1, and
X
@vj(w . p1 . . .pn) = w . p1 . . .~d(pk, . .,.pk+d-1) . . .pn,
d,k
where the sum ranges over d = 2, . .,.nj and k = n1+ . .+.nj-1+ 1, . .,.n1+ . .*
*+.
nj-1+ nj- d + 1.
Finally, our construction together with lemma 1.2.6 gives the following resul*
*t:
1.3.5. Lemma. The dg-modules OpPB(r, s) defined in paragraph 1.3.3 together with
the differentials supplied by claim 1.3.4 form a differential graded prop Op PB*
*. The
map P 7! OpPB(r, s) defines a functor on the category of operads P equipped wit*
*h a
morphism K ! P, where K denotes an A1 -operad. Moreover, the canonical maps
: OpPB(r, s) ,! End PB(r, s)
define a natural morphism of differential graded props, which is an isomorphism*
* if
the operad P is *-projective or if the ground field F is infinite.
The definition of Op PBis motivated by the following invariance property whi*
*ch
is not satisfied by the endomorphism operad EndPB.
1.3.6. Lemma. The functor P 7! OpPBmaps a weak-equivalence of operads under
a fixed A1 -operad K to a weak-equivalence of props.
Q
Proof.Recall that Op PB(r, s)mn denotes the product m*,n* P n1+...+ns(m1 +
. .+.mr) over indices m*, n* such that m1 + . .+.mr = m and n1 + . .+.ns = n.
We equip OpPBwith the decreasing filtration
Y
FpOp PB(r, s) = OpPB(r, s)mn .
n-m -p
We obtain a spectral sequence Er(Op PB) such that
Y
E0d*(Op PB) = P n1+...+ns(m1 + . .+.mr),
m*,n*
where the product ranges over all indices (m1, . .,.mr), (n1, . .,.ns) such tha*
*t (n1+
. .+.ns) - (m1 + . .+.mr) = -d, and where d0 = ffi the internal differential of*
* P.
Hence, an operad equivalence OE : P -~! Q induces an isomorphism
E1(OE) : E1(Op PB) -'!E1(Op QB).
16 BENOIT FRESSE
Recall that P(0) = 0 by convention. Consequently, we have P n (m) = 0 for
n > m, so that F0Op PB= OpPBand Erd*= 0 for d < 0. Equivalently, our spectral
sequence Er(Op PB) is associated to the tower of fibrations of dg-modules
. .!.Op PB=FpOp PB! . .!.Op PB=F1Op PB! Op PB=F0Op PB= 0.
Let us mention that Op PB= limpOpPB=FpOp PBby definition of Op PB. In this con-
text, the arguments of [8, Chapter IX] imply that an operad equivalence OE : P *
*-~! Q
induces an isomorphism
OE* : H*(Op PB) -'!H*(Op QB),
since OE induces an isomorphism at the E1 level of the spectral sequence.
Proof of theorem 1.A. In this paragraph, we forget about prop structures, we co*
*n-
sider the endomorphism operad of the bar construction EndPB(r) = EndPB(r, 1) and
the suboperad such that Op PB(r) = OpPB(r, 1). As mentioned in the introduction,
the shuffle product, which provides the bar construction of a commutative algeb*
*ra
B(A) with the structure of a commutative algebra, functorially in A, is equival*
*ent
to an operad morphism r : C ! End CB. We observe in the next subsection that
this morphism factors through OpCB.
For an E1 -operad E, the natural morphism Op EB! End EBis an isomorphism,
since E is supposed to be *-cofibrant, and hence *-projective. As a consequen*
*ce,
for any operad Q equipped with a morphism Q ! C, an operad morphism Q !
End EBwhich provides the bar construction of an E-algebra B(A) with the structu*
*re
of a Q-algebra as in theorem 1.A is equivalent to a lifting in the operad diagr*
*am
p7OpEB__'__//EndEB7.
pp | |
p p | |
p p fflffl| fflffl|
Q _____//C____//OpCB____//EndCB
By lemma 1.3.6, the augmentation map E _~////_Cof an E1 -operad induces an
acyclic fibration Op EB_~////_OpCB. Therefore, the lifting exists as long as Q*
* is a
cofibrant operad and has the left-lifting property with respect to acyclic fibr*
*ations.
Let us notice that this lifting is unique up to a left-homotopy (see [32, xI.1]*
*).
1.4. The prop of bar operations for commutative algebras.
1.4.1. The prop of commutative bialgebras. In this section, we let B denote the*
* prop
of commutative bialgebras. By definition, the prop B is generated by a 2-invar*
*iant
operation ~ 2 B(2, 1) with 2 inputs and 1 output and by an operation 2 B(1, 2)
with 1 input and 2 outputs together with the classical relations of the product*
* and
the coproduct of a commutative bialgebra. Explicitly:
(1a) the associativity relation: ~ O (~ 1) = ~ O (1 ~),
(1b) the commutativity relation: ( 1) O = (1 ) O ,
(1c) and the distribution relation: O ~ = (~ ~) O o23O ( ).
Accordingly, a prop morphism OE : B ! End is uniquely determined by elements
~ = OE(~) and = OE( ) which satisfy the relations above, and hence, a repre*
*sen-
tation of B is equivalent to the structure of a commutative bialgebra.
THE BAR CONSTRUCTION OF AN E-INFINITE ALGEBRA 17
In fact, we have an isomorphism
M1
A r(m)_ m C s(m) -'!B (r, s),
m=0
where A _ denotes the cooperad of coassociative coalgebras dual to A , because,
according to the distribution relation, any bialgebra operation ` 2 B(r, s) adm*
*its a
unique factorization
` = (~1 . . .~s) O w* O ( 1 . . . r)
such that
r --1-...--r-! m
is an r-fold coproduct represented by a tensor product of operations 1 . . . *
*r 2
A(m1)_ . . .A(mr)_,
*
m w--! m
is a tensor permutation represented by a permutation w 2 m , and
m ~1-...-~s-----! s
is an s-fold product represented by a tensor product of operations ~1 . . .~s*
* 2
C(n1) . . .C(ns).
The prop B has a completion B^ such that
Y1
A r(m)_ m C s(m) -'!B ^(r, s)
m=0
and which operates on connected commutative bialgebras. To be precise, we equip
the prop B with the decreasing filtration
M
FpB (r, s) = A r(m)_ m C s(m).
m p
One observes that FpB is a prop ideal of B, so that B^ = limpB =FpB defines a
prop together with the properties above. In the following paragraphs, we consid*
*er
this completed version of B since the bar construction of a commutative algebra
forms a connected commutative bialgebra.
This section is devoted to the proof of theorem 1.B. Namely, we prove that the
classical bialgebra structure on the bar construction of commutative algebras B*
*(A)
is associated to a composite morphism of props
B^ -r! OpCB,! End CB
and the morphism r which occurs in this construction defines a weak-equivalence
from the prop of connected commutative bialgebras B^ to the prop of bar opera-
tions OpCB.
1.4.2. Observation. The natural operation rA : B(A) B(A) ! B(A) given by
the shuffle product of tensors comes from an element r 2 OpCB(2, 1).
The natural operation A : B(A) ! B(A) B(A) given by the deconcatenation
of tensors comes from an element 2 OpCB(1, 2).
18 BENOIT FRESSE
Proof.For m1 + m2 = n1, we let
rm1m2 2 C n1(m1 + m2) = Ind m1+m2xn1 C(1) n1 = F[ m1+m2 ]
1
P
denote the element rm1m2 = ww, where w 2 n1 ranges over the set of (m1, m2)-
shuffles. The associated transformation
(rm1m2 ) : ( A) m1+m2 ! ( A) n1
reduces to the sum of tensor permutations w* : ( A) m1+m2 -'! ( A) n1, for
the permutations w which occur in the expansion of rm1m2 . Hence, the natural
transformation
M M
(r**)A : Bm1(A) Bm2(A) ! Bn1(A)
m1,m2 n1
associated to the collection
Y
(rm1m2 )m1m2 2 C n1(m1 + m2)
m1+m2=n1
represents the shuffle product.
For m1 = n1 + n2, we let
n1n22 C n1+n2(m1) = Ind m1xn1+n2 C(1) n1+n2 = F[ m1]
1
denote the element represented by the identity permutation 1m1 2 m1. The
associated transformation
( n1n2) : ( A) m1 ! ( A) n1+n2
is the identity morphism. Hence, the natural transformation
M M
( **)A : Bm1(A) ! Bn1(A) Bn2(A)
m1 n1,n2
associated to the collection
Y
( n1n2)n1n22 C n1+n2(m1)
m1=n1+n2
is the deconcatenation coproduct of B(A).
1.4.3. Fact. We have a morphism of props r : B ! Op CBwhich maps the gener-
ating operations ~ 2 B(2, 1) and 2 B(1, 2) to r 2 OpCB(2, 1) and 2 OpCB(1, *
*2).
Proof.One proves classically that the shuffle product and the deconcatenation
product provides B(A) with the structure of a commutative algebra. Consequently,
the elements r and satisfy the relations of the product and the coproduct of a
commutative bialgebra in the prop OpCB ,! End CB, so that the map ~ 7! r, 7!
yields a morphism of props r : B ! Op CB.
THE BAR CONSTRUCTION OF AN E-INFINITE ALGEBRA 19
1.4.4. The filtration of OpCB. We equip the module OpCBwith the decreasing filt*
*ra-
tion Y
FpOp CB(r, s) = Op CB(r, s)mn .
m p , n
Recall that
Y
Op CB(r, s)mn = C n1+...+ns(m1 + . .+.mr),
m*,n*
where the product ranges over indices m*, n* such that m1 + . .+.mr = m and
n1 + . .+.ns = n, and
C n1+...+ns(m1 + . .+.mr)
,! Hom__A2CAlg(Bm1(A) . . .Bmr(A), Bn1(A) . . .Bns(A)).
Furthermore, we have C n(m) = 0 if n > m and hence OpCB(r, s)mn = 0 if n > m.
Let us observe that FpOp CBdefines a filtration of Op CBby prop ideals and t*
*hat
Op CB= limpOpCB=FpOp CBby definition of Op CB. In fact, the composition product
of Op CBmaps Op CB(t, s)mn OpCB(r, t)pq into Op CB(r, s)pn if q = m and vanis*
*hes
otherwise. Moreover, in this formula, we have necessarily p q = m n, since
Op CB(r, s)mn = 0 for n > m. Our observation follows from these properties.
1.4.5. Lemma. Our morphism r : B ! Op CBpreserves filtrations and induces a
morphism of filtered props r : B^ ! Op CB.
Proof.By definition, if an element ` 2 B(r, s) has filtration p, then the ass*
*ociated
operation is given by a composite
m1 ... mr w* rn1 ...rnr
B(A) r ----------! B(A) m --! B(A) m --------! B(A) s
such that m = m1+ . .+.mr = n1+ . .+.ns p, where mi : B(A) ! B(A) mi ,
respectively rnj : B(A) nj ! B(A), denotes the mi-fold coproduct, respectively
nj-fold product, of B(A).
Recall that B0(A) = 0 by convention (see paragraph 0.7). Consequently, the
components Bd1(A) . . .Bdm(A) of the middle term B(A) m satisfy d1 + . .+.
dm m p. This relation implies that the composite above has filtration p in
Op CB.
1.4.6. The spectral sequence of Op CB. We consider the second quadrant spectral
sequence defined by the filtration of paragraph 1.4.4:
Y
FpOp CB(r, s) = Op CB(r, s)mn .
m p , n
Thus, we have:
Y
E0mn(Op CB(r, s)) = OpCB(r, s)mn ' C n1+...+ns(m1 + . .+.mr).
m1+...+mr=mn
1+...+ns=n
We use the notion of a strictly polynomial transformation recalled in paragraph
1.2.3. We have:
C n(m) ' Hom V 2 (FMod)(V m, C(V ) n ).
20 BENOIT FRESSE
Consequently, for our spectral sequence, we obtain
Y
E0mn(Op CB) ' Hom V 2 (FMod)(V m1+...+mr, C(V ) n1+...+ns)
m*,n*
Y
' Hom V 2 (FMod)(V m1+...+mr, Bn1E(V ) . . .BnsE(V )),
m*,n*
where E(V ) denote the graded commutative algebra generated by V in degree 1
(equivalently, we set E(V ) = C( -1V )). Moreover, the differential
d0 : E0m*(Op CB) ! E0m*-1(Op CB)
is induced by the bar differential on the target of this module of natural tran*
*sfor-
mations. This observation allows to deduce the E1 term of our spectral sequence
from classical results. Namely:
L
1.4.7. Fact (see x7.3 in [29] for instance). Let K(V ) = n Kn(V ) denote eith*
*er
the exterior algebra K(V ) = (V ) in characteristic 2 or the symmetric algebra
K(V ) = C(V ) otherwise. The morphism ~ : Kn(V ) ! BnE(V ) such that
X
~(v1. .v.n) = vw(1) . . .vw(n)2 BnE(V )
w2 n
define a quasi-isomorphism from the graded module K(V ) equipped with a trivial
differential to the bar complex BE(V ).
As a corollary, we obtain:
1.4.8. Lemma. We have isomorphisms
Y
E1(Op CB(r, s))'-! H*(Hom V 2 (FMod)(V m1+...+mr, BE(V ) s))
m1,...,mr
-' Y Hom (V m1+...+mr, K(V ) s)
~ m1,...,mr V 2 (FMod)
-' Y C s (m + . .+.m )
m1,...,mr 1 r
which maps the element
1n1 . . .1ns 2 Ind nn1x...xCns(n1) . . .C(ns) C s(m1 + . .+.mr),
where n = n1 + . .+.ns = m1 + . .+.mr, to the class of the element
X
w1 . . .ws 2 F[ n1+...+ns] = C n1+...+ns(m1 + . .+.mr),
w1,...,ws
where the sum ranges over permutations w 2 n such that w = w1 . . .ws 2
n1x . .x. ns. In particular, we have E1mn(Op CB) = 0 if n 6= m, and our spectr*
*al
sequence degenerates at E1.
According to lemma 1.4.5, the morphism r : B ! Op CBpreserves filtrations and
hence gives rise to a morphism of spectral sequences Er(r) : Er(B ) ! Er(Op CB),
where Er(B ) = E0(B ), since the prop B has no differential. Theorem 1.B is an
immediate corollary of the following claim. Notice simply that the spectral seq*
*uence
Er(Op CB) converges to H*(Op CB) since this spectral sequence degenerates at E1*
* and
Op CB= limpOpCB=FpOp CB.
THE BAR CONSTRUCTION OF AN E-INFINITE ALGEBRA 21
1.4.9. Claim. We claim that E1(r) is an isomorphism.
Proof.Recall that
E0mmB (r, s) = A r(m)_ m C s(m).
As A (mi)_ ' F[ mi] and Ind mm1x...x mrA(m1)_ . . .A(mr)_ ' F[ m ], the
regular representation, we have
Y Y
E0mmB (r, s) ' C s(m) = Ind mn1x...x nsC(n1) . . .C(ns),
m* m*,n*
where m* and n* ranges over indices such that m1+ . .+.mr = n1+ . .+.ns = m.
The tensors w.1n1 . . .1ns 2 Ind mn1x...xCns(n1) . . .C(ns) of the component
indexed by (m*, n*) are associated to the natural transformations
V m ,! BE(V ) r -`!BE(V ) s
such that ` is given by the composite
m1 ... mr w* rn1 ... rns
BE(V ) r __________//_BE(V ) m____//BE(V ) m__________//_BE(V ),s
where mi : BE(V ) ! BE(V ) mi , respectively rnj : BE(V ) nj ! BE(V ),
denotes the mi-fold coproduct, respectively the nj-fold product, of BE(V ). We
have a commutative diagram
m1 ... mr w* rn1 ... rns
BE(VO)Or ____________//BE(VO)Om____//_BE(VO)Om__________//BE(VO)Os
| | | = |
| | | |
| = | w* | rn1 ... rns |
V m1+...+mr____________//V m________//_V m____________//BE(V ) s
Furthermore, on V m BE(V ) m , the operation
n1 ... rns
V m -r--------! Bn1E(V ) . . .BnsE(V ) ' V n1 . . .V ns
can be identified with the norm map
X
v1 . . .vm 7! (w1 . . .ws)*(v1 . . .vm )
w1,...,ws
where the sum ranges over permutations w 2 n such that w = w1 . . .ws 2 n1x
. .x. ns. Consequently, we deduce from lemma 1.4.8 that the morphism E1(r)
maps generators of E1(B ) = E0(B ) to generators of E1(Op CB). The conclusion
follows.
As explained above, this claim achieves the proof of theorem 1.B.
Remark. As mentioned in the introduction, our arguments can be carried out for
End AB, the endomorphism prop of the bar construction for associative algebras.*
* In
this case, we have a quasi-isomorphism V ~-!B(A ( -1V )), for any F-module V .
Accordingly, in fact 1.4.7, the functor K(V ) is replaced by the identity funct*
*or
K(V ) = V . Consequently, in our spectral sequence, we obtain E1(Op AB(r, s)) '
A r(s)_, and we conclude that EndAB' OpABis equivalent to the prop of connected
coalgebras K^ as claimed.
22 BENOIT FRESSE
2.The uniqueness theorem
2.1. Introduction. In this section, we prove the uniqueness part of theorem A. *
*To
be more precise, we prove the uniqueness assertion in an apparently more general
context, as in the existence theorem 1.A:
Theorem 2.A. The structure supplied by the existence theorem 1.A is homotopi-
cally unique. To be more precise, let E and Q denote E1 -operads and let
ae0A, ae1A: E ! End B(A)
denote operad morphisms which provide the bar complex of an E-algebra B(A)
with the structure of a Q -algebra as in theorem 1.A. The algebras (B(A), ae0A)*
* and
(B(A), ae1A) can be connected by weak-equivalences of Q -algebras
(B(A), ae1A) -~ . -~! (B(A), ae0A)
functorially in A.
According to the results of section 1, the morphisms ae0A, ae1A: Q ! End B(*
*A),
A 2 EAlg, are given by composites
__ae0//_ '
Q ____//_OpEB___//EndEB____//EndB(A)
ae1
such that ae0, ae1 : Q ! Op EBrepresent solutions to the lifting problem
p7OpEB777.
ae0pppppppp
ppppppp ~||
ppppppae1pppp fflfflfflffl|
pp C
Q ____//_C_r_//_OpB
This property implies that the morphisms ae0 and ae1, are left-homotopic (see [*
*32,
xI.1]), and, as a consequence, so are ae0Aand ae1A. Therefore, the uniqueness a*
*ssertion
is a corollary of the following general theorem:
Theorem 2.B. Let P be an operad. Let ae0, ae1 : P ! End A denote a pair of
operad morphisms which provide A with the structure of a P-algebra. The operad
morphisms ae0, ae1 are left-homotopic in the model category of operads if and o*
*nly
if the P-algebras (A, ae0) and (A, ae1) are equivalent in the homotopy category*
* of
P-algebras.
In the simplicial setting, this result is a corollary of the main theorem of*
* [33]
which asserts that the nerve of a category of P-algebra equivalences is homotopy
equivalent to an operadic mapping space. Nevertheless, we would like to give a
different proof of theorem 2.B, because we can make this result more effective *
*in
the differential graded context. Namely, for certain canonical cofibrant operad*
*s Q,
we make explicit a good cylinder object
___d0//_ s0
Q _____//CylQ_____//Q
d1
THE BAR CONSTRUCTION OF AN E-INFINITE ALGEBRA 23
together with a one-to-one correspondence between left-homotopies oe : CylQ !
EndA and diagrams of weak-equivalences
OEoe 0
FQ (A, ae1)~__//FQ (A, ae,)
|~| |~|
fflffl| fflffl|
(A, ae1) (A, ae0)
where FQ (A, aei) denotes a canonical resolution of (A, aei), the Q-algebra def*
*ined by
the morphism aei= oedi: Q ! End A. The functoriality claim in theorem 2.B is an
immediate consequence of this more precise result (lemma 2.3.2).
We use the bar duality of operads in our construction. We recall briefly this
theory in appendix A.2. For more details, we refer to our article [13], from wh*
*ich we
take our conventions, and to the original articles of Getzler-Jones [15] and Gi*
*nzburg-
Kapranov [16].
2.2. Bar duality for operads and cylinder objects. In fact, we consider a
quasi-free operad Q such that Q = Bc(D ), the cobar construction of a cooperad
D , and the aim of this section is to make explicit a cylinder object CylQ for
such operads. Recall that any operad P is equivalent to an operad of this form
Q = Bc(D ). More precisely, for D = B(P), the operadic bar construction of P, t*
*he
operad Q = Bc(B(P)) is endowed with a canonical weak-equivalence ffl : Q -~! P
(see paragraph A.2.21).
2.2.1. Construction. Let I[-1] denote the graded module I[-1] = F0 F1 F01,
where the elements 0, 1 have degree -1 and the element 01 has degree 0. We
consider the free operad
CylQ = F(I[-1] "D).
Thus, we have by definition |0 fl| = |1 fl| = |fl| - 1 and |01 fl| = |fl|*
* in CylQ .
We generalize the definition of the differential of the operadic cobar const*
*ruction
(see recalls in paragraph A.2.2) in order to equip CylQ with a differential. To*
* begin
with, we equip the module D O D(n) of formal composites w . fl0(fl001, . .,.fl0*
*0r) with
a weight grading given by the number of factors in "D. Thus, to be more explici*
*t,
a formal composite w . fl0(fl001, . .,.fl00r) has weight d + 1 2 if we have f*
*l0 2 "D(r),
fl00i2 "D(ni) for d indices i = i1, . .,.id and fl00i= 1 for i 6= i1, . .,.id. *
*Notice that we
can assume i1 = r - d + 1, . .,.id = r by equivariance. We let
X
aed+1(fl) = w . fl0(1, . .,.1, fl00r-d+1, . .,.fl00r)
(fl)d+1
denote the components of the coproduct of fl of weight d + 1.
We consider the derivation @ : CylQ ! Cyl Q defined on generators by the
formulas
X X
@(0 fl) = - w . 0 fl0Or 0 fl00, @(1 fl) = - w . 1 fl0Or 1*
* fl00
(fl)2 (fl)2
24 BENOIT FRESSE
X
and @(01 fl)= 1 fl - 0 fl + w . 01 fl0Or 1 fl00
(fl)2
X X
- w . 0 fl0(1, . .,.1, 01 fl00r-d+1, . .,.01 *
*fl00r) .
d 1(fl)d+1
The unspecified signs are yielded by tensor permutations. To be precise, in the
formula of @(0 fl), respectively @(1 fl), we assume that the element 0, respect*
*ively0
1 crosses the tensor fl0, and this tensor transposition produces the sign = (*
*-1)fl.
Our conventions yield the same sign for the terms w .01 fl0Or1 fl00in the formu*
*la
of @(01 fl).
We check that @ defines a differential and provides CylQ with the structure of
a quasi-free operad.
2.2.2. Claim. The derivation above @ : CylQ ! CylQ commutes with the internal
differential of D and satisfies @@ = 0.
Proof.The first assertion is immediate. For the second assertion, it suffices t*
*o check
that @@ vanishes on the generators of CylQ .
In fact, the differentials @(0 fl) and @(1 fl), where fl 2 "D, can be identi*
*fied with
the cobar differential of fl in F ( -1D"). Therefore, we have clearly @@(0 f*
*l) =
@@(1 fl) = 0 in F (I[-1] "D). The identity @@(01 fl) = 0, follows from a
generalization of the arguments involved in the vanishing of the cobar differen*
*tial
(see proof of claim A.2.3). Explicitly, we deduce from the associativity of the
cooperad coproduct that the terms in the expansion of @@(01 fl) agree two by two
and cancel to each other according to the sign conventions of differential grad*
*ed
calculus. This straightforward verification is left to the reader.
From claim 2.2.2, we conclude:
2.2.3. Lemma. The pair CylQ = (F (I[-1] "D), @) defines a quasi-free dg-opera*
*d.
We prove now that CylQ defines a cylinder object for the operad Q = Bc(D ) =
(F ( -1D"), @), defined by the cobar construction of D. The definition of the c*
*ylinder
faces and degeneracy follows from the following observation:
2.2.4. Observation. We have operad morphisms
___d0//_ s0
Q _____//CylQ_____//Q
d1
such that d0(fl) = 0 fl, d1(fl) = 1 fl, s0(0 fl) = s0(1 fl) = fl, and s0(01 fl)*
* = 0,
for all fl 2 "D. Moreover, these morphisms satisfy s0d0 = s0d1 = IdQ.
Proof.We have to check that the morphisms induced by the maps above commute
with differentials. In fact, it suffices to check this property for generators *
*of Q =
F ( -1D") and CylQ = F(I[-1] "D).
We have mentioned already that the differential of 0 fl and 1 fl in CylQ*
* can
be identified with the cobar differential of fl. Therefore, we have clearly d0(*
*@fl) =
@(d0(fl)), d1(@fl) = @(d1(fl)), s0(@(0 fl)) = @(s0(0 fl)), and s0(@(1 fl)) = @(*
*s0(1
fl)). The identity s0(@(01 fl)) = 0 is also immediate from the definitions.
THE BAR CONSTRUCTION OF AN E-INFINITE ALGEBRA 25
2.2.5. Claim. If the cooperad D is *-cofibrant, then the morphism
(d0, d1) : Q _ Q ! CylQ
is a cofibration.
Proof.We observe that this morphism fits in a sequence
Q _ Q '-!F (sk0M) ,! . .,.! F (skdM) ,! . . .
. .,.! F (colimdskdM) = F(I[-1] "D)
where each map is induced by a cofibration of *-modules skd-1M ,! skdM and
such that @(skdM) F(skd-1M): clearly, the modules
(
"D(r) 1 "D(r) 01 "D(r),if r , d + 1
skdM(r) = 0
0 "D(r) 1 "D(r), otherwise,
satisfy these conditions. These assertions imply that (d0, d1) is a cofibration.
2.2.6. Claim. If D"(0) = "D(1) = 0, then the morphisms of observation 2.2.4 are
weak-equivalences.
Proof.We use the spectral sequence of a quasi-free operad
Er(F (M), @) ) H*(F (M), @)
introduced in [13, x3.6]. Recall that E0 = (F (M), ~@), where ~@: F(M) ! F (M)
denotes the indecomposable component of @ : F (M) ! F(M), which satisfies
~@(M) M.
The assumption "D(0) = "D(1) = 0 ensures that the spectral sequence converges
for Q = F( -1D") and for CylQ = F(I[-1] "D). For Q = F( -1D"), we obtain
~@= 0. For Q = F(I[-1] "D), we obtain ~@= @I, where @I(01 fl) = 1 fl - 0 *
* fl
and @I(0 fl) = @I(1 fl) = 0. Clearly, the morphisms d0, d1, s0 induce quas*
*i-
isomorphisms on generators. In fact, the diagram
__d0//_ s0
-1D" __d1//_I[-1] _"D_//_ -1D"
represents the classical cylinder object of -1D" in the category of dg-modules.
According to loc. cit., this property implies that d0, d1, s0 induce weak-equiv*
*alences
on the associated free operads, and hence that d0, d1, s0 yield isomorphisms at*
* the
E1 level of the spectral sequence. The conclusion follows.
We can now conclude:
2.2.7. Lemma. The dg-operad CylQ defines a cylinder-object for the operad Q .
Furthermore, this cylinder is very good if D is a *-cofibrant cooperad. Explic*
*itly,
in the diagram
(d0,d1) s0
Q _ Q______//_CylQ______//Q,
the map s0 is an acyclic fibration, and we have s0d0 = s0d1 = IdQ. Furthermore,
if D is *-cofibrant, then the map (d0, d1) is an operad cofibration.
Proof.This statement is a direct corollary of the claims 2.2.5 and 2.2.6 above.
26 BENOIT FRESSE
2.3. Cylinder objects and algebra equivalences. In this section, we relate the
cylinder defined in the previous section to algebra equivalences and we deduce *
*the
proof of theorem 2.B from this relationship. Recall that an operad morphism ae :
Bc(D ) ! V , which provides the dg-module V with the structure of an algebra ov*
*er
Bc(D ), is equivalent to a quasi-cofree coalgebra (D (V ), @ae) (see observatio*
*n A.2.8).
We extend this construction to the cylinder object CylQ . We obtain the followi*
*ng
result:
2.3.1. Lemma. We have a one-to-one correspondence between left homotopies
(ae0,ae1)
Q_ Q _____//EndV::
uu
(d0,d1)||uu
fflffl|hu
CylQ
and morphisms of quasi-cofree coalgebras OEoe: (D (V ), @~1) ! (D (V ), @~0) su*
*ch
that OEoe|V : V ! V is the identity morphism, where (D (V ), @~0), respecti*
*vely
(D (V ), @~1), denotes the quasi-cofree D -coalgebra associated to the Bc(D )-a*
*lgebra
(V, ae0), respectively (V, ae1).
Proof.We have by definition ~0(v) = ~1(v) = 0 for v 2 V and
~0(fl(v1, . .,.vn)) = ae0(fl)(v1, . .,.vn)
= h(d0(fl))(v1, . .,.vn) = h(0 fl)(v1, . .,.vn),
~1(fl(v1, . .,.vn)) = ae1(fl)(v1, . .,.vn)
= h(d1(fl))(v1, . .,.vn) = h(1 fl)(v1, . .,.vn),
if fl 2 "D(n).
We consider the map oe : D(V ) ! V such that oe(v) = v for v 2 V
and oe(fl(v1, . .,.vn)) = h(01 fl)(v1, . .,.vn), if fl 2 "D(n).
We prove that this map oe satisfies the equation of lemma A.2.11, explicitly
X
ffi(oe)(fl(v1, . .,.vn)) + ~0fl0 oefl001(v_1), . .,.oefl00r(v_r)
(fl)
X
- oefl0 v_0, ~1fl00(v_00) - oe~1(fl(v1, . .,.vn))*
* = 0,
(fl)2
if and only if the map h : F (I[-1] "D) ! End V commutes with differentials.
Notice that the equation above is obviously satisfied for a generator v 2 V . I*
*n the
case fl 2 "D, one observes easily that this equation is equivalent to the follo*
*wing
relation in EndV :
ffi(h(01 fl)) - h(01 ffi(fl))
X X
+ h(0 fl) + w . h(0 fl0) 1, . .,.1, h(01 fl001), . .,.h(01 *
*fl00r)
d 1(fl)d+1
X
- w . h(01 fl0) Or ae(1 fl00) - h(1 fl) = 0,
(fl)2
THE BAR CONSTRUCTION OF AN E-INFINITE ALGEBRA 27
and hence, to the identity ffi(h(01 fl)) - h(01 ffi(fl)) - h(@(01 fl)) = *
*0. Our
claim follows.
2.3.2. Lemma. Suppose given operad morphisms ae0, ae1 : P ! EndV , as in
theorem 2.B, which provide the dg-module V with P-algebra structures. We let
D = B(P), Q = Bc(D ), and we assume that P is *-cofibrant. For any left-
homotopy
(ae0,ae1)
Q _ Q_____//P _ P____//EndV55j,
j j
(d0,d1)|| j j j
fflffl|jjj h
CylQ
the associated morphism of quasi-cofree coalgebras OEoe: (D (V ), @ae1) ! (D (V*
* ), @ae0)
supplied by lemma 2.3.1 yields an equivalence of P-algebras OEoe: FP (D (V ), @*
*ae1) -~!
FP (D (V ), @ae0), so that we have a diagram of P-algebras
FP (D (V ), @ae1)~OEoe//_FP (D (V ), @ae0)
|~| |~|
fflffl| fflffl|
(V, ae1) (V, ae0)
in which all morphisms are weak-equivalences.
Proof.This claim is a corollary of lemmas A.2.20 and A.2.17 and of the observat*
*ions
of lemma A.2.22.
3. The homotopy interpretation of the bar construction
3.1. Introduction. In this section, we prove the homotopy interpretation stated
in theorem B. To be more precise, as in the previous sections, we prove the ass*
*ertion
of this theorem in an apparently more general context:
Theorem 3.A. Let E denote an E1 -operad. Let Q denote a cofibrant E1 -operad
together with an operad equivalence Q -~! E, so that any E-algebra form a Q -
algebra by restriction of structure. Let FA _~_////_A denote a cofibrant resolu*
*tion of
an E-algebra A in the category of Q -algebras. Suppose that the bar constructi*
*on
B(A) is equipped with the structure of a Q -algebra as in theorem 1.A. Then, we
have a weak-equivalence of Q -algebras
FA -~! B(A),
where FA denotes the suspension of FA in the closed model category of Q-algebr*
*as.
Let us recall that the homotopy category of algebras over an E1 -operad E do*
*es
not depend on E. Therefore, we can prove the theorem for well chosen operads
E and Q . Precisely, we let E denote the chain operad of the simplicial operad
introduced Barratt and Eccles in [3]. In fact, this operad has E(0) = F, so tha*
*t we
may consider a connected version ~E, where this component is removed. Then, we
let Q = Bc(B(~E)), the quasi-free operad defined by the cobar-bar construction *
*of
~E. Let us mention that the Barratt-Eccles operad is a good operad: the category
of E-algebras is equipped with the structure of a closed model category defined*
* as
usual (see [5, 6]).
28 BENOIT FRESSE
The canonical operad equivalence ffl : Q -~! E induces an equivalence of homo-
topy categories. In particular, the associated (derived) extension functor carr*
*ies the
cofibrant resolution of an E-algebra A in the category of Q-algebras to the cof*
*ibrant
resolution of A in the category of E-algebras and preserve suspensions. Therefo*
*re,
in regard to the theorem above, we can consider the suspension of FA , the cofi*
*brant
resolution of A in the category of E-algebras, and we prove that this E-algebra*
* FA
is connected to B(A) by equivalences of Q-algebras. Let us notice in addition t*
*hat
the bar construction preserves all equivalences of E-algebras. In particular, w*
*e ob-
tain B(FA ) -~!B(A) for the cofibrant resolution of an E-algebra in the categor*
*y of
E-algebras. Furthermore, for the uniqueness theorem 2.A, it is sufficient to pr*
*ovide
the bar construction of quasi-free algebras B(F ) with the structure of a Q-alg*
*ebra,
functorially in F 2 QF ree(E Alg), because, according to lemma 1.2.7, we have
EndEB(r) = Hom A2EAlg(B(A) r, B(A)) ' Hom F2QF ree(EAlg)(B(F ) r, B(F )).
Consequently, in the proof of this theorem 3.A, we can restrict ourself to quas*
*i-free
E-algebras F = FA .
We deduce the theorem above from a comparison result between the classical bar
construction B(F ) and a categorical bar construction B_E(F ) in which the tens*
*or
product is replaced by the coproduct in the category of E-algebras. According
to Mandell (see [27, x3]), we have a natural equivalence B(F ) -~! B_E(F ). We
give another proof of this assertion for our purposes. We observe precisely th*
*at
B(F ) forms a strong deformation retract of B_E(F ), for any quasi-free E-algeb*
*ra
F , functorially in F . The categorical bar construction B_E(F ) is endowed wit*
*h the
structure of an E-algebra. By a transfer argument, we obtain that B(F ) can be
equipped with the structure of an algebra over Q which makes B(F ) equivalent
to B_E(F ) in the homotopy category of Q -algebras. Moreover, we can check that
this Q -algebra structure satisfies the assumptions of our existence and unique*
*ness
theorem (theorem A).
Finally, the conclusion of theorem 3.A is a direct consequence of a general r*
*esult
of [27, x14]. Namely, for any (good) operad P, the categorical bar construction
B_P(F ) is equivalent to F in the homotopy category of P-algebras. By uniquene*
*ss,
we conclude that the classical bar construction B(F ) is equivalent to F as a *
*Q -
algebra, for any Q-algebra structure that satisfies the assumptions of the exis*
*tence
and uniqueness theorem.
3.2. Coproduct of algebras over the Barratt-Eccles operad.
3.2.1. Operads and augmented algebras. Incidentally, in this section, we consid*
*er
augmented algebras with unit and operads P such that P(0) 6= 0. In particular,
we assume now that C denotes the operad of unital, associative and commutative
algebras, which has a component C(0) = F.
In general, for an operad as above P, the dg-module P(0) is equipped with the
structure of a P-algebra since the operad composition products yields morphisms
P(r) P(0) r ! P(0), for r 1. Furthermore, this algebra structure, also denoted
by *, represents the initial object in the category of P-algebras since the eva*
*luation
product of a P-algebra yields a morphism jA : P(0) ! A, for any P-algebra A.
In this context, an augmented algebra denotes an object of P Alg=*, the comma
category of P-algebras over *, or more explicitly, a P-algebra A equipped with *
*an
augmentation fflA : A ! * which is a morphism of P-algebras. Notice that this
assertion implies fflA jA = Id. In particular, we have A = * ~A, where ~A= ke*
*rfflA .
THE BAR CONSTRUCTION OF AN E-INFINITE ALGEBRA 29
u0 Ok v0__//_u1 Ok_v0//_._._.//_ud Ok v0
| | |
fflffl| fflffl| fflffl|
u0 Ok v1__//_u1 Ok_v1//_._._.//_ud Ok v1
| | |
fflffl| fflffl| fflffl|
.. . .
. .. ..
| | |
fflffl| fflffl| fflffl|
u0 Ok ve__//_u1 Ok_ve//_._._.//_ud Ok ve
Figure 1. The composition product in the Barratt-Eccles operad
Let ~Pdenote the operad such that
(
~P(n) = 0, if n =,0
P(n), otherwise.
One observes that ~Ais endowed with the structure of an algebra over ~P. Furthe*
*r-
more, as for associative algebras, the map A 7! ~Adefines an equivalence from t*
*he
category of augmented P-algebras to the category of ~P-algebras.
3.2.2. The Barratt-Eccles operad. For our purposes, we consider P = C, the oper*
*ad
of unital associative and commutative algebras, and P = E, the Barratt-Eccles
operad.
This E1 -operad E is defined by E(r) = N*(E r), the normalized chain complex
of the simplicial set E r such that
(E r)n = { (w0, . .,.wn) 2 xn+1r}
together with the classical faces and degeneracies given respectively by the om*
*ission
and the repetition of components. Explicitly:
di(w0, . .,.wn)= (w0, . .,.cwi, . .,.wn),for i = 0,,. .,.n
sj(w0, . .,.wn)= (w0, . .,.wj, wj, . .,.wn),for j = 0,.. .,.n
The simplicial sets E r, r 2 N, are endowed with operad composition products
E r x E n1x . .x.E nr ! E n1+...+nr
induced by composition products on permutations. The composition products of E
are obtained by composition of the composition products above with the Eilenber*
*g-
Zilber equivalence:
N*(E r) N*(E n1) . . .N*(E nr)
! N*(E r x E n1x . .x.E nr) ! N*(E n1+...+nr).
Explicitly, for simplices u = (u0, . .,.ud) 2 E(m)d and v = (v0, . .,.ve) 2 E(n*
*)e,
the partial composite u Ok v 2 E(m + n - 1)d+e, is the sum of the d + e-simplic*
*es
formed by the paths of the diagram of figure 1 together with a sign determined *
*by
the signature of a shuffle permutation.
The augmentations N0(E r) ! F, r 2 N, induce an operad equivalence E -~!
C, so that Eforms a *-cofibrant E1 -operad. Moreover, the degree 0 component of
E can be identified with the operad A+ of unital associative algebras. Conseque*
*ntly,
30 BENOIT FRESSE
we have an operad embedding A + ,! E and any E-algebra is endowed with the
structure of a honest associative algebra.
Notice that E(0) = F according to our definition.
3.2.3. Coproducts of algebras over the Barratt-Eccles operad. The coproduct in a
category is denoted by _. For quasi-free algebras over an operad F1 = (P(V1), @*
*1)
and F2 = (P(V2), @2), we have F1 _ F2 = (P(V1 V2), @1 + @2).
For algebras over the Barratt-Eccles operad, we have a natural morphism
F1 . . .Fd EM--!F1 _ . ._.Fn
which maps a tensor a1 . . .ad 2 F1 . . .Fd to the product ~(a1, . .,.ad) of
a1 2 F1, . .,.ad 2 Fd in F1 _ . ._.Fd, where ~ 2 E(d) denotes the operation that
represents the d-fold associative product in the Barratt-Eccles operad. Accordi*
*ng
to [27, x3], this map is a quasi-isomorphism. As mentioned in the introduction *
*of
this section, we give another proof of this assertion. Namely, we observe that *
*the
map above is an instance of an Eilenberg-Zilber equivalence. As a consequence, *
*we
obtain a stronger result, which permits to use the transfer argument of section*
* A.3.
Namely:
3.2.4. Lemma. Let F1, . .,.Fr denote quasi-free algebras over the Barratt-Eccles
operad. We have a strong deformation retract
________________________________*
*_________
oAWo_ _________________________________*
*_______________________________
F1 . . .Fr____//F1 _ . ._.Fr _H____________________________*
*___,
EM ``_______________________________*
*_____________________________________________________
where H satisfies the side conditions H . EM = AW . H = H . H = 0, and such
that EM, AW, H are functorial in F1, . .,.Fr 2 QF ree(E Alg).
Moreover, for free algebras Fk = E(Vk), the augmentation map E ! C induces
a morphism of strong deformation retracts:
____________________________*
*_________________________
AWoo_ _____________________________*
*_________________________
E(V1) . . .E(Vr)___//_E(V1 . . .Vr)_H________________________*
*_______.
EM ``___________________________*
*______________________________________
| | _______________________
| | ____________________________*
*___________
fflffl| ' fflffl|_____________________________*
*__________________________________
C(V1) . . .C(Vr)'__//_C(V1oo_. . .Vr)`0`_______________________*
*______________________________
_____________________________*
*___________________________________________
Let us mention that the last assertion of the lemma can be extended to quasi-
free algebras, but this is not necessary for our purposes. The rest of the sect*
*ion is
devoted to the proof of this lemma.
3.2.5. An Eilenberg-Zilber equivalence. We let Fi = (E(Vi), @i). We have weight
decompositions:
F1 _ . ._.Fr
= E(V1 . . .Vr)
M
= E (n1 + . .+.nr) V1 n1 . . .Vr nr n x...x n,
n1,...,nr 1 r
F1 . . .Fr
= E(V1) . . .E(Vr)
M
= E (n1) . . .E(nr) V1 n1 . . .Vr nr n x...x n.
n1,...,nr 1 r
THE BAR CONSTRUCTION OF AN E-INFINITE ALGEBRA 31
u0 v0___//u1 v0_//_._._.//ud v0
| | |
fflffl| fflffl| fflffl|
u0 v1___//u1 v1_//_._._.//ud v1
| | |
fflffl| fflffl| fflffl|
.. . .
. .. ..
| | |
fflffl| fflffl| fflffl|
u0 ve___//u1 ve_//_._._.//ud ve
Figure 2. The simplices of EM(u v) for a 2-tensor u v =
(u0, . .,.ud) (v0, . .,.ve) 2 E(m) E(n)
For each collection (n1, . .,.nr), we specify equivariant maps
____________________________*
*_________________________
oAWo_ _____________________________*
*_________________________
E(n1) . . .E(nr)___//E(n1 + . .+.nr) _H________________________*
*_______
EM ``___________________________*
*_____________________________________________________________
which induce the morphisms involved in lemma 3.2.4. We let n = n1 + . .+.nr.
Let Ik denote the interval Ik = {n1+. .+.nk-1+1, . .,.n1+. .+.nk-1+nk}. In
the weight decompositions above, the cartesian product n1x. .x. nr is identifi*
*ed
with the Young group I1x . .x. Ir n. The permutation of n associated
to w1 2 n1, . .,.wr 2 nr is denoted by w1 . . .wr 2 n. We consider the
map of simplicial sets - . . .- : E n1 x . .x.E nr ! E n induced by this
embedding. We let w|Ik2 Ik denote the restriction of a permutation w 2 n to
the subset Ik {1, . .,.n}, and for a simplex ss = (w0, . .,.wd) 2 E(n)d, we s*
*et
ss|Ik= (w0|Ik, . .,.wd|Ik).
The map EM : E(n1) . . .E(nr) ! E(n1+. .+.nr) is defined by the composite:
N*(E n1) . . .N*(E nr) ! N*(E n1x . .x.E nr) ---...-----!N*(E n),
where the first map is given by the classical Eilenberg-Mac Lane equivalence. H*
*ence,
for r = 2, and simplices u = (u0, . .,.ud) 2 E(m)d and v = (v0, . .,.ve) 2 E(n)*
*e the
element EM(u v) 2 E(m + n)d+e is the sum of the d + e-simplices formed by the
paths of the diagram of figure 2 together with a sign determined by the signatu*
*re of
a shuffle permutation as usual. The map AW : E(n1+. .+.nr) ! E(n1) . . .E(nr)
is defined by the composite:
N*(E n) ! N*(E n) . . .N*(E n)
|I1 ... |Ir
-------! N*(E I1) . . .N*(E Ir)
' N*(E n1) . . .N*(E nr),
where the first map is given by the classical Alexander-Whitney diagonal. Hence,
for r = 2 and for a simplex ss = (w0, . .,.wd), we have
Xd
AW (ss) = (w0|I, . .,.wp|I) (wp|J, . .,.wd|J) 2 N*(E I) N*(E J).
p=0
32 BENOIT FRESSE
For a simplex ss = (w0, . .,.wd), we set
Xn
H(w0, . .,.wd) = (-1)p(w0, . .,.wp, EM . AW (wp, . .,.wd)),
p=0
where on the right hand side we consider the concatenation of (w0, . .,.wp) with
the simplices of EM . AW (wp, . .,.wd).
The next assertion is classical in the setting of the Eilenberg-Zilber equiva*
*lence
for normalized chain complexes (compare with [35]).
3.2.6. Claim. The maps above define a strong deformation retract:
____________________________*
*_________________________
AWoo_ _____________________________*
*_________________________
E(n1) . . .E(nr)____//E(n1 + . .+.nr) _H_______________________*
*________
EM ``___________________________*
*_____________________________________________________________
Explicitly, we have AW . EM = Idand Id-EM . AW = ffiH + Hffi. In addition, the
chain homotopy H satisfies the side conditions H . EM = AW . H = H . H = 0.
The augmentation E ! F is preserved by the maps EM and AW and cancelled
by the chain-homotopy H. Consequently, the augmentation E ! F induces a
morphism of strong deformation retracts:
____________________________*
*_________________________
AWoo_ _____________________________*
*_________________________
E(n1) . . .E(nr)____//E(n1 + . .+.nr) _H_______________________*
*________
EM ``___________________________*
*______________________________________
| | _______________________
| | __________________________________*
*_______
fflffl|oo_=_________ fflffl|_____________________________*
*______________________________
F_________=_________//F``_0______________________________*
*_______________________
___________________________________*
*__________________________
We observe that the maps EM, AW, H are compatible with operad composition
products. We obtain the following result:
3.2.7. Claim. Let ae 2 E(s). For i = 1, . .,.nk, we set i0= n1+ . .+.nk-1 + i. *
*The
diagram
_______________________*
*___________________________________________
oo__AW______ ________________________*
*_____________________
E (n1) . . .E(nk) . . .E(nk)_________//E(n) H___________________*
*__________
EM ``______________________*
*_______________________________________________________________________
1 ... -Oiae ...|1| |-Oi0ae|_______________
|fflffl AW fflffl|____________________*
*___________________________________________________________________
E(n1) . . .E(nk + s - 1) . . .E(nk)_//_E(no+os_- 1) _H_______________*
*______________________
EM ``___________________*
*__________________________________________________________________________
defines a morphism of strong deformation retracts. Thus, the maps EM, AW, H
commute with partial composites.
Proof.We prove that EM commutes with partial composites. Let 1r denote the
identity permutation of r, and by an abuse of notation, the associated vertex *
*in
E(r). For permutations w1 2 n1, . .,.wr 2 nr, the composite 1r(w1, . .,.wr)
is represented by the direct sum w1 . . .wr 2 n. Hence, for simplices ss1 2
E(n1), . .,.ssr 2 E(nr), the composite 1r(ss1, . .,.ssr) 2 E(n) can be identifi*
*ed with
EM(ss1 . . .ssr) 2 E(n). By associativity of the operad composition product, we
have:
1r(ss1, . .,.ssk Oiae, . .,.ssr) = 1r(ss1, . .,.ssr) Oi0ae.
Hence, for a given tensor ss1 . . .ssr 2 E(n1) . . .E(nr), we have the iden*
*tity
EM(ss1 . . .ssk Oiae . . .ssr) = EM(ss1 . . .ssr) Oi0ae.
THE BAR CONSTRUCTION OF AN E-INFINITE ALGEBRA 33
( w0 Oi0ae0|I1_//_._././_w* Oi0ae0|I1) . . .
( w* Oi0ae0|Ik-1_//._._.//_wp Oi0ae0|Ik-1)
0 1
wp Oi0ae0|Ik_//_._._.//wq Oi0ae0|Ik
BB C
BB fflffl|| fflffl||CCC
BB .. .. CC
BB . . CC
@ fflffl|| fflffl||A
wp Oi0aee|Ik_//_._._.//wq Oi0aee|Ik
( wq Oi0aee|Ik+1//_._._.//_w* Oi0ae0|Ik+1) . . .
( w* Oi0aee|Ir_//_._././_wd Oi0ae0|Ir)
Figure 3. The terms of AW (ss Oi0ae)
We prove that AW commutes with partial composites. We assume
ss = (w0, . .,.wd) 2 E(n)d and ae = (ae0, . .,.aee) 2 E(s)e.
We have w* Oi0ae*|Ik = w*|IkOiae* for k = i and w* Oi0ae*|Ik = w*|Ik for k 6= i.
Consequently, for k 6= i, the restriction |Ik of an edge wx Oi0aey ! wx Oi0aey+1
yields a degenerate edge. Hence, the element AW (ss Oi0ae) is the tensor produc*
*t of
simplices defined by tensor product of paths represented in figure 3 since the *
*other
terms are degenerate. The identity
X
AW (ss Oi0ae) = AW1(ss) . . .AWk(ss) Oiae . . .AWr(ss),
P
where AW (ss) = AW1(ss) . . .AWr(ss) 2 E(n1) . . .E(nr) denote the
expansion of AW (ss), follows from these observations.
We prove that H commutes with partial composites. As above, we assume
ss = (w0, . .,.wd) 2 E(n)d and ae = (ae0, . .,.aee) 2 E(s)e. The element H(ss O*
*i0ae) is
given by the sum of composite paths in the diagrams of figure 4, where p, q ran*
*ge
over p = 0, . .,.d and q = 0, . .,.e. We have proved that EM and AW preserve
operadic composition products. Consequently,Pwe have the identity represented in
figure 5. Then, let EM . AW (wp, . .,.wd) = (w0p, . .,.w0q) denote the expans*
*ion
of EM . AW (wp, . .,.wd). We deduce that the simplices of figure 4 are given by
the paths of figure 6. We observe immediately that these simplices represent t*
*he
operadic composites
X
H(w0, . .,.wd) Oi0(ae0, . .,.aee) = (w0, . .,.wp, w0p, . .,.w0d) Oi0(ae0,*
* . .,.aee)
and the conclusion follows. Namely, we have the identity: H(ssOi0ae) = H(ss)Oi0*
*ae.
3.2.8. The induced strong deformation retract for coproducts. For elements ssk(*
*v_k) 2
E(Vk), where ssk 2 E(nk) and v_k2 Vk nk, we set
EM(ss1(v_1) . . .ssr(v_r)) = EM(ss1, . .,.ssr)(v_1 . . .v_r).
P
For ss 2 E(n), we let AW (ss) = AW1(ss) . . .AWr(ss) 2 E(n1) . . .E(nr)
denote the expansion of AW (ss). For a tensor v_1 . . .v_r2 V1 n1 . . .Vr nras
34 BENOIT FRESSE
w0 Oi0ae0__//._._.//_wp Oi0ae0
| |
fflffl| fflffl|
.. .
. ..
| |
fflffl| fflffl|
w0 Oi0aeq__//._._.//_wp_Oi0aeq___
____________________________
__________________
____________________
_____________________wp_Oi0aeq//_././._wd1Oi0aeq *
* 0
_______________________
__________________________| |
_________________________________ C *
* B
________________________fflffl| fflffl|C *
* B
___//_________________________________________________*
*_________EM...AW.BB.CC
. . C *
* B@
| | A
fflffl| fflffl|
wp Oi0aee__//_._._.//_wd Oi0aee
Figure 4. The simplices of H(ss Oi0ae)
0 wp Oi0aeq__//_._././_wd Oi0aeq1
B | | C
B fflffl| fflffl|C
EM . AW BBB ... ... CC
@ | | CA
fflffl| fflffl|
wp Oi0aee__//_._._.//wd Oi0aee
= EM . AW (wp, . .,.wd) Oi0(aeq, . .,.aee).
Figure 5. The map EM . AW preserves operadic composites
w0 Oi0ae0__//_._._.//wp Oi0ae0
| |
fflffl| fflffl|
.. .
. ..
| |
fflffl| fflffl|
w0 Oi0aeq__//_._._.//wp Oi0aeq
KK
KKK
K%%K
w0pOi0aeq__//._._.//_w0dOi0aeq
| |
fflffl| fflffl|
.. .
. ..
| |
fflffl| fflffl|
w0pOi0aee__//._._.//_w0dOi0aee
Figure 6. The simplices of H(ss Oi0ae) after reduction
THE BAR CONSTRUCTION OF AN E-INFINITE ALGEBRA 35
above, we set
X
AW (ss(v_1 . . .v_r)) = AW1(ss)(v_1) . . .AWr(ss)(v_r),
and similarly:
H(ss(v_1 . . .v_r)) = H(ss)(v_1 . . .v_r).
The next statement is mentioned as a remark. In fact, this observation is not
necessary for our purposes.
3.2.9. Observation. The map EM : E(V1) . . .E(Vr) ! E(V1 . . .Vr) induced
by the maps EM : E(n1) . . .E(nr) ! E(n1 . . .nr) defined in paragraph 3.2.5
can be identified with the map EM defined in paragraph 3.2.3.
Proof.This assertion is a consequence of an observation stated in the proof of
claim 3.2.7. Namely, for elements ss1 2 E(n1), . .,.ssr 2 E(nr), we have
EM(ss1 . . .ssr) = 1r(ss1, . .,.ssr),
the operadic composite of ss1, . .,.ssr with the identity permutation 1r, ident*
*ified
with a vertex of E(r). In fact, the operation 1r 2 E(r) represent precisely the*
* r-fold
associative product ~ for E-algebras. Consequently, for elements ssk(v_k) 2 E(V*
*k),
we obtain the identities:
EM(ss1(v_1) . . .ssr(v_r)) = EM(ss1 . . .ssr)(v_1 . . .v_r)
= 1r(ss1, . .,.ssr)(v_1 . . .v_r) = ~(ss1(v_1), . .,.ssr(*
*v_r)).
This proves our observation.
Remark. We can also give a categorical interpretation of the map AW : E(V1)_. .*
*_.
E(Vr) ! E(V1) . . .E(Vr). We let F1 = E(V1), . .,.Fr = E(Vr). One observes
that the Barratt-Eccles operad is endowed with a coassociative diagonal : E(r*
*) !
E(r) E(r) induced by the classical Alexander-Whitney diagonal for the simplic*
*ial
set E r. Moreover, this diagonal defines an operad morphism. This property
implies that the Barratt-Eccles operad operates naturally on a tensor product of
E-algebras. In this context, we observe that the tensor product F1 . . .Fr is
endowed with natural morphisms of E-algebras
Fk ! F1 . . .Fr
which maps an element ak 2 Fk to the tensor product 1 . . .ak . . .1, where 1
denote the algebra units of F1, . .,.Fr. From this observation, we deduce a nat*
*ural
map from the categorical sum to the tensor product
F1 _ . ._.Fr ! F1 . . .Fr
which, for quasi-free algebras, can be identified with the map AW specified in
paragraph 3.2.8.
3.2.10. Claim. Let F1 = (E(V1), @1), . .,.Fr = (E(Vr), @r) denote quasi-free al*
*ge-
bras. The maps EM, AW, H commute with the differentials @1, . .,.@r of F1, . .,*
*.Fr.
36 BENOIT FRESSE
Proof.This assertion is a direct consequence of claimP3.2.7. Explicitly, let ss*
*k(v_k) =
ssk(v1k, . .,.vmk) 2 E(Vk). If we let @k(vik) = aei(v_0ik), then we obtain
X
@k(ssk(v1k, . .,.vmk)) = ssk(v1k, . .,.@k(vik), . .,.vmk)
i X
= ssk Oiaei(v1k, . .,.v_0ik, . .,.vmk*
*).
i
Consequently, for the map EM, we obtain
EM(ss1(v_1) . . .@k(ssk(v_k)) . . .ss1(v_1))
X
= EM(ss1(v_1) . . .ssk Oiaei(v1k, . .,.v_0ik, . .,.vmk) . . .ss1(v*
*_1))
Xi
= EM(ss1 . . .ssk Oiaei . . .ssr)(v_1 . . .v_0ik . . .v_m)
Xi
= (EM(ss1 . . .ssr) Oi0aei)(v_1 . . .v_0ik . . .v_m)
Xi
= EM(ss1 . . .ssr)(v_1 . . .aei(v_0ik) . . .v_m)
i
= @kEM(ss1 . . .ssr)(v_1 . . .v_m) = @kEM(ss1(v_1) . . .ssm (v_m)).
The relations
AW (@k(ss(v_1 . . .v_r))) = @kAW (ss(v_1 . . .v_r))
and H(@k(ss(v_1 . . .v_r))) = @kH(ss(v_1 . . .v_r))
are obtained similarly.
3.2.11. Claim. The maps EM, AW, H commute with morphisms of quasi-free al-
gebras OEk : Fk ! Fk0.
Proof.The arguments are similar to the proof of the claim above. Explicitly, for
ssk(v_k) = ssk(v1k, . .,.vmk) 2 E(Vk), we let
X X
OEk(v1k) = aei(v_01k), . .,.OEk(vmk) = aei(v_0mk).
We obtain
X
OEk(ssk(v1k, . .,.vmk)) = ssk(OEk(v1k), . .,.OEk(vmk))
X
= ssk(ae1, . .,.aem )(v_01k, . .,.v_0*
*mk).
Hence, the relation
EM(ss1(v_1) . . .OEk(ssk(v_k)) . . .ss1(v_1)) = OEkEM(ss1(v_1) . . .ssm *
*(v_m))
is a consequence of claim 3.2.7 (namely, the map EM preserve operadic composite*
*s),
as in the proof of claim 3.2.10.
The identities
AW (OEk(ss(v_1 . . .v_r))) = OEkAW (ss(v_1 . . .v_r))
and H(OEk(ss(v_1 . . .v_r))) = OEkH(ss(v_1 . . .v_r))
are obtained similarly.
THE BAR CONSTRUCTION OF AN E-INFINITE ALGEBRA 37
These assertions complete the proof of lemma 3.2.4. Namely, the strong defor-
mations retract identities are corollaries of the equivalent identities for the*
* maps
EM, AW, H defined in paragraph 3.2.5 (claim 3.2.6). The claim 3.2.10 implies th*
*at
EM, AW, H define morphisms of quasi-free algebras and the claim 3.2.11 implies
that these maps EM, AW, H are functorial.
3.3. The categorical bar construction.
3.3.1. The categorical bar construction. We recall the definition of the catego*
*rical
bar construction B_P(A), for A an augmented algebra over an operad P, as in
paragraph 3.2.1. One considers the simplicial P-algebra B__P(A) such that
B__P(A) = A_n,
where _ denote the categorical coproduct in the category of P-algebras, together
with faces and degeneracies defined by
8
>A_i-1_ r _ A_n-i-1, for i = 1, . .,.n,- 1
:A_n-1 _ fflA , for i = n,
sj = A_i_ jA _ A_n-i, for j = 0, . .,.n,
where r : A _ A ! A denotes the fold map, jA : * ! A the algebra unit, and
fflA : A ! * the algebra augmentation. Then, we set
B_P(A) = N*(B__P(A)),
the normalized chain complex of this simplicial categorical bar construction. T*
*his
chain complex is equipped with the structure of an (augmented) P-algebra, like
the normalized chain complex of any simplicial algebra over an operad, which is
defined by the composite of the Eilenberg-Mac Lane equivalence with the evaluat*
*ion
product of B__P(A).
Let us recall that the classical bar construction of an (augmented) associat*
*ive
algebra B(A) is also defined by the normalized chain complex of a simplicial mo*
*dule
B_(A). We have explicitly
B_(A) = A n ,
together with faces and degeneracies defined by
8
>A i-1 ~ A n-i-1, for i = 1, . .,.n,- 1
:A n-1 fflA , for i =,n
sj= A i jA An-i, for j = 0, . .,.n,
where ~ : A A ! A denotes the product of A. Notice that Nn(B_(A)) = ~A n,
the tensor power of the augmentation ideal of A, so that this bar complex B(A)
coincides with the bar complex of the non-unital algebra ~Aconsidered in previo*
*us
sections. Let us mention that for P = C we have a canonical isomorphism B_(A) '
B__C(A), since the coproduct is realized by the tensor product for associative *
*and
commutative algebras with unit.
38 BENOIT FRESSE
According to Mandell (see [27, x3]), for a quasi-free algebra F over an E1 -
operad E, the natural map EM : F n ! F _ndefined in paragraph 3.2.3 yields an
equivalence of simplicial dg-modules
EM__ : B_(F ) -~!B__E(F ).
Consequently, we have a quasi-isomorphism of dg-modules EM : B(F ) -~!B_E(F ).
We improve this result in order to apply the transfer argument of section A.3.
Namely, for algebras over the Barratt-Eccles operad, we prove that Mandell's eq*
*uiv-
alence fits in a strong deformation retract of dg-modules:
3.3.2. Lemma. For any quasi-free algebra F over the Barratt-Eccles operad E, we
have a strong deformation retract of dg-modules
______________________________________*
*___
oAW*o_ _______________________________________*
*_________________________
B(F )_____//B_E(F ) _H*_________________________________,
EM ``_____________________________________*
*___________________________________________________
functorial in F 2 QF ree(E Alg), and where H* satisfies the side conditions H* .
EM = AW* . EM = H* . H* = 0.
Moreover, for a free algebra F = E(V ), the augmentation map E ! C induces
a morphism of strong deformation retracts:
___________________________________*
*______
oAW*o_ _____________________________________*
*___________________________
B(E(V ))____//B_E(E(V )) _H*______________________________*
*___.
EM _``__________________________________*
*______________________________
| | _______________________
| | ____________________________________*
*____
fflffl|' fflffl|__________________________________*
*____________________________
B(C(V ))_'__//B_C(C(Vo))o_0``_____________________________*
*_______________________
_____________________________________*
*____________________________________
Proof.The construction of paragraph 3.2.8 yields a strong deformation retract
____________________________________
oAWo_ ________________________________________*
*_____________________________
F n_EM__//F _n``_H___________________________________*
*____________________,
________________________________________*
*________________
for any n 2 N. One observes easily that EM commutes with simplicial faces, but
not the maps AW, H. Thus, we obtain a deformation retract of dg-modules
______________________________________*
*_______________
oAWo_ _______________________________________*
*________________
B(F )____//_B_E(F ) H_______________________________,
EM ``_____________________________________*
*___________________________________________________
where B(F ) and B_E(F ) are only equipped with internal differentials ffi. We l*
*et @
denote the bar differential of B_E(F ). We apply the basic perturbation lemma
(see [17, 18]) in order to deduce a strong deformation retract
__________________________________*
*_______
oAW*o_ ___________________________________*
*_____________________________
(B(F ), @*)___//(B_E(F ), @)H*_____________________________*
*_____
EM* ``_________________________________*
*_______________________________________________________
from this construction. The resulting maps EM*, AW*, H* (and the resulting diff*
*er-
ential @*) are clearly functorial by construction: recall that these maps are d*
*efined
by the formulas
X X
@* = EM . @ . H_._@_.-.H..z@___".AW, EM* = H_._@_.-.H..z@___".EM,
n 0 n n 0 n
X X
AW* = AW . @_._H_.-.@..zH___"and H* = H_._@_.-.H..z@___".H.
n 0 n n 0 n
THE BAR CONSTRUCTION OF AN E-INFINITE ALGEBRA 39
The second assertion is also immediate from the definition of EM*, AW*, H*. The*
*re-
fore, the lemma is a consequence of the next claim.
3.3.3. Claim. The map EM* above, supplied by the basic perturbation lemma,
reduces to the given map EM. So does the differential @* : B(F ) ! B(F ), which
reduces to the classical bar differential @ : B(F ) ! B(F ).
Proof.This claim is a consequence of the side condition and of the observation
recalled above: the map EM preserves bar differentials. Explicitly, for n 1,
because of the side condition H . EM = 0, we have the identity:
H_._@_.-.H..z@___".EM = H_._@_.-.H..z@___"H . EM . @ = 0,
n n-1
from which we deduce EM* = EM and @* = AW . @ . EM = AW . EM . @ = @.
Proof of theorem 3.A. As mentioned in the introduction, we consider the operad
Q = Bc(D ), where D = B(~E) denotes the operadic bar construction of the re-
duced Barratt-Eccles operad E, in which the component E(0) = F is removed (see
paragraph 3.2.1).
According to lemma A.3.4, the action of Q on the categorical bar construction
of a quasi-free E-algebra B_E(F ) can be transferred to the classical bar const*
*ruction
B(F ) through the strong deformation retract of lemma 3.3.2. Moreover, the resu*
*lt-
ing algebra (B(F ), ss) is related to the former (B_E(F ), ae) by weak-equivale*
*nces of
Q -algebras.
We check that our construction satisfies the requirements of the uniqueness *
*the-
orem (theorems 1.A-2.A). Since the strong deformation retract associated to a
quasi-free algebra F is functorial in F 2 QF ree(E Alg), the resulting action on
End B(F)is still functorial in F 2 QF ree(E Alg). Consequently, the transfer co*
*n-
struction yields a morphism ss : Q ! End EB.
We check that ss : Q ! End EBlifts the classical map r : C ! End CB. Recall
that
EndEB(r)= Hom A2EAlg(B(A) r, B(A))
' Hom A2QF ree(EAlg)(B(A) r, B(A))
' Hom A2Free(EAlg)(B(A) r, B(A)).
Let A = C(V ) denote a free commutative algebra. Consider the free E-algebra F =
E(V ) endowed with the augmentation map ffl : E(V ) ! C(V ). Notice that ffl de*
*fines
a surjective morphism of E-algebras. Consequently, the action of an operation q*
* 2
Q (r) on B(C(V )) deduced from our morphism ss : Q ! End EBcan be characterized
by the functoriality diagram:
ss(q)
B(E(V )) r_____//B(E(V )).
ffl|| ffl||
fflffl|ss(q) fflffl|
B(C(V )) r_____//B(C(V ))
40 BENOIT FRESSE
On the other hand, recall that ffl yields a_morphism_of_strong_deformation_re*
*tracts:________________________________
____________________________________*
*_____________________________
B(E(V ))____//_B_E(E(Vo))o_``_____________________________*
*_______________________.
____________________________________*
*_____________________________________
ffl|| |ffl| ____________________
fflffl|' fflffl|__________________________________*
*________________________________________________
B(C(V ))__'_//_B_C(C(Vo))o_0``____________________________*
*________________________
____________________________________*
*_____________________________________
For the bottom strong deformation retract, the transfer provides B(C(V )) with *
*the
structure of a Q-algebra that reduces to the classical commutative algebra stru*
*cture
of B(C(V )), since this strong deformation retract is trivial. By functoriality*
* of the
transfer construction, the action of an operation q 2 Q(r) on B(E(V )) and B(C(*
*V ))
makes the following diagram commute:
ss(q)
B(E(V )) r_____//B(E(V )),
ffl|| ffl||
fflffl|r(q) fflffl|
B(C(V )) r_____//B(C(V ))
where r(q) denotes the image of q under the composite
Q ~-!E -~! C r-!EndCB.
Thus, we conclude that ss(q) = r(q) for a commutative algebra.
The conclusion of theorem 3.A follows from the following arguments. According
to [27, x3], the algebra (B_E(F ), ae) is equivalent to F , the suspension of *
*F in
the category of E-algebras. Hence, we conclude that (B(F ), ss) is equivalent *
*as
a Q -algebra to F , where the E-algebra F is equipped with the structure of a
Q-algebra by restriction through the augmentation ffl : Q -~! E.
This achieves the proof of theorem 3.A.
4. The relationship with the cochain algebra of a loop space
4.1. Introduction. In this section, we study the relationship between nFX , the
iterated suspension of a cofibrant resolution of the cochain algebra of a point*
*ed
space X, and C*( nX), the cochain algebra of the iterated loop space nX. Let
us recall our statements.
Theorem 4.A. We let C*(X) denote the cochain algebra of a pointed space X
with coefficients in a field F of characteristic p > 0. We let FX _~_////_C*(X)*
* denote
a cofibrant resolution of C*(X) in the category of E-algebras. We assume that X
is connected p-complete, nilpotent and of finite p-type (as in [27]). Then, for*
* any
n 0, the natural map
nFX ! C*( nX)
defines a weak-equivalence of E-algebras provided that ssn(X) is a finite p-gro*
*up.
We deduce theorem 4.A from results of [27]. Namely, the constructions of loc.
cit. allows to prove our theorem by induction on the Postnikov tower of X. To be
precise, for a connected space X such that nX remains connected, the statement
above is contained in the main result of [27]. Thus, we check simply that certa*
*in
constructions of loc. cit. can be extended to spaces with finitely many connect*
*ed
components. Anyway, we may assume that the original space X is connected or
THE BAR CONSTRUCTION OF AN E-INFINITE ALGEBRA 41
not, because the loop space X sees only the connected component of the base
point of X, as well as the bar construction BC*(X) and so does the suspension
FX , since FX ~ BC*(X).
As mentioned in the introduction, we can generalize theorem 4.A to spaces such
that ssn(X) is not a finite group. In this situation, we consider Bousfield-Ka*
*n'
tower {RsX} (see [8]) which supplies an approximation of X by spaces RsX that
satisfy the finiteness assumption of the theorem as long as the cohomology mod-
ules H*(X, Fp) are degreewise finite. According to [8, xIII.6] and [10], we ha*
*ve
colimsH*(RsX, F) ' H*(X, F) provided that X is connected. We deduce from this
property that the colimit FX = colimsFRsX of the cofibrant resolutions of C*(Rs*
*X)
defines a cofibrant resolution of C*(X). Consequently, we have a weak-equivalen*
*ce
nFX = colims nFRsX -~! colimsC*( nRsX),
and we obtain
H*( nFX , F) = colimsH*( nRsX, F).
This analysis yields the following result stated in the introduction:
Theorem 4.B. We can let F = Fp. We assume that X is a pointed space whose
cohomology modules H*(X, Fp) are degreewise finite. We let RsX denote Bousfield-
Kan' tower of X (for R = Fp). We fix a cofibrant resolution FX of C*(X), as in
theorem C above. We have
^
H0( nFX ) = Fssn(R1pX)p,
the module of maps ff : ssn(R1 X) ! F pwhich are continuous in regard to the
p-profinite topology and
H*( nFX ) = H0( nFX ) colimsH*( n0RsX, Fp),
where n0X denotes the connected component of the base point of X.
4.2. Proofs. We can assume F = Fp, as in theorem 4.B above. For a space X, we
let FX denote any cofibrant resolution of C*(X) in the category of E-algebras. *
*The
suspension FX is defined by the cofiber of any cofibration FX //_//CFX that f*
*its
in a factorization
FX //__//CFX _~__////_*
of the augmentation map FX ! *. The natural map FX ! C*( X) fits in a
commutative diagram that relates the cofiber sequence FX //_//CFX ! X to
the path space fibration X ! P X _~_////_X and is characterized by this proper*
*ty
up to homotopy. Explicitly, we have
FX //____//_CFXO______//_OFX ,
~ || ~ O O
fflfflfflffl|fflfflO fflfflO
C*(X) _____//C*(P X)____//C*( X)
where the middle vertical arrow is deduced from the lifting diagram
FX ______//C*(P X).
fflffl|~t::t
| tt ~||
fflffl|t~ fflfflfflffl|
CFX ________//*
42 BENOIT FRESSE
The arguments of [27, x5] imply that the resulting map FX ! C*( X) forms a
weak-equivalence provided that the Eilenberg-Moore spectral sequence
*(X,F ) *
TorH* p(Fp, Fp) ) H ( X, Fp)
converges. This condition is satisfied if ss0( X) is a finite p-group and if H**
*( X, Fp)
is degreewise finite. By induction, one can deduce that nFX ! C*( nX) is a
weak-equivalence as mentioned in the introduction, but strong finiteness assump-
tions are required for this argument. For our arguments, we need only the follo*
*wing
special instance:
4.2.1. Claim (compare with Proposition 9.4 in [27]). For any m 2 Z, we have a
weak-equivalence FK(Z=p,m)-~! FK(Z=p,m-1), where by convention K(Z =p, m) ~
* for m < 0. Consequently, theorem 4.A holds for any Eilenberg-Mac Lane space
X = K(Z =p, m), m 2 Z.
Our induction argument is supplied by the following claim.
4.2.2. Claim. Let K(Z =p, m) ! E ! B denote a principal fibration of spaces
that satisfy the assumptions of theorem 4.A. To be precise, as mentioned in the
introduction, the space B need not be connected, but we assume at least that B *
*has
finitely many connected components.
If theorem 4.A holds for X = B, then theorem 4.A holds for X = E as well.
Explicitly, if the natural map nFB ! C*( nB) forms a weak-equivalence, then
so does the map nFE ! C*( nE).
Proof.This claim is an easy consequence of [27, Lemma 5.2]. Explicitly, the pri*
*n-
cipal fibration is equivalent to a cartesian square
E _____//L(Z =p, m +,1)
| ||
fflfflfflffl||fflfflfflffl|
B __k_//_K(Z =p, m + 1)
in which vertical maps are fibrations and such that*L(Z =p, m + 1) ~ *. Fix a
lifting FK(Z=p,m+1)! FB of the morphism C*(B) -k! C*(K(Z =p, m + 1)) and a
factorization FK(Z=p,m+1) //_//CFK(Z=p,m+1)_~_////_C*(L(Z =p, m + 1)) of the map
FK(Z=p,m+1)_~_////_C*(K(Z =p, m + 1)) ! C*(L(Z =p, m + 1)). Let FE denote the
pushout
FE = FB _FK(Z=p,m+1)CFK(Z=p,m+1).
We have then a cocartesian square
FK(Z=p,m+1)_____//_FB
fflffl fflffl|
| |
fflffl| fflffl|
CFK(Z=p,m+1)_____//FE
in which vertical maps are cofibrations together with weak-equivalences
CFK(Z=p,m+1) oo______ooFK(Z=p,m+1)_________//FB .
~ || ~ || ~||
fflfflfflffl| fflfflfflffl|* fflfflfflffl|
C*(L(Z =p, m + 1))oo__C*(K(Z =p, s + 1))__k//_C*(B)
THE BAR CONSTRUCTION OF AN E-INFINITE ALGEBRA 43
In this situation, lemma 5.2 in loc. cit. asserts precisely that the induced *
*map
FE ! C*(E) is a weak-equivalence, provided that the Eilenberg-Moore spectral
sequence *
TorH*(K(Z=p,m+1))(H*(B), F) ) H*(E)
converges. This condition is satisfied under the assumptions of theorem 4.A.
As, on one hand, the functor X 7! nX preserves fibrations and cartesian
squares and, on the other hand, the functor F 7! nF preserves cofibrations and
cocartesian squares, we conclude that the map
nFE ! C*( nE)
forms also a weak-equivalence of E-algebras for the same reason, since the map
nFK(Z=p,m+1) ! C*( nK(Z =p, m + 1)) is a weak-equivalence by claim 4.2.1,
the map nFB ! C*( nB) by assumption, and the map nFL(Z=p,m+1) !
C*( nL(Z =p, m + 1)) because L(Z =p, m + 1) ~ *.
4.2.3. Claim. Theorem 4.A holds for any connected Eilenberg-Mac Lane space X =
K(Z ^p, m), m 6= 0, provided that n 6= m. Explicitly, the map nFK(Z^p,m)!
C*( nK(Z ^p, m)) is a weak-equivalence provided that n 6= m.
Proof.By induction, we deduce from claim 4.2.2 that theorem 4.A holds for X =
K(Z =ps, m). To be more precise, our construction yields a tower of cofibrant
resolutions
* //__//FK(Z=p,m)//_//. ././//_FK(Z=ps-1,m)//__//_FK(Z=ps,m)////_.,. .
| | | |
|= ~| |~ ~|
fflfflfflffl||fflfflfflffl| fflfflfflffl| fflfflfflffl|
*__//_C*(K(Z =p, m))//_._././_C*(K(Z =ps-1,_m))//_C*(K(Z =ps,_m))//_. . .
such that FK(Z=ps-1,m)//_//FK(Z=ps,m)is a cofibration, and the induced maps
nFK(Z=ps,m)! C*( nK(Z =ps, m))
are all weak-equivalences.
For m 6= 0, we have colimsH*(K(Z =ps, m)) ' H*(K(Z ^p, m)). Therefore,
according to [27], the colimit F = colimsFK(Z=ps,m)is endowed with a weak-
equivalence
F -=! colimsFK(Z=ps,m)-~!colimsC*(K(Z =ps, m)) -~!C*(K(Z ^p, m)),
and defines a cofibrant resolution of C*(K(Z ^p, m)). The suspension functor n
preserves colimits. Consequently, we obtain:
nF ' colims nFK(Z=ps,m)-~!colimsC*( nK(Z =ps, m)).
As above, for m 6= n, we have colimsC*( nK(Z =ps, m)) ' C*(K(Z =ps, m-n)) -~!
C*(K(Z ^p, m - n)) and therefore we obtain a weak-equivalence
nF -~! C*( nK(Z ^p, m)).
This proves our claim.
We can now proceed to the proof of our theorem:
4.2.4. Claim. Let X be any connected nilpotent p-complete space of finite p-typ*
*e.
The map nFX ! C*( nX) forms a weak-equivalence provided that ssn(X) is a
finite p-group. Thus, theorem 4.A holds for such spaces X.
44 BENOIT FRESSE
Proof.By assumption, the Postnikov tower of X can be refined to a tower of prin*
*ci-
pal fibrations X = limsXs with Fs = K(Z =p, ns) or Fs = K(Z ^p, ns) as fibers. *
*Fur-
thermore, we have colimsH*(Xs) ' H*(X) and the natural map colimsC*(Xs) !
C*(X) forms a weak-equivalence of E-algebras. For the loop space, we obtain
nX = lims nXs with nFs as fibers. We have obviously Fs = nK(Z =p, ns) =
K(Z =p, ns- n) or nFs = nK(Z ^p, ns) = K(Z ^p, ns- n). Hence, the tower nXs
satisfies the same property for the space nX, except that nXs may have finite*
*ly
many connected components. Anyway, we obtain colimsH*( nXs) ' H*( nX).
As for claim 4.2.3, we prove by induction that theorem 4.A holds for Xs. To be
precise, we obtain a tower of cofibrant resolutions FXs -~! C*(Xs) such that the
induced maps nFXs ! C*( nXs) are all weak-equivalences. According to the
discussion above, we have a weak-equivalence
colimsFXs -~! colimsC*(Xs) -~!C*(X),
so that F = colimsFXs defines a cofibrant resolution of C*(X). Moreover, for the
suspension, we obtain:
nF ' colims nFXs -~! colimsC*( nXs) -~!C*(X).
This proves our claim.
This claim achieves the proof of theorem 4.A.
As mentioned in the introduction of this section, we use the Bousfield-Kan to*
*wer
in order to generalize theorem 4.A:
4.2.5. Claim. Let X be a connected space whose cohomology modules H*(X, Fp)
are degreewise finite. We have weak-equivalences
nFX ~- colims nFRsX -~! colimsC*( nRsX),
where {RsX} denotes the classical Bousfield-Kan tower of X (see [8]).
Proof.According to [8, xIII.6] and [10], we have colimsH*(RsX) ' H*(X) for any
connected space X. Consequently, as in the proof of claim 4.2.4, we have a tower
of cofibrant resolutions FRsX ~-!C*(RsX), and any cofibrant resolution of the
cochain algebra C*(X) is equivalent to the colimit F = colimsFRsX .
The finiteness assumption implies that the spaces RsX satisfy the condition
of claim 4.2.4. Consequently, theorem 4.A holds for RsX. As in the proof of
claim 4.2.4, we deduce a weak-equivalence
nF ' colims nFRsX -~! colimsC*( nRsX).
Then, we obtain:
4.2.6. Claim. We assume that X is a connected space together with degreewise
finite cohomology modules H*(X, Fp) as above. With the notation of theorem 4.B,
we obtain colimsH*( nRsX) ' colimsH0( nRsX) colimsH*( n0RsX) and
^
colimsH0( nRsX) ' Fssn(R1 X)p.
Proof.We have nRsX = n0RsX x ss0( nRsX) = n0RsX x ssn(RsX). As a
consequence, under our finiteness assumption, we obtain:
colimsH*( nRsX) ' colimsFssn(RsX) colimsH0( n0RsX).
THE BAR CONSTRUCTION OF AN E-INFINITE ALGEBRA 45
Furthermore, the finiteness assumption implies:
lim1sssn(RsX) = 0 and limsssn(RsX) = ssn(R1 X)
(see [8, Chapter^IX]). Consequently, the colimit colimsFssn(RsX)can be identifi*
*ed
with Fssn(R1 X)p.
This assertion achieves the proof of theorem 4.B.
Appendix A. Operads, bar duality and transfer
A.1. Operads. The purpose of the next paragraphs is to recall some fundamental
results and conventions on operads. We refer to [13] for a more comprehensive
introduction and for further references to the literature.
A.1.1. Operads. In this article, we consider symmetric operads in the category *
*of
dg-modules. Accordingly, an operad consists of a sequence of dg-modules P(r),
r 2 N, together with unital and associative composition products
P(r) P(n1) . . .P(nr) ! P(n1 + . .+.nr),
defined for r 1 and n1, . .,.nr 0. In addition, each module P(r) is equipped
with an action of the symmetric group r and the composition products above
are assumed to be equivariant. The operadic composite of p 2 P(r) with q1 2
P(n1), . .,.qr 2 P(nr) is denoted by p(q1, . .,.qr). We consider also partial c*
*ompos-
ites pOiq 2 P(r+s-1), for p 2 P(r), q 2 P(s), defined by pOiq = p(1, . .,.q, . *
*.,.1),
where q is composed at the ith entry of p. The unit of P is defined by an opera*
*tion
1 2 P(1) such that 1(p) = p and p(1, . .,.1) = p for all p 2 P(r). A morphism of
operads is an equivariant map f : P ! Q which preserves composition products
and operad units.
As mentioned in section 0, we consider only connected operads, such that P(0*
*) =
0 and P(1) = F1, and we assume tacitely that this condition is satisfied.
A.1.2. Free and quasi-free operads. Recall that a (differential graded) *-modu*
*le M
consists of a sequence of (differential graded) modules M(r), r 2 N, equipped w*
*ith
an action of the symmetric groups r. Let us recall furthermore that any *-
module M has an associated free operad, denoted by F(M), which is characterized
by the classical universal property. Equivalently, we have a forgetful functor *
*from
the category of (differential graded) operads to the category of (differential *
*graded)
*-modules U : dg Op ! dg *Mod and the free operad associated to a *-
module is defined by the left adjoint to this functor F : dg *Mod ! dg Op. For
our purposes, we can assume that the free operad F(M) is the object spanned by
formal operadic composites of elements of M F(M).
In the differential graded context, the free operad is equipped with a canon*
*ical
differential ffi : F (M) ! F(M) induced by the internal differential of M and
such that ffi|M (M) M in F (M). A quasi-free operad denotes a free operad
F (M) equipped with a non-canonical differential defined by a homogeneous map
@ : F (M) ! F (M) of degree -1, which satisfies a derivation relation in regard
to operadic composites, and such that ffi(@) + @2 = 0, so that the pair (F (M),*
* @)
defines a dg-operad. The derivation relation implies that @ : F (M) ! F (M) is
determined by a restriction @|M : M ! F (M), but we do not have @|M (M) M
in general, unless F(M) is a free-operad.
46 BENOIT FRESSE
The projection of @|M : M ! F (M) onto M, denoted by ~@: M ! M, defines
the indecomposable component of the differential @. This map satisfies the iden*
*tity
ffi(~@) + ~@2= 0, and hence, the pair (M, ~@) defines a dg-module.
A.1.3. The closed model category of operads. The category of dg- *-modules is
equipped with the structure of a cofibrantly generated closed model category in
which a morphism f : M ! N is a weak-equivalence if the morphisms of dg-
modules f : M(r) ! N(r) are quasi-isomorphisms, and a fibration if the mor-
phisms f : M(r) ! N(r) are surjective. We recall the structure of cofibrations *
*in
paragraph A.1.4.
The category of connected operads is equipped with the structure of a closed
model category obtained by transfer through the adjunction
_F___//
dg *Mod oo___ dgOp .
U
Explicitly, an operad morphism f is a weak-equivalence, respectively a fibratio*
*n, if
and only if U(f) defines a weak-equivalence, respectively a fibration, in the c*
*ategory
of dg- *-modules. This category is cofibrantly generated and we recall the stru*
*cture
of cofibrant operads in paragraph A.1.5. These results are borrowed from [6, 20*
*].
A.1.4. Cofibrations of *-modules. In fact, the category of dg- *-modules is a *
*par-
ticular instance of a closed model category of dg-modules over an algebra for w*
*hich
we refer to [23]. In this setting, one observes that a morphism of dg- *-modules
OE : M ! N is a cofibration if and only if this morphism can be decomposed in a
sequence
M -'! sk0N ,! . .,.! skdN ,! . .,.! colimdskdN = N,
such that ffi(skdN) skd-1N and where skd-1N ,! skdN is a split injective
morphism of dg- *-modules with a projective cokernel. In fact, if M and N are
non-negatively graded, then we can consider a degreewise filtration and these c*
*on-
ditions can be simplified. Explicitly, any morphism of non-negatively graded dg*
*- *-
modules OE : M ! N with a projective cokernel forms a cofibration in the catego*
*ry
of *-modules. (Notice in particular that a cofibrant *-module is a projective
object in the category of *-modules.)
A.1.5. Cofibrant operads. The category of operads is cofibrantly generated by m*
*or-
phisms of free operads OE : F (M) ! F (N) associated to a set of generating cof*
*i-
brations of dg- *-modules OE : M ! N.
In particular, an operad Q is cofibrant if and only if Q is the retract of a*
* quasi-
free operad (F (M), @) where the *-module M can be equipped with an increasing
filtration
0 = sk0M ,! . .,.! skdM ,! . .,.! colimdskdM = M
such that @(skdM) F (skd-1M) and where skd-1M ,! skdM defines a cofi-
bration of *-modules. If M(0) = M(1) = 0 (equivalently, if the operad F (M)
is connected) and ~@= 0, then these conditions can be simplified. In fact, und*
*er
the assumption above M(0) = M(1) = 0, a decomposable element fl 2 F (M)(r)
is composed of elements x 2 M(n) such that n < r. Therefore, if we consider the
THE BAR CONSTRUCTION OF AN E-INFINITE ALGEBRA 47
filtration (
skdM(r) = M(r), if r d,
0, otherwise,
then we have automatically @(skdM) F (skd-1M). Consequently, a connected
operad Q is cofibrant if and only if Q is the retract of a quasi-free operad (F*
* (M), @)
such that M is a cofibrant *-module.
Let us mention that any cofibrant operad Q is *-cofibrant, explicitly any co*
*fi-
brant operad Q forms a cofibrant object in the category of *-modules, but the
converse assertion does not hold (see [6]).
A.1.6. Algebras over an operad. Recall that an algebra over an operad P is a dg-
module A equipped with r-equivariant evaluation products
P(r) A r ! A,
defined for r 0, and which are unital and associative with respect to the ope*
*rad
composition products.
Equivalently, the structure of an algebra over an operad P is defined by an *
*operad
morphism ae : P ! End Awhere EndA denotes the endomorphism operad of the dg-
module A, defined by EndA (r) = Hom__(A r, A). In general, we omit the morphism
ae in our notation and a dg-algebra is specified by its underlying dg-module A.*
* But,
if we consider a non-canonical structure, then the resulting P-algebra is denot*
*ed
by the pair (A, ae).
By an abuse of notation, the map associated to an operation p 2 P(r) is also
denoted by p : A r ! A.
A.1.7. Free algebras over an operad. The free algebra over an operad P generated
by a dg-module V is denoted by P(V ). Recall that
1M
P(V ) = (P(r) V r) r.
r=0
The universal morphism jV : V ! P(V ) identifies an element v 2 V with the
tensor 1 v 2 P(1) V in P(V ). The element of P(V ) represented by the tensor
p v1 . . .vr 2 P(r) V ris denoted by p(v1, . .,.vr) 2 P(V ), since this t*
*ensor
represents the image of v1 . . .vr 2 V runder the operation p : P(V ) r ! P(V *
*).
The free algebra P(V ) is equipped with a canonical differential ffi : P(V )*
* ! P(V )
induced by the internal differential of V and P. Notice that ffi|V (V ) V in *
*P(V ).
As for operads, a quasi-free algebra denotes a free algebra F = P(V ) equipped *
*with
a non-canonical differential defined by a P-algebra derivation @ : P(V ) ! P(V )
such that ffi(@) + @2 = 0. The derivation formula implies that @ : P(V ) ! P(V )
is determined by a restriction @|V : V ! P(V ), but we do not have @|V (V ) *
*V ,
unless (P(V ), @) is a free algebra.
A.1.8. The closed model category of algebras over an operad. If P is a good ope*
*rad
(for instance, if P is a cofibrant operad), then the category of P-algebras is *
*equipped
with the structure of a cofibrantly generated closed model category in which a
morphism f : A ! B is a weak-equivalence, respectively a fibration, if f is a
quasi-isomorphism of dg-modules, respectively a surjective morphism. Moreover,
48 BENOIT FRESSE
a P-algebra A is cofibrant if and only if A is the retract of a quasi-free alge*
*bra
(P(V ), @), where the dg-module V is equipped with a filtration
0 = sk0V ,! sk1V ,! . .,.! skdV ,! . .,.! colimdskdV = V
such that @(skdV ) P(skd-1V ) and where skd-1V ,! skdV is a cofibration of
dg-modules. As usual, these conditions can be simplified if V is non-negative*
*ly
graded.
Let us mention that the category of algebras over a *-cofibrant operad do not
form a closed model category in general. Nevertheless, this problem can be arra*
*nged
by the introduction of semi-model structures (see [39]).
A.2. Bar duality for operads. The cobar construction of a cooperad supplies a
quasi-free operad Q = Bc(D ) together with a nice interpretation of the structu*
*re
of a Q-algebra which permits to construct easily morphisms in the homotopy cat-
egory of Q-algebras. The purpose of this section is to recall this setting, bor*
*rowed
from [15, x2]. In fact, the theory was settled in characteristic zero in the or*
*iginal
reference. Therefore, we give a careful survey in order to check that results o*
*f loc.
cit. can be generalized to (Z -graded) operads defined over a ring, but this se*
*ction
does not contain any original idea. The construction of section 2, where we ext*
*end
the operadic cobar construction in order to define good cylinder objects for op*
*er-
ads and the transfer construction of section A.3 provide our motivations for pr*
*ecise
recalls: the technical results of this appendix are used in these constructions.
A.2.1. Cooperads. As in [13, x1.2.17], a cooperad denotes a *-module D, such t*
*hat
D (0) = 0 and D(1) = F 1, together with n-equivariant coproducts
M1 M
D(n) -ae! D (r) Ind nn1x...x nrD(n1) . . .D(nr) r
r=0 n1+...+nr=n
dual to the composition products of an operad. The right hand side module is
also denoted by D O D, because the functor associated to this compositeL *-modu*
*le
satisfies the relation D O D(V ) = D(D (V )). Notice that D O D(n) = 1r=0D(r)
D r(n) r. Thus, according to our conventions, the coproduct of an element fl 2
D (n) is represented by a sum of formal composites
X
ae(fl) = w . fl0(fl001, . .,.fl00r),
(fl)
where w 2 n, fl02 D(r), and fl0012 D(n1), . .,.fl00r2 D(nr).
As for operads, a cooperad D is called *-cofibrant, respectively *-project*
*ive, if
D forms a cofibrant, respectively projective, object in the category of *-modu*
*les.
A.2.2. The operadic cobar construction. The operadic cobar construction of a co-
operad Q = Bc(D ), introduced in [15, x2.1], is a quasi-free operad such that
Q = F ( -1D"), where "Ddenotes the coaugmentation coideal of D , together with
a differential @ : F ( -1D") ! F ( -1D") determined by the coproduct of D . We
recall the definition of this differential more explicitly.
We have by definition
(
"D(n) = 0, if n = 0,,1
D (n), otherwise,
THE BAR CONSTRUCTION OF AN E-INFINITE ALGEBRA 49
and we consider the components
ae h M ioe
D"(n) ! "D(r) Ind n1x...x sx...x11 . .".D(s) . . .1
___________r+s-1=n_________-z__________________________" r
D O D(n)
of the coproduct of D. Thus, for an element fl 2 D, we let
X
ae2(fl) = w . fl0(1, . .,.fl00, . .,.1)
(fl)2
denote the components of ae(fl) in which only one factor fl00= fl00ibelongs to *
*"D.
As for operads, the formal composite w . fl0(1, . .,.fl00, . .,.1) is also deno*
*ted by
w . fl0Oifl00. Notice that we can assume i = r by r-equivariance.
Let @ : -1D" ! F ( -1D") be the map such that
X
@(fl) = - w . fl0Or fl00.
(fl)2
To be precise, suspensions are omitted in this formula, but these suspensions g*
*ives
to @ the degree d = -1, since one suspension occurs on the left hand-side of the
formula while two suspensions occur on the right-hand side (one suspension for *
*each
factor "D). Moreover, our construction requires a tensor permutation
-2(D"(r) "D(s)) ' ( -1D"(r)) ( -1D"(s)),
which produces the unspecified sign0of the formula of @(fl). Hence, this sign *
*is
given explicitly by = (-1)|fl.| The additional minus sign is motivated by the
relationship of Bc(D ) with a simplicial version of this construction (see [13,*
* x4]).
A.2.3. Claim (compare with proposition 2.2 in [15]). The derivation
@ : F( -1D") ! F ( -1D")
induced by the map above commutes with the internal differential of D and satis*
*fies
the identity @@ = 0.
Proof.The first assertion is immediate. The identity @@ = 0 follows from the
associativity of the coproduct of the cooperad D. Explicitly, for a generator f*
*l 2 "D,
one deduces from the associativity of the cooperad coproduct that the terms in *
*the
expansion of @@(fl) agree two by two and cancel to each other according to the *
*sign
conventions of differential graded calculus. We refer to loc. cit. for a detail*
*ed proof
in the dual case of the bar construction of an operad.
To conclude, our construction gives the following result:
A.2.4. Proposition. The pair Bc(D ) = (F ( -1D"), @) defines a quasi-free opera*
*d,
which is cofibrant if the cooperad D forms a cofibrant object in the category o*
*f *-
modules.
(The cofibrant claim follows from the observations of paragraph A.1.5.) Let *
*us
recall that any operad P is equivalent to an operad of this form Q = Bc(D ) for
D = B(P), the operadic bar construction of P (see paragraph A.2.21).
50 BENOIT FRESSE
A.2.5. Cofree and quasi-cofree coalgebras over a cooperad. As for algebras over*
* an
operad, the functor V 7! D(V ) defined by the formula
1M
D (V ) = (D (r) V r) r
r=0
associates to any dg-module V the cofree coalgebra cogenerated by V over the
cooperad D . To be precise, the direct sum implies that D (V ) forms a connected
coalgebra and the coinvariants imply that D (V ) is a D -coalgebra with divided
symmetries (see [11]), but we do not care about these subtleties in this articl*
*e,
especially if we assume that the cooperad D is *-projective, in which case the*
*re
is no difference between invariants and coinvariants.
A quasi-cofree coalgebra denotes a cofree coalgebra D(V ) equipped with a no*
*n-
canonical differential defined by a map @ : D(V ) ! D (V ). As usual, we assume
that ffi(@) + @2 = 0, so that ffi + @ defines a differential on D(V ) and we de*
*note the
resulting dg-coalgebra by the pair (D (V ), @).
Let us mention that the differential of a coalgebra over a cooperad is suppo*
*sed
to satisfy a coderivation relation. This property implies that the differential*
* of a
quasi-cofree coalgebra is determined by a homogeneous map : D(V ) ! V which
satisfies an equation equivalent to the identity ffi(@) + @2 = 0. This relatio*
*nship
is made more precise in the next statements. Nevertheless, we do not recall the
definition of a coderivation: for our purposes, we can take the next assertion *
*as a
definition.
A.2.6. Proposition (see proposition 2.14 in [15]). We have a one-to-one corre-
spondence between the set of coderivations @ : D(V ) ! D (V ) and the set of ho*
*mo-
geneous maps : D(V ) ! V . The map associated to a coderivation @ is given
by the composite of @ : D(V ) ! D (V ) with the projection D(V ) ! V .
Conversely, the coderivation associated to , also denoted by @ = @ , is det*
*er-
mined by the formula
X
@ (fl(v1, . .,.vn)) = fl0 v_0, fl00(v_00) , for any fl(v1, . .,.vn) *
*2,D(V )
(fl)2
where v_0and v_00denote appropriate groupings of variables.
P
Recall that ae2(fl) = (fl)2w . fl0Or fl00denotes the quadratic component o*
*f the
coproduct of fl 2 D, defined in paragraph A.2.2. The permutation w is performed
on the tensor power V n, and therefore, does not appear explicitly in the form*
*ula
above. Hence, the groupings v_0and v_00are given by v_0= vw(1) . . .vw(r-1)
and v_00= vw(r) . . .vw(r+s-1), or equivalently, by the tensor decomposition
w*(v1 . . .vn) = vw(1) . . .vw(n)= v_0 v_00.
A.2.7. Lemma. A coderivation @ : D (V ) ! D(V ) of degree -1 satisfies the
identity ffi(@ ) + @2 = 0, so that the pair (D (V ), @ ) defines a quasi-cofree*
* coalgebra,
if and only if the associated map : D(V ) ! V satisfies the relation
X
ffi( )(fl(v1, . .,.vn)) + fl0 v_0, fl00(v_00) = 0, for fl(v1, . .,.*
*vn) 2.D(V )
(fl)2
Proof.This lemma can be proved directly or can be deduced from proposition A.2.*
*6.
To be precise, one can observes that the square of a coderivation of degree -1 *
*defines
a coderivation. Equivalently, one can deduce from the associativity of a cooper*
*ad
THE BAR CONSTRUCTION OF AN E-INFINITE ALGEBRA 51
coproduct that the composite @ @ agrees with the coderivation @ o associated to
the map o such that
X
o (fl(v1, . .,.vn)) = fl0 v_0, fl00(v_00) .
(fl)2
Similarly, one observes that the map ffi(@ ) agrees with the coderivations @ffi*
*(a)s-
sociated to the differential of . Therefore, the lemma is a consequence of the
relationship of proposition A.2.6. (Remark: compare our proof with the char-
acteristic 0 arguments of [15, Proposition 2.10], which involve the commutator
[@ , @ ] = @ @ + @ @ = 2@ @ .)
Then, we have the following result:
A.2.8. Observation (see proposition 2.15 in [15]). A morphism of dg-operads ae :
Bc(D ) ! End V is equivalent to a map : D(V ) ! V which satisfies the equation
of paragraph A.2.7 and such that the restriction |V vanishes.
Proof.Since Bc(D) = (F ( -1D"), @) is a quasi-free operad, an operad morphism
ae : Bc(D ) ! End V is determined by a map of *-modules ae : -1D" ! End V,
and hence by a homogeneous map : D (V ) ! V of degree -1, defined by
(fl(v1, . .,.vn)) = ae(fl)(v1, . .,.vn). We have (v) = 0 since ae is defined*
* on the
coaugmentation coideal of D.
Then, one checks readily that, for a generator fl 2 "D(n), the identity ffi(*
*ae(fl)) =
ae((ffi + @)(fl)) is equivalent to the equation of lemma A.2.7. Consequently, t*
*he map
ae : -1D" ! End V induces a morphism of dg-operads ae : (F ( -1D"), @) ! End V
if and only if the associated map : D(V ) ! V satisfies this equation.
A.2.9. Bar duality and morphisms of quasi-cofree coalgebras. Thus, according to
the observation above, a Bc(D )-algebra (V, ae) is equivalent to a quasi-cofree*
* coal-
gebra (D (V ), @ ). Moreover, one observes easily that a morphism of dg-modules
ff : U ! V defines a morphism of Bc(D )-algebras ff : (U, ss) ! (V, ae) if and *
*only
if the induced morphism ff : D(U) ! D (V ) defines a morphism of dg-coalgebras
ff : (D (U), @~) ! (D (V ), @ ), for the quasi-cofree coalgebras associated to *
*(U, ss)
and (V, ae). Thus, the category of Bc(D )-algebras is equivalent to the catego*
*ry
formed by quasi-cofree coalgebras (D (V ), @ ), together with the morphisms of *
*coal-
gebras OEff: (D (U), @~) ! (D (V ), @ ) which are induced by morphisms of dg-
modules ff : U ! V .
However, one deduces from the universal property of cofree coalgebras that m*
*or-
phisms of dg-coalgebras OEff: (D (U), @~) ! (D (V ), @ ) are associated to ma*
*ps
ff : D (U) ! V , such that ff does not necessarily vanish on "D(U) D (U). (We
make this relationship more explicit in the next statements.) We will observe t*
*hat
these morphisms give morphisms in the homotopy category of Bc(D )-algebras.
As for coderivations, we do not recall the definition of a morphism of coalg*
*ebras
over a cooperad: for our purposes, we can take the following assertion as a def*
*inition.
A.2.10. Proposition. We have a one-to-one correspondence between morphisms of
cofree coalgebras OE : D(U) ! D (V ) and morphisms of dg-modules ff : D(U) ! V .
The map ff associated to a morphism OE is given by the composite of OE : D(U) !
D (V ) with the projection D(V ) ! V .
52 BENOIT FRESSE
Conversely, the morphism associated to ff, also denoted by OE = OEff, is dete*
*rmined
by the formula
X
OEff(fl(u1, . .,.un)) = fl0 ff(fl001(u_1)), . .,.ff(fl00r(u_r)) ,
(fl)
for any fl(u1, . .,.un) 2,D(U)
where u_i2 V nidenote appropriate groupings of variables.
P
To be precise, recall that ae(fl) = (fl)w .fl0(fl001, . .,.fl00r) denotes t*
*he coproduct of
an element fl 2 D, as in paragraph A.2.1. The permutation w is performed on the
tensor power V n, and therefore, does not appear explicitly in the formula abo*
*ve
(as in the formula of proposition A.2.6). Hence, the groupings u_i2 V niwhich
occur in the formula are defined by the tensor decomposition w*(u1 . . .un) =
uw(1) . . .uw(n)= u_1 . . .u_r.
A.2.11. Lemma. A morphism of cofree coalgebras OEff: D (U) ! D (V ) satisfies
the relation (ffi + @ )OEff= OEff(@~ + ffi) and hence, defines a morphism of qu*
*asi-cofree
coalgebras OEff: (D (U), @~) ! (D (V ), @ ), if and only if we have the relation
X
ffi(ff)(fl(u1, . .,.un)) + fl0 fffl001(u_1), . .,.fffl00r(u_r)
(fl)
X
- fffl0 u_0, ~fl00(u_00) - ff(~fl(u1, . .,.un)) = 0,
(fl)2
for fl(u1, . .,.un) 2.D(U)
Proof.Like lemma A.2.7, the assertion above can be proved directly or can be
deduced from a suitable generalization of proposition A.2.6. To be precise, li*
*ke
morphisms and coderivations, the maps @ OEff, OEff@~ and ffi(OEff) = ffiOEff- O*
*Effffi can be
written in term of their projection D(U) ! V , which are precisely represented *
*by
terms of the equation above. The lemma follows from this observation.
In fact, explicit formulas for @ OEff, OEff@~ and ffi(OEff) can be deduced f*
*rom the
associativity of the cooperad coproduct. We omit this straightforward verificat*
*ion,
since these formulas are not used elsewhere in the article.
A.2.12. Construction of quasi-free resolutions. In the following paragraphs, we*
* as-
sume that (D (U), @~) and (D (V ), @ ) are quasi-free coalgebras associated to *
*Bc(D )-
algebras, (U, ss) and (V, ae) respectively, and we aim to prove that a morphism*
* of
quasi-free coalgebras OEff: (D (U), @~) ! (D (V ), @ ) yields a morphism in the*
* ho-
motopy category of Bc(D )-algebras between (U, ss) and (V, ae).
Let Q = Bc(D ). For our purpose, we associate first a quasi-free Q -algebra
FQ (D (V ), @ ) to any quasi-cofree coalgebra (D (V ), @ ). Explicitly, we con*
*sider
the free Q-algebra Q(D (V ), @ ) generated by the underlying dg-module of the p*
*air
(D (V ), @ ). Then, we consider the derivation @D : Q(D (V )) ! Q (D (V )) indu*
*ced
by the composite map
D(V@)___ae//_A_D(D (V_))//_QOO(D,(V ))
_______________________aeoe|
@D|D(V )
THE BAR CONSTRUCTION OF AN E-INFINITE ALGEBRA 53
where the first map is induced by the cooperad coproduct ae : D ! D O D and the
second one by the canonical morphism
D ! "D '-! -1D" ,! F ( -1D") = Q .
Equivalently,Pfor a generator fl(v1, . .,.vn) 2 D (V ), we perform the coproduct
ae(fl) = (fl)w . fl0(fl001, . .,.fl00r) of fl 2 D (n). Then, we let q0 2 Q (*
*r) denote the
image of the root factor of this coproduct fl02 D(r) in Q = F( -1D"), and we set
X
@D (fl(v1, . .,.vn)) = q0(fl001(v_1), . .,.fl00r(v_r)),
(fl)
where v_1, . .,.v_rdenote appropriate groupings of variables.
A.2.13. Claim. The derivation above @D : Q(D (V ), @ ) ! Q (D (V ), @ ) commutes
with the internal differential ffi and with @ . Moreover, if we let @Q denote*
* the
differential of the cobar construction Q = Bc(D ) defined in paragraph A.2.2, t*
*hen
we have the identity @Q @D + @D @Q + @D @D = 0.
Proof.The commutation relation ffi@D + @D ffi = 0 is immediate. For a generator
fl(v1, . .,.vn) 2 D(V ), the relations
(@ @D + @D @ )(fl(v1, . .,.vn)) = 0 and (@Q @D + @D @D )(fl(v1, . .,.vn)*
*) = 0
can be deduced from the associativity of the cooperad coproduct by a straightfo*
*r-
ward (and omitted) verification. This claim implies that @D satisfies the relat*
*ions
@ @D + @D @ = 0 and @Q @D + @D @Q + @D @D = 0 on Q(D (V )).
As a consequence, we obtain the following result:
A.2.14. Proposition. The free Q-algebra Q(D (V ), @ ) can be equipped with a to*
*tal
differential given by the sum ffi + @ + @Q + @D . The quasi-free Q -algebra re*
*sulting
from this construction is denoted by FQ (D (V ), @ ).
In addition:
A.2.15. Proposition. We assume |V = 0 as in observation A.2.8. The quasi-free
algebra FQ (D (V ), @ ) is a cofibrant Q -algebra if the dg-module V is cofibra*
*nt and
the cooperad D is *-cofibrant.
Proof.We equip the dg-module D(V ) with the filtration such that
M
skdD (V ) = (D (r) V r) r.
r d
We have then @D (skdD (V )) Q(skd-1D (V )) and similarly, the assumption |V =
0 implies @ (skdD (V )) skd-1D (V ), because for a non-trivial composite w .
fl0(fl001, . .,.fl00r) 2 D O D(n), the factors fl00i2 D (ni) satisfy ni < n and*
* we have
also r < n. Moreover, the assumptions imply that each dg-module (D (r) V r) r
is cofibrant. The proposition follows.
A.2.16. Claim. Let (V, ae) denote an algebra over Q = Bc(D ) and consider the
associated quasi-cofree coalgebra (D (V ), @ ). The canonical projection r : D(*
*V ) !
V induces a morphism of Q -algebras
r : FQ (D (V ), @ ) ! (V, ae).
The canonical inclusion i : V ! Q(D (V )) defines a morphism of dg-modules
i : V ! FQ (D (V ), @ ) such that ri = Id.
54 BENOIT FRESSE
Proof.We prove that the algebra morphism r : Q (D (V )) ! V specified in the
claim satisfies the identity r(@Q + @ + @D ) = 0. Since this map commutes
clearly with internal differentials, we conclude that r maps the total differen*
*tial of
FQ (D (V ), @ ) to the differential of V and hence, defines a morphism of dg-al*
*gebras
r : FQ (D (V ), @ ) ! (V, ae), as claimed.
We can check the identity above r(@Q + @ + @D ) = 0 for a generator x =
fl(v1, . .,.vn) 2 D(V ). By definition, we have r(x) = v, for x = v 2 V , and r*
*(x) = 0,
for generators x = fl(v1, . .,.vn) 2 D(V ) such that fl 2 "D. The identity is c*
*learly
satisfied for x = v, since in this case all differential vanishes. In the other*
* case x =
fl(v1, . .,.vn) 2 D(V ), all components of the differential @ (fl(v1, . .,.vn))*
* 2 D(V )
are cancelled by r, except (fl(v1, . .,.vn)) 2 V . (Notice that we assume impl*
*icitly
|V = 0.) Consequently, we obtain
r@ (fl(v1, . .,.vn)) = (fl(v1, . .,.vn)) = ae(fl)(v1, . .,.vn).
Similarly, all components of the differential @D (fl(v1, . .,.vn)) 2 D(V ) are *
*cancelled
by r, except the term q(v1, . .,.vn) 2 Q(V ), where q 2 Q(n) denotes the image *
*of
fl 2 "Dunder the canonical map "D,! F ( -1D"). The morphism r : Q(D (V )) !
V maps the product q(v1, . .,.vn) to the corresponding operation in V , which is
represented by the expression ae(fl)(v1, . .,.vn) 2 V . Thus, we obtain
r@ (fl(v1, . .,.vn)) = r@D (fl(v1, . .,.vn))
and our assertion follows from this relation, since the derivation @Q does not *
*occur
for generators.
The second assertion of the claim is immediate since the differentials @Q , @*
* and
@D vanish on i(v) = v 2 V .
Remark. The constructions above can be compared with the setting of [15, Propo-
sition 2.18]. The map r : FQ (D (V ), @ ) ! (V, ae) is the adjunction augmentat*
*ion
of loc. cit..
A.2.17. Lemma (compare with theorem 2.19 in [15]). If the cooperad D is *-
cofibrant, then the morphism r : Q (D (V ), @ ) ! V defines a weak-equivalence *
*of
Q -algebras.
Proof.We consider in this proof the functor V 7! M(V ) associated to a *-module
M. Recall that 1
M
M(V ) = (M(r) V r) r.
r=0
Let us equip this functor with the increasing filtration
M
FdM(V ) = (M(r) V r) r.
r d
The functor V 7! Q(D (V )) is associated to a composite *-module Q O D (see [1*
*3,
x1.2]), and for M(V ) = FP (D (V ), @ ), the filtration above gives rise to a r*
*ight-hand
half-plane homological spectral sequence Er(FP (D (V ), @ )) such that
(E0, d0) = (Q (D (V )), @Q + @D ).
The map i : V ! Q(D (V )) is induced by a morphism of dg- *-modules i :
I ! Bc(I, D, D), where I denote the *-module such that I(V ) = V , the identity
functor. Clearly, this map preserves filtrations and yields a morphism from the
trivial spectral sequence E1(V ) = E2(V ) = . .=.H*(V ) to Er(FP (D (V ), @ )).
THE BAR CONSTRUCTION OF AN E-INFINITE ALGEBRA 55
We prove that i induces an isomorphism at the E1 level of the spectral sequence*
*s.
Consequently, the spectral sequence Er(FP (D (V ), @ )) degenerates and hence c*
*on-
verges. Then, we can also conclude that i induces an isomorphism i : H*(V ) -'!
H*(FP (D (V ), @ )). The lemma follows since r is left-inverse to i.
One can observe precisely that the dg-module Q(D (V )) together with the diff*
*er-
ential ffi + @Q + @D can be identified with the functor (M(V ), @) associated t*
*o the
dg- *-module (M, @) = Bc(I, D, D) defined in [13, x4.8.1]. Furthermore, accord-
ing to loc. cit., the morphism i : I ! Bc(I, D, D) defines a weak-equivalence of
dg- *-modules for any cooperad D . Actually, for a Z-graded cooperad, the proof
of this assertion requires the arguments of [15]. To be precise, one can deduce
this weak-equivalence from the chain-homotopy introduced in the proof of theo-
rem 2.19 in loc. cit.. (Notice that the spectral sequence involved in this pr*
*oof
converges, because, for fixed n 2 N , the module Bc(I, D, D)(n) has a bounded
filtration.) One proves easily that Bc(I, D, D) forms a cofibrant object in the
category of *-modules if the cooperad D satisfies this property. Consequently,
under this assumption, the map i : I ! Bc(I, D, D) induces a weak-equivalence
of dg-modules i : V ! (Q (D (V )), @Q + @D ) for any dg-module V . Thus, we co*
*n-
clude that i induces an isomorphism from E1(V ) = H*(V ) to E1(FP (D (V ), @ ))*
* =
H*(Q (D (V )), @Q + @D ).
The following immediate observation permits to achieve the aim of this sectio*
*n.
A.2.18. Observation. The map (D (V ), @ ) 7! Q(D (V ), @ ) defines a functor fr*
*om
the category of quasi-cofree coalgebras over D to the category of Q -algebras. *
* Ex-
plicitly, any morphism of dg-coalgebras OEff: (D (U), @~) ! (D (V ), @ ) yiel*
*ds a
morphism of quasi-free Q -algebras OEff: Q(D (U), @~) ! Q (D (V ), @ ).
To be more precise, from lemma A.2.17 we conclude:
A.2.19. Proposition. Let D be a *-cofibrant cooperad. Suppose given a morphism
of quasi-cofree coalgebras OEff: (D (U), @~) ! (D (V ), @ ), where (D (U), @~) *
*and
(D (V ), @ ) denote quasi-cofree coalgebras associated to Bc(D )-algebras, (U, *
*ss) and
(V, ae) respectively. (Notice that we assume implicitly ~|U = |V = 0.)
Let Q = Bc(D ). The morphism of Q -algebras induced by OEfffits in a diagram
FQ (D (U), @~)OEff//_FQ (D (V ),,@ )
|~| |~|
fflffl| fflffl|
(U, ss) (V, ae)
in which the vertical maps are weak-equivalences of Q -algebras. Accordingly, t*
*his
map yields a morphism from (U, ss) to (V, ae) in the homotopy category of Q-alg*
*ebras.
We record the following statement for the purposes of section 2 and section *
*A.3.
A.2.20. Lemma. Assume that D is a *-cofibrant operad. Let OEff: (D (U), @~) !
(D (V ), @ ) be a morphism of quasi-cofree coalgebras, as in proposition A.2.19*
* above,
associated to a map ff : D(U) ! V .
If the restriction ff|U defines a weak-equivalence of dg-modules ff|U : U -~*
*! V ,
then OEffinduces a weak-equivalence of Q -algebras
OEff: FQ (D (U), @~) -~!FQ (D (V ), @ ).
56 BENOIT FRESSE
Proof.Clearly, we have a commutative diagram of dg-modules
FQ (D (U),O@~)OEff//_FQO(DO(VO), @ )
~ i|| ~ |i|
| ff|U |
(U, ss)__________//_(V, ae)
in which all vertical maps are equivalences by lemma A.2.17. The lemma follows
from this observation.
A.2.21. Operadic bar duality. The bar construction of an operad is a cooperad
D = B(P) defined dually to the cobar construction of a cooperad. One proves
that the composite cobar-bar construction Bc(B(P)) is endowed with a canonical
operad morphism ffl : Bc(B(P)) -~! P which is a weak-equivalence for any operad
P. For a non-negatively graded operad, this weak-equivalence can be deduced from
the arguments of [13, x4.8]. The proof of [16, Theorem 3.2.16] works for a Z-gr*
*aded
operad. As consequence, the construction Q = Bc(B(P)) supplies a canonical
quasi-free resolution for any operad P. Let us mention that Q = Bc(B(P)) is
cofibrant if the operad P is *-cofibrant.
The map ffl : Q -! P yields a restriction functor ffl! : P Alg ! Q Alg, whi*
*ch
associates to any P-algebra A the same dg-module with the Q-algebra structure o*
*b-
tained by restriction through ffl. Consider the extension functor ffl!: P Alg !*
* Q Alg
which represents a left-adjoint of ffl!. For a quasi-free algebra F = (Q (V ), *
*@), we have
ffl!F = P(V ) together with the derivation induced by the composite map V @|V-*
*-!
Q (V ) -ffl!P(V ). In particular, for the quasi-free algebra F = FQ (D (V ), @*
* ), we
obtain ffl!F = FP (D (V ), @ ), where FP (D (V ), @ ) = P(D (V ), @ ).
For our purposes, we record the following statement:
A.2.22. Lemma. Assume that P is a cofibrant operad. Let D = B(P) denote
the associated bar construction, and Q = Bc(D ) denote the associated cobar-bar
construction.
a. The morphism ffl : Q -~! P induces a natural weak-equivalence of Q-algebr*
*as
FQ (D (V ), @ ) -~!FP (D (V ), @ )
that commutes with any morphism of quasi-cofree coalgebras as in lemma
A.2.20.
b. The morphism r : FQ (D (V ), @ ) ~-!(V, ae) factors through FP (D (V ), *
*@ )
by a weak-equivalence of P-algebras. Consequently, we have a commutative
diagram
FQ (D (V ), @_)____~_______//FP (D (V ),,@ )
MM qq
MMM qqq
~MMMM&&M xxqqq~qq
(V, ae)
in which all morphisms are weak-equivalences.
Proof.The morphism FQ (D (V ), @ ) ! (V, ae) factors clearly through FP (D (V )*
*, @ )
by adjunction. One can adapt the proof of lemma A.2.17 in order to prove that t*
*he
resulting morphism of P-algebras r : FP (D (V ), @ ) ! (V, ae) is a weak-equiva*
*lence.
Explicitly, one observes that the functor V 7! (P(D (V ), @ ), @D ) is associat*
*ed to
THE BAR CONSTRUCTION OF AN E-INFINITE ALGEBRA 57
the *-module B(P, P, I) defined in [13, x4.4]. As in lemma A.2.17, we have a w*
*eak-
equivalence of *-modules i : I -~! B(P, P, I) and we conclude by the same argu-
ments that i induces a weak-equivalence of dg-modules i : (V, ae) ! FP (D (V ),*
* @ )
provided that P is *-cofibrant. The claim follows since r is left-inverse to *
*this
map.
One deduce from this result that the map FQ (D (V ), @ ) ! FP (D (V ), @ ) is*
* also
a weak-equivalence. Thus, we are done.
A.3. Transfer of operad actions. In this section, we prove that the action of t*
*he
cobar operad Q = Bc(D ) can be transferred through strong deformation retracts.
In fact, the transferred operad action is defined by effective formulas and, as*
* a
consequence, carries the functoriality required for the construction of section*
* 3.
For this purpose, we generalize the classical inductive construction of [19] in*
* the
framework of operads. Let us mention that the arguments of [17, 18] involving t*
*he
basic perturbation lemma should not work in our situation. To be precise, the t*
*ensor
trick of loc. cit. can hardly be generalized in the framework of symmetric oper*
*ads
(over a field of positive characteristic), because of the equivariance requirem*
*ents.
We refer to [9] for a short historical overview of perturbation techniques. We *
*refer
to [7] and [28] for other transfer arguments in the context of operads.
A.3.1. Transfer data. Let D be a cooperad. We are given a strong deformation
retract
oor__ __________________________________________*
*_______________
U __i__//Vffh______________________________________*
*__________________________,
where the chain-homotopy h, such that ffih+hffi = ir-Id, satisfies the side con*
*ditions
hi = rh = hh = 0, and a differential @ * : D (V ) ! D (V ) associated to a map
* : D (V ) ! V such that *|V = 0. Hence, according to observation A.2.8, the
differential @ * is equivalent to an operad morphism ae : Bc(D ) ! End V, which
provides the dg-module V with the structure of an algebra over Q = Bc(D ).
As mentioned above, the purpose of this section is to prove that this Q-alge*
*bra
structure can be transferred to U through the deformation retract above. More
explicitly, in the next paragraph, we define a map ~* : D (U) ! U such that
(D (U), @~*) defines a quasi-cofree coalgebra, and a map i* : D(U) ! V which yi*
*elds
a morphism of quasi-cofree coalgebras OEi* : (D (U), @~*) ! (D (V ), @ *). Mor*
*e-
over, we observe that the induced morphism of Q-algebras OEi*: FQ (D (U), @~*) !
FQ (D (V ), @ *) is a weak-equivalence provided that D is *-cofibrant. Consequ*
*ently,
the Q-algebra (U, ss) determined by the pair (D (U), @~*) is related to (V, ae)*
* by weak-
equivalences of Q-algebras.
In our notation, the `*' refers to the weight grading of the functor D(V ). *
*Explic-
itly, we consider the homogeneous components Dr(V ) = (D (r) V r) r of D(V ),
and we let r denote the restriction of * to Dr(V ). We adopt similar conventi*
*ons
for the maps ~* and i*. In fact, the maps ~* and i* are defined by induction on
* 1.
58 BENOIT FRESSE
A.3.2. Transfer construction. Let i* : D (U) ! V and ~* : D (U) ! U be the
maps defined recursively by i1 = i, ~1 = 0, and
in(fl(u1, . .,.un)) = h(~nfl(u1, . .,.un))
~n(fl(u1, . .,.un)) = r(~nfl(u1, . .,.un))
X
where ~nfl(u1, . .,.un) = *fl0 i*fl001(u_1), . .,.i*fl00r*
*(u_r) .
(fl)
(Let us mention that this construction differs from [19] because we assume *|V*
* =
0.) Notice that i* and ~* depends functorially of the transfer data of para-
graph A.3.1.
A.3.3. Claim. The coderivation @~* : D(U) ! D (U) associated to ~* : D(U) ! U
satisfies the equation of lemma A.2.7, so that the pair (D (U), @~*) defines a *
*quasi-
cofree coalgebra. The map i* : D(U) ! V satisfies the equation of lemma A.2.11
and hence yields a morphism of quasi-cofree coalgebras
OEi*: (D (U), @~*) ! (D (V ), @ *).
Notice that i*|U = i by construction.
Proof.As in [19], this claim follows from a tedious but straightforward inducti*
*ve
verification left to the reader.
A.3.4. Lemma. We assume now that D is a *-cofibrant cooperad. If we let Q =
Bc(D ), then the construction of paragraph A.3.2 gives rise to a diagram of Q -
algebras
FQ (D (U), @~)~__//FQ (D (V ),,@ )
|~| |~|
fflffl| fflffl|
(U, ss) (V, ae)
which depends functorially of the transfer data of paragraph A.3.1, and where a*
*ll
morphisms are weak-equivalences.
Proof.The claim is a direct corollary of lemma A.2.17 and lemma A.2.20.
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Universit'e de Lille 1, UMR 8524 du CNRS, 59655 Villeneuve d'Ascq C'edex (Fr*
*ance)
E-mail address: Benoit.Fresse@math.univ-lille1.fr
URL: http://math.univ-lille1.fr/~fresse