THE HOMOTOPY THEORY OF E1 ALGEBRAS
MICHAEL A. MANDELL
Abstract.Let k be a commutative ring and let C be the operad of differen*
*tial
graded kmodules obtained as the singular kchains of the linear isometr*
*ies
operad [4, xV.9]. We show that the category of Calgebras is a proper cl*
*osed
model category. We use the amenable description of the coproduct in this
category [4, V.3.4] to analyze the coproduct of and develop a homotopy t*
*heory
for algebras over an arbitrary E1 operad. Draft: January 26, 1998, 17:26.
Introduction
Recent years have brought increasing interest in operads and in algebras over
operads. The monograph [4] provides an exposition of the basic theory of operads
and makes a particular study of the algebras over E1 operads and the modules ov*
*er
these algebras. For a particular E1 operad C, [4] obtains a good homotopy theo*
*ry
on the categories of modules. In this paper, we study the "homotopy theory" of
E1 algebras themselves.
Ideally, we would like to show that for an E1 operad E, the category of E
algebras forms a closed model category. Although we have not succeeded in this
for a general E1 operad, we do provide a closed model structure on the category
of Calgebras, for the E1 operad C of [4, xV.9]. In general, we prove that the
category of Ealgebras has many of the useful properties expected from a closed
model category. We construct and make sense of "homotopies" of E1 algebras;
these allow us to analyze the homotopical properties of adjoint functors as done
for model categories in [1, 7]. In addition, we analyze the coproducts and push*
*outs
of E1 algebras and find sufficient conditions for them to behave as expected: *
*we
prove that in favorable cases the pushout has homology given by the "E1 torsion
product" of [4].
We present an application of this theory in [5], to provide a characteristic p
version of Sullivan's theorem comparing the unstable rational homotopy category*
* to
the category of commutative differential graded algebras over the rational numb*
*ers.
Over the rational numbers, E1 algebras are essentially the same as commutative
differential graded algebras; for a precise statement see [4, II.1.5]. Over a f*
*ield of
characteristic p, the category of commutative differential graded algebras cann*
*ot
have a good homotopy theory; for example, the free commutative differential gra*
*ded
algebra on a contractible differential graded module is never quasiisomorphic *
*to the
ground field. On the other hand, the homotopy theory we develop in this paper f*
*or
categories of E1 algebras allows us to deduce in [5] that the homotopy categor*
*y of
connected pcomplete nilpotent spaces of finite ptype are contravariantly equi*
*valent
to a full subcategory of the homotopy category of E1 Fpalgebras, where Fpdenot*
*es
the algebraic closure of the field with p elements.
____________
1991 Mathematics Subject Classification. Primary 18G55; Secondary 18G15.
The author was supported by an Alfred P. Sloan Dissertation Fellowship.
1
2 MICHAEL A. MANDELL
1. Summary of Results
We take this section to summarize for easy reference the main results proved *
*in
this paper. Throughout, k denotes a fixed but arbitrary commutative ring that we
take as ground ring. All constructions are formed in the category of kmodules,
differential graded kmodules, or their more structured subcategories. In parti*
*cular
"" always denotes the tensor product over k.
We begin with the operad C of [4, xV.9]. For C, the fundamental result is the
following theorem, proved in Section 3.
Theorem 1.1. Let k be a commutative ring and let C be the E1 operad of differ
ential graded kmodules of [4, xV.9]. The category of Calgebras is a proper cl*
*osed
model category, with weak equivalences the quasiisomorphisms and fibrations the
surjections.
The cofibrations in this category also admit a straightforward description as
the retracts of "relative cell inclusions". The relative cell inclusions are th*
*e maps
obtained by a cell attachment process:
Definition 1.2.Let G be an operad and let G denote the free Galgebra functor.
A map of Galgebras f :A ! B is relative cell Galgebra inclusion if there exis*
*ts a
sequence of Galgebra maps A = A0 i0!A1 i1!. .s.uch that
(i)B ~=Colimin under A.
(ii)Each map in is formed as a pushout of Galgebras
GXn _____//GCXn
 
 
fflffl fflffl
An __in__//_An+1
where Xn is a free differential graded kmodule with zero differential, CX*
*n is
the cone on Xn [4, p. 58], and Xn ! CXn is the canonical inclusion.
We say that a Galgebra A is a cell Galgebra if the initial map G(0) ! A is a
relative cell inclusion.
As mentioned in the introduction, we are presently unable to obtain closed mo*
*del
structures on the categories of algebras over E1 operads other than C. The prob*
*lem
is the difficulty in analyzing the coproduct. To explain this, consider the sim*
*plest
contractible differential graded kmodule Ck[n] that consists of k in degrees n*
* and
n + 1 and zero in all other degrees. The obstruction for the existence of a clo*
*sed
model structure on the category of Galgebras is the behavior of the coproduct *
*of
GCk[n] with arbitrary Galgebras. In Section 2, we prove the following proposit*
*ion.
Proposition 1.3.Let G be an operad. The following are equivalent.
(i)For every Galgebra A and every integer n, the inclusion A ! A q GCk[n] is
a quasiisomorphism.
(ii)The category of Galgebras is a closed model category with weak equivalenc*
*es
the quasiisomorphisms and fibrations the surjections.
We are able to verify condition 1.3.(i) for the E1 operad C, but not for any
other E1 operad. On the other hand, if we restrict to the case of cell algebra*
*s, we
can analyze coproducts in the category of Ealgebras for an arbitrary E1 operad
E. The following theorem is proved in Section 6.
THE HOMOTOPY THEORY OF E1 ALGEBRAS 3
Theorem 1.4. Let E be an E1 operad. If A and B are cell Ealgebras then there
is a canonical isomorphism H*(A q B) ~=Tor*(A; B).
We can also prove a relative version of the previous result. Part V of [4] d*
*e
scribes a generalization of the differential torsion product over a differentia*
*l graded
kalgebra to an "E1 torsion product" over an E1 kalgebra. For a Calgebra,
this is constructed directly; for an E1 algebra over a different E1 operad, t*
*his is
constructed by first replacing by an equivalent Calgebra. We prove the followi*
*ng
theorem in Section 6.
Theorem 1.5. Let E be an E1 operad. If A is a cell Ealgebra and A ! B
and A ! C are relative cell inclusions, then there is a canonical isomorphism
H*(B qA C) ~=TorA*(B; C).
An EilenbergMoore spectral sequence for the calculation of this E1 torsion
product is given in [4, V.7.3]. In certain cases, this type of spectral sequenc*
*e can be
constructed directly from the Ealgebras, using the bar construction. For Ealg*
*ebras
A, B, C and Ealgebra maps A ! B, A ! C, we can form a simplicial Ealgebra
fiEo(B; A; C) that in degree n is given by
fiEn(B; A; C) = B q A_q_._.q.Az____"qC:
n factors
The face maps are induced by the codiagonal maps AqA ! A, BqA ! BqB ! B,
and A q C ! C q C ! C. The degeneracies are induced by the inclusions of
the form X q Y ! X q A q Y for each factor of A in the coproduct. From a
simplicial differential graded kmodule, we obtain a differential graded kmodu*
*le
by normalization (called "totalization" in [4, xII.V]), the obvious generalizat*
*ion
of forming the normalized chain complex of a simplicial abelian group. In the
case of a simplicial Ealgebra, the normalization obtains a Ealgebra structure*
* via
the shuffle map [4, p. 51]. Write fiE(B; A; C) for the Ealgebra obtained as t*
*he
normalization of fiEo(B; A; C). Regarding B qA C as a constant simplicial Ealg*
*ebra,
the natural map B q C ! B qA C induces a natural map of simplicial Ealgebras
fiEo(B; A; C) ! B qA C and therefore a map of Ealgebras fiE(B; A; C) ! B qA C.
In Section 7 we prove the following theorem giving a sufficient condition for t*
*his
map to be a quasiisomorphism.
Theorem 1.6. Let E be an E1 operad. If A, B, and C are cell Ealgebras and
A ! B and A ! C are relative cell inclusions, then the natural map fiE(B; A; C)*
* !
B qA C is a quasiisomorphism.
Since fiE(B; A; C) is the normalization of a simplicial differential graded k
module, it has a canonical filtration by differential graded kmodules. We ther*
*efore
obtain from Theorem 1.6 a strongly convergent spectral sequence for the calcula
tion of H*(B qA C). When k is a field or when H*A and H*B are flat kmodules,
Theorem 1.4 allows the easy identification of the E2term as TorH*A*;*(H*B; H*C*
*).
In order to apply the previous theorems in general, we need to be able to rep*
*lace
arbitrary Ealgebras with quasiisomorphic cell Ealgebras and to replace arbit*
*rary
maps with quasiisomorphic relative cell inclusions. For this, we provide the *
*fol
lowing observation, proved in Section 2 by the small object argument.
4 MICHAEL A. MANDELL
Proposition 1.7.If A ! B and A ! C are maps of Ealgebras, then we can form
a commutative diagram
B0 oo__oA0o__//_//_C0
  
~ ~ ~
fflfflfflfflfflfflfflfflfflfflfflffl
B oo___A_______//C
~
where A0, B0, and C0 are cell Ealgebras, the maps labeled "i " are surjective *
*quasi
isomorphisms, and the maps labeled "ae" are relative cell inclusions.
The use of the symbol "~" and the arrows "ae", "i" in the previous proposition
provide a convenient shorthand notation in diagrams. We formalize this here for
use in the rest of this paper.
Notation 1.8. The arrow "ae" indicates a map that is known to be or is assumed
to be a relative cell inclusion. The arrow "i" indicates a map that is known to*
* be
or is assumed to be a fibration. The symbol "~" decorating an arrow indicates a
map that is known to be or is assumed to be a quasiisomorphism.
Although not enough to establish a model category structure, these results on
coproducts are sufficient to develop a notion of (Quillen left) homotopy in the
category for which cell Ealgebras satisfy the analogue of the Whitehead theore*
*m.
Aside from this, the most useful feature of a closed model structure would prov*
*ide is
the criterion [7, Theorem 43] for an adjoint pair of functors to induce an adj*
*unction
on homotopy categories; we show in Section 5 that the categories of E1 algebras
satisfy the following version of this theorem.
Theorem 1.9. Let L: E ! M and R: M ! E be left and right adjoints between
the category E of algebras over an E1 operad E and a closed model category M .
(i)If L converts relative cell inclusions between cell Ealgebras to cofibrat*
*ions and
R converts fibrations to surjections, then the left derived functor of L a*
*nd the
right derived functor of R exist and are adjoint. Moreover, L converts qua*
*si
isomorphisms between cell Ealgebras to weak equivalences, and the restric*
*tion
of the left derived functor of L to the cell Ealgebras is naturally isomo*
*rphic
to the derived functor of the restriction of L.
(ii)Suppose that (i) holds and in addition for any cell Ealgebra A and any fi*
*brant
object Y in M , a map A ! RY is a quasiisomorphism if and only if the
adjoint LA ! Y is a weak equivalence. Then the left derived functor of L
and the right derived functor of R are inverse equivalences.
We have in addition the following useful variants. For simplicity of statemen*
*t, we
say that a map is an acyclic relative cell inclusion if it is both a relative c*
*ell inclusion
and a quasiisomorphism. Likewise, we say that a map is an acyclic surjection i*
*f it
is both a surjection and a quasiisomorphism.
Theorem 1.10. The hypothesis of 1.9.(i) is equivalent to each of the following.
(i)L converts relative cell inclusions between cell Ealgebras to cofibration*
*s and
acyclic relative cell inclusions between cell Ealgebras to acyclic cofibr*
*ations.
(ii)R converts fibrations to surjections and acyclic fibrations to acyclic sur*
*jec
tions.
A dual version of Theorem 1.9 also holds with slightly stronger hypotheses. T*
*he
exact statement is given in Section 5 as Theorem 5.3.
THE HOMOTOPY THEORY OF E1 ALGEBRAS 5
Although we are primarily interested in E1 algebras, Proposition 1.3 can be u*
*sed
to obtain closed model category structures on categories of algebras over other*
* kinds
of operads. The following theorem proved in Section 13 implies in particular th*
*at
when G is an A1 operad, the category of Galgebras is a closed model category.
The proof is independent of the work on E1 algebras of Sections 312.
Theorem 1.11. If G is the operad associated to a nonSigma operad [4, I.1.2.(i*
*)],
then the category of Galgebras is a closed model category with weak equivalenc*
*es
the quasiisomorphisms and fibrations the surjections.
2. Relative Cell Inclusions and the Small Object Argument
In this section, G denotes an arbitrary operad of differential graded kmodul*
*es.
We prove Propositions 1.3 and 1.7 and develop some of the properties of relative
cell inclusions needed for the proof of the remaining results. The main step is*
* given
by Quillen's small object argument [7, p. II.3.34]. We use it in the following*
* form.
Proposition 2.1.A map of Galgebras f :A ! B can be factored f = p O i where
i is a relative cell inclusion and p is an acyclic surjection.
Proof.Let A0 = A and let p0: A0 ! B be the map f. We construct An+1, pn+1
inductively as follows. We form An+1 as a pushout An qGXn GCXn, where Xn is
a free differential graded kmodule with zero differential. We choose Xn to hav*
*e a
generator in degree m for each pair (x; y) where x is a degree m element of An *
*and
y is an element of B whose differential is pnx. The map GXn ! An is induced by
the tautological map Xn ! An, and the map pn+1: An+1 ! B is induced by the
tautological map CXn ! B: the maps that send a basis element to the element x
or y that indexes it. Write A0for ColimAn. Let i: A ! A0be the inclusion, and l*
*et
p: A0! B be the colimit of the maps pn. By definition i is a relative cell incl*
*usion.
The map p1 contains in its image all the cycles of B, and so p2 is a surjection.
The map p is therefore both surjective and surjective on homology. It is inject*
*ive
on homology since H*A0= ColimH*An and the kernel of the map H*An ! H*B __
maps to zero in H*An+1. __
In commuting homology with the sequential colimit of Galgebras in the previo*
*us
proof, we have implicitly used the following fact.
Proposition 2.2.Filtered colimits of Galgebras are formed in the category of d*
*if
ferential graded kmodules.
Proof.The monad G that defines Galgebras commutes with filtered colimits. _*
*__
Proposition 1.7 is an immediate consequence of Proposition 2.1: Form A0 by
factoring the initial map G(0) ! A. We obtain B0, C0and the relative cell inclu*
*sions
A0! B0 and A0! C0 by factoring the composite maps A0! A ! B and A0!
A ! C. The following proposition implies that B0 and C0 are cell Galgebras.
Proposition 2.3.The composite of relative cell inclusions is a relative cell in*
*clu
sion.
Proof.The proposition is proved by rearranging the order in which the "cells" a*
*re
attached as in the proof of [4, III.1.2]. Consider relative cell inclusions A *
*! B
and B ! C. Choose An and Xn as in Definition 1.2, and choose Bn and Yn
such that B0 = B, Bn+1 = Bn qGYn GCYn and C ~=Colim Bn. Let Dm;0 = Am
6 MICHAEL A. MANDELL
and let Zm;0 = Xm . Then ColimDm;0 ~=B = B0; we form Dm;n+1 and Zm;n+1
with Bn+1 ~= ColimmDm;n+1 inductively as follows. Let D0;n+1= A. Choose
a basis for Yn+1. For each basis element y choose my such that the image of y
under the map Yn ! Bn is in the image of Dmy;n. Let Zm;n+1 be the direct
sum of Zm;n and the submodule of Yn generated by those basis elements y with
my = m, and let Dm+1;n+1 = Dm;n+1 qGZm;n+1GCZm;n+1. A check of universal
properties identifies Colimm Dm;n+1 as Bn+1. Let Zm = ColimnZm;n, D0 = A,
and Dm+1 = Dm qGZm GCZm . A check of universal properties identifies Dm as
ColimnDm;n and therefore identifies Colimm Dm as ColimnBn ~=C. This shows __
that the map A ! C is a relative cell inclusion. __
We begin the proof of Proposition 1.3 with the following fact. Recall [1, 3.1*
*.2]
that given a solid diagram of the form
A _____//X>>___
____
i_____p___
fflfflfflffl____
B _____//Y
the map i is said to have the left lifting property with respect to the map p i*
*f there
exists a map B ! X represented by the dotted arrow above that makes the whole
diagram commute.
Proposition 2.4.A map has the left lifting property with respect to the collect*
*ion
of acyclic surjections if and only if it is the retract of a relative cell incl*
*usion.
Proof.The retracts of relative cell inclusions have the left lifting property w*
*ith
respect to the acyclic surjections of Galgebras since the maps Xn ! CXn have
the left lifting property with respect to the acyclic surjections of differenti*
*al graded
kmodules.
On the other hand let f be a map that has the left lifting property with resp*
*ect
to the acyclic surjections of Galgebras. Factor f as pOi as in Proposition 2.1*
*. Then
we can find a lift in the following diagram
A _//i_//_A0>>_
_______
f ____~p__
fflfflfflfflfflffl_____
B __id_//B:
This expresses f as a retract of i, a relative cell inclusion. *
* ___
Proposition 2.1 gives one factorization required by the axioms of a closed mo*
*del
category. The following proposition provides the other factorization in the ca*
*se
when G satisfies condition 1.3.(i).
Proposition 2.5.A map of Galgebras f :A ! B can be factored f = pOi where p
is a surjection and i is a relative cell inclusion that has the left lifting pr*
*operty with
respect to the collection of surjections. If G satisfies 1.3.(i) then i can be *
*chosen to
be a quasiisomorphism.
Proof.Let X be the free differential graded kmodule with zero differential that
has one generator xb in dimension n for each element b of B in dimension n + 1.
We use Cxb to denote the unique element of the differential graded kmodule CX
whose differential is the image of xb under the canonical inclusion X ! CX. We
THE HOMOTOPY THEORY OF E1 ALGEBRAS 7
obtain a surjection of differential graded kmodules CX ! B, induced by sending
Cxb to b. The map A q GCX ! B is then a surjection. The map A ! A q GCX
clearly has the necessary lifting property. Writing A ! A q GCX as the composite
of A ! A q GX and A q GX ! A q GCX shows that it is a relative cell inclusion.
We show that the map A ! A q GCX is a quasiisomorphism when G satisfies
1.3.(i). Consider the filtered system of finite subsets of homogeneous el*
*ements
of B ordered by inclusion. Letting Xffbe the free differential graded kmodule *
*with
zero differential that has one generator xb in dimension n for each element b o*
*f Bff
that is in dimension n + 1 in B, we obtain a filtered system Aff= A q GCXffwhose
colimit is AqGCX. Any map in this system Aff! Afican be factored into a finite
sequence of maps,
Aff= Aff1! . .!.Affm= Afi
in which each Bffj+1has a single element not in Bffj. Then Affj+1~=Affjq GCk[n]
for some n depending on j. Since we are assuming that condition 1.3.(i) holds, *
*all
maps in the filtered system are quasiisomorphisms, and it follows that_t*
*he
map from A to the colimit A q GCX is a quasiisomorphism. __
The previous proposition allows the proof of the following lifting property b*
*y the
standard retraction argument, as in the proof of Proposition 2.4.
Proposition 2.6.Suppose G satisfies 1.3.(i). If f :A ! B is an acyclic relative
cell inclusion, then f satisfies the left lifting property with respect to the *
*collection
of surjections.
Proposition 1.3 is now an easy consequence.
Proof of Proposition 1.3.Assume that (i) holds. We take the cofibrations to be
the retracts of the relative cell inclusions; we need to verify MC15 of [1, p.*
* 12].
Conditions MC13 are wellknown. Condition MC4 is given by Propositions 2.4
and 2.6. Condition MC5 is given by Propositions 2.1 and 2.5.
Assume that (ii) holds. The map A ! A q GCk[n] has the left lifting prop
erty with respect to the collection of surjections, and must therefore_be_a qua*
*si
isomorphism. __
3. The Proof of Theorem 1.1
The proof of Theorem 1.1 relies heavily on the theory of modules over a Calg*
*ebra
developed in [4]. The category underlying these is the category of C(1)modules;
these are the modules for k regarded as a Calgebra. The special properties of *
*the
operad C allow the construction of a symmetric weakly monoidal product . For
C(1)modules M and N, there is a canonical map Tor*(M; N) ! H*(M N) that
is an isomorphism in favorable cases [4, V.1.9].
The product satisfies all the axioms of a symmetric monoidal product with the
minor modification that the given natural transformation k () ! Idis not an
isomorphism. To correct the minor difficulties this causes, [4] introduces the *
*cat
egory of "unital" C(1)modules. A unital C(1)module is a C(1)module M with a
chosen C(1)module map k ! M thought of as a unit. This allows the construction
of "mixed products" C, B and a "unital product" with stronger unit properties;
see [4, xV.2] for details. In particular, the product is a symmetric monoidal *
*prod
uct on the category of unital C(1)modules [4, V.2.11]. The commutative monoids
8 MICHAEL A. MANDELL
for are exactly the Calgebras [4, V.3.9], and therefore provides the coprodu*
*ct
of Calgebras.
For a Calgebra A, an Amodule is a C(1)module M with a suitably associative
and unital map A C M ! M [4, V.3.3]. The category of Amodules has a sym
metric weak monoidal product A , and the E1 torsion product TorA*(M; N) of
Amodules M and N is defined as the homology of the left derived functor of A .
Note that for A = k, "Tork*(M; N)" in this sense is canonically isomorphic to t*
*he
usual differential torsion product of the underlying differential graded kmodu*
*les
[4, V.1.9].
A unital Amodule is an Amodule M with a chosen Amodule map A ! M.
A unital version A of the A product makes the category of unital Amodules
symmetric monoidal. For unital Amodules M and N, there is a canonical map
M A N ! M A N and therefore a canonical map TorA*(M; N) ! H*(M A N).
A commutative monoid B in the category of unital Amodules is the same thing
as a Calgebra B and a map of Calgebras A ! B. It follows that for maps of
Calgebras A ! B and A ! C, the pushout B qA C in the category of Calgebras
is given by B A C. Thus, we have a canonical map TorA*(B; C) ! H*(B qA C).
Theorem 3.1. Let A ! B and A ! C be maps of Calgebras and assume that
A ! B is a relative cell inclusion. Then the natural map TorA*(B; C) ! H*(BqA C)
is an isomorphism.
Theorem 1.1 is an immediate consequence of the previous theorem. The map
k ! CCk[n] is a quasiisomorphism and a relative cell inclusion, so the map
H*A ~=Tor*(A; k) ! Tor*(A; CCk[n]) ~=H*(A q CCk[n])
is an isomorphism. Proposition 1.3 gives the closed model structure. Left prope*
*r
ness follows from the fact that for a relative cell inclusion A ! B, a quasi
isomorphism A ! C induces an isomorphism
H*B ~=TorA*(B; A) ! TorA*(B; C) ~=H*(B qA C):
Right properness follows from the fact that limits of Calgebras are created in
the category of differential graded kmodules and the fact that the fibrations *
*are
surjections. The proof of Theorem 3.1 occupies the remainder of this section.
We prove Theorem 3.1 by identifying a large class of unital Amodules for whi*
*ch
the A product is particularly wellbehaved.
Definition 3.2.We say that a unital Amodule M is A flat if the canonical map
TorA*(M; N) ! H*(M A N) is an isomorphism for every unital Amodule N.
The following proposition lists some elementary properties of A flat modules
that can be proved by the standard arguments.
Proposition 3.3.Let A be a Calgebra and let M be a unital Amodule.
(i)A is A flat.
(ii)If M is the retract of a A flat unital Amodule then M is A flat.
(iii)If M is Amodule chain homotopy equivalent rel A to a A flat unital A
module then M is A flat.
(iv)If M is a filtered colimit of A flat unital Amodules, then M is A flat.
(v) If M is the normalization of a simplicial unital Amodule that is A flat *
*in
each degree then M is A flat.
(vi)If M ~=A N for a flat unital C(1)module N then M is A flat.
THE HOMOTOPY THEORY OF E1 ALGEBRAS 9
(vii)If M ~=N B P for some A flat unital A Bmodule N and some B flat
unital Bmodule P , for some Calgebra B, then M is A flat.
The following lemma is proved in Section 8.
Lemma 3.4. If Z is a bounded below free differential graded kmodule (or more
generally, a cell kmodule [4, III.1.1]), then the unital C(1)module CZ is fl*
*at.
We actually use the following variation.
Proposition 3.5.If Z is a free differential graded kmodule with zero different*
*ial,
then CCZ is CZflat.
Proof.Consider the simplicial Calgebra fio = fiCo(CZ; CZ; CCZ), and write fi f*
*or
the normalization, the Calgebra fiC(CZ; CZ; CCZ). The inclusion of CZ into the
coproduct CZ q CCZ = fi0 makes fi a Calgebra under CZ and therefore in partic
ular a unital CZmodule. In simplicial degree n, fio is given by
fin = CZ CZ__._._.CZz_____"CCZ:
n factors
It therefore follows from Lemma 3.4 and 3.3.(iv)(vi) that fi is CZflat. There
is a canonical map of Calgebras fi ! CCZ. To prove that CCZ is CZflat, we
construct a section as a map of Calgebras under CZ. Let ffo be the simplicial
differential graded kmodule
ffn = Z Z__._._.Zz____"CZ;
n factors
and let ff be its normalization. Then there is a canonical identification of f*
*io as
Cffo and therefore a canonical map of differential graded kmodules ff ! fi. On
the other hand,
ff ~=(Z I) [Z CZ
where I is the "unit interval differential graded kmodule" [4, p. 58]. The inc*
*lusion
of CZ in ff above is not a map under Z, but it is straightforward to construct *
*a map
CZ ! ff that is a map under Z and with the further property that the composite
CZ ! ff ! fi ! CCZ
is the usual inclusion. This induces a map CCZ ! Cff = fi of Calgebras under_
CZ whose composite with the map fi ! CCZ is the identity. __
Theorem 3.1 is an immediate consequence of the following proposition.
Proposition 3.6.Let A and B be Calgebras and let A ! B be a relative cell
inclusion. Then B is A flat.
Proof.Let An and Xn be as in Definition 1.2. By 3.3.(iv), it suffices to show t*
*hat __
each An is A flat. This follows by induction from Proposition 3.5 and 3.3.(vii*
*). __
4.The Homotopy Theory of E1 Algebras
In this section, E denotes an arbitrary E1 operad. We show that for cell E
algebras, the factorization and lifting properties in the axioms for a closed m*
*odel
category hold. We use these properties to construct a homotopy theory of E
algebras for which the analogue of the Whitehead theorem holds. This allows us
10 MICHAEL A. MANDELL
to describe the derived category E [Q1] obtained by formally inverting the qua*
*si
isomorphisms in terms of the category cell Ealgebras and homotopy classes of m*
*aps.
We use this description to prove the adjunction theorems in the next section.
The following lemma is the main building block we need for the results of this
section. It is proved in Section 6 by comparing the Ealgebra coproduct with the
Calgebra coproduct via the functor V of [4, V.1.7].
Lemma 4.1. If B is a cell Ealgebra then the inclusion B ! B q ECk[n] is a
quasiisomorphism for every n.
Lemma 4.1 leads to the following proposition.
Proposition 4.2.Let A be a cell Ealgebra. A map f :A ! B has the left lifting
property with respect to the collection of surjections if and only if f is the *
*retract
of an acyclic relative cell inclusion.
Proof.Factor f as p O i using Proposition 2.5. Using Lemma 4.1, the second part
of the proof of Proposition 2.5 shows that i is a quasiisomorphism. Now if we
assume that f is an acyclic relative cell inclusion, then applying the argument*
* of
Proposition 2.6 with this f, p, i shows that f is a retract of i, and so has the
required lifting property. Conversely, if f has the lifting property then we ca*
*n find
a lift in the following diagram
A _//i~//_A0>>_
_______
f _____p_
fflfflfflfflfflffl_____
B __id_//B:
Such a lift expresses f as a retract of i, an acyclic relative cell inclusion. *
* ___
As a consequence of Proposition 4.2, we obtain the following result.
Proposition 4.3.Let A be a cell Ealgebra and let A ! B and A ! C be relative
cell inclusions. If A ! B is a quasiisomorphism, then so is the map C ! B qA C.
Proof.Applying Proposition 4.2, we see that the map C ! B qA C has the left lif*
*t
ing property with respect to the collection of surjections. Applying Propositio*
*n_4.2
again, we conclude that C ! B qA C is a quasiisomorphism. __
Definition 4.4.Let A be a Ealgebra. A cylinder object for A consists of a E
algebra IA together with a relative cell inclusion @0q @1: A q A ! IA and a weak
equivalence oe :IA ! A such that the composites oe O @0 and oe O @1 are each the
identity on A. We say that a cylinder object is nice if the map oe is a surject*
*ion. If
f0, f1 are maps A ! B, a (nice) homotopy from f0 to f1 is a map f :IA ! B for
some (nice) cylinder object IA such that fi = f O @i, i = 0; 1. We say that maps
f0 and f1 are (nicely) homotopic if there exists a (nice) homotopy from f0 to f*
*1.
Write ss(A; B) for the set of Ealgebra maps from A ! B modulo the equivalence
relation generated by "nicely homotopic".
It follows from Proposition 2.1 that nice cylinder objects always exist. We h*
*ave
used the equivalence relation generated by nice homotopies in the definition of
ss(A; B), since, as we shall see, nice homotopies have better composition prope*
*rties
than the more basic notion of homotopies. On the other hand, we have made the
definition of homotopies above in analogy with the definition of (left) homotop*
*ies in
THE HOMOTOPY THEORY OF E1 ALGEBRAS 11
the motivating example of closed model categories. This weaker notion is essent*
*ial,
just as it is for closed model categories, because functors that preserve "cofi*
*bra
tions" tend not to preserve "fibrations" and then fail to preserve a relation l*
*ike
"nice homotopy". In the special case when A is a cell Ealgebra, it turns out t*
*hat
homotopies and nice homotopies are essentially equivalent and two maps A ! B
represent the same element of ss(A; B) if and only if they are nicely homotopic*
*. We
state this as the following proposition.
Proposition 4.5.Let A be a cell Ealgebra. Then maps f0; f1: A ! B are homo
topic if and only if they are nicely homotopic. Moreover, "nicely homotopic" is*
* an
equivalence relation on the set of maps A ! B.
Proof.Clearly nicely homotopic maps are homotopic. Let IA ! B be a homotopy
from f0 to f1. Factor the map IA ! A using Proposition 2.5 as IA ! I0A ! A.
Then I0A is a nice cylinder object and the map IA ! I0A has the left lifting
property with respect to the collection of surjections by Proposition 4.2. It f*
*ollows
that we can lift the map IA ! B to a map I0A ! B that provides a nice homotopy.
For the second statement, clearly "homotopic" is reflexive (factor through A) a*
*nd
symmetric (reverse the indexes of @0 and @1); Proposition 4.3 with the argument
of [7, Lemma 13] allows us to glue cylinder objects and shows that the_relatio*
*n is
symmetric. __
Proposition 4.6.If g0; g1: B ! C are (nicely) homotopic maps then for any map
h: C ! D, h O g0 and h O g1 are (nicely) homotopic. If g0; g1 are nicely homoto*
*pic
maps, then for any map f :A ! B, g0 O f and g1 O f are nicely homotopic.
Proof.Let g :IB ! C be a homotopy from g0 to g1; then h O g is a homotopy
from h O g0 to h O g1. Now assume that IB is a nice cylinder object. Choose a n*
*ice
cylinder object for A. Proposition 2.4 allows us to construct a lift in the fol*
*lowing
diagram.
fqf
A qfAf____//_lfflB_q_B//IB
 
 ~
fflffl fflfflfflffl
IA ________//A___f___//_B
Such a lift is a nice homotopy from g0 O f to g1 O f. *
*___
It follows that there is a welldefined composition law
ss(A; B) x ss(B; C) ! ss(A; C);
and we can therefore form a category ssE whose objects are the Ealgebras and
whose set of morphisms from A to B is given by ss(A; B). Here we have done slig*
*htly
better than the general theory of closed model categories since the map from any
Ealgebra to the final object 0 is a surjection. This puts us in the analogue o*
*f the
situation when all objects are fibrant.
The following proposition is the analogue of [7, Lemma 17].
Proposition 4.7.Let A be a cell Ealgebra. If g :B ! C is an acyclic surjection,
then ss(A; g): ss(A; B) ! ss(A; C) is a bijection.
Proof.By Proposition 2.4, the map is a surjection. By Proposition 4.5, it suffi*
*ces
to show that maps f0; f1: A ! B are homotopic whenever g O f0 and g O f1 are
12 MICHAEL A. MANDELL
homotopic. Let h: IA ! C be a homotopy from g O f0 to g O f1. A lift in the
following diagram
f0qf1
A qfAf____//_lfflB;;____
 ________
 ______~
fflffl__fflfflfflffl__
IA __h___//_C
gives a homotopy from f0 to f1. ___
Proposition 4.8.If A and B are cell Ealgebras and f :A ! B is a quasi
isomorphism, then the image of f in ssE is an isomorphism.
Proof.It follows from Proposition 4.7 that the proposition holds in the special
case when f is an acyclic surjection. By factoring f as in Proposition 2.1, it
therefore suffices to consider the case when f is an acyclic relative cell incl*
*usion.
By Proposition 4.2, we can find a lift g in the following diagram.
Afflfflid//_A??__
______
~f______g
fflffl_fflfflfflffl____
B _____//0
Then g is a quasiisomorphism, and we can factor it as an acyclic relative cell
inclusion g0:B ! A0 followed by an acyclic surjection h: A0 ! A. Arguing as
above, the map g0 has a retraction f0:A0 ! B0, and we obtain the following
commutative diagram of quasiisomorphisms.
Afflfflid____//_A9999
 tttttth
f 0 9A0J9 0
 gtttt JJfJ
fflffl99tt J%%J
B _____id_____//_B
Since h is an acyclic surjection between cell Ealgebras, it is an isomorphism *
*in_ssE ,
and it follows that f and g are inverse isomorphisms in ssE . *
*__
The Ealgebra analogue of the Whitehead theorem is an easy consequence.
Theorem 4.9. (The Whitehead Theorem) Let A be a cell Ealgebra. A quasi
isomorphism f :B ! C induces an isomorphism ss(A; B) ! ss(A; C).
Proof.By Proposition 1.7, we can form a commutative diagram
f0
B0_//~_//_C0
 
b ~ ~c
fflfflfflfflfflfflfflffl
B __f~_//C;
where B0and C0are cell Ealgebras. The theorem now follows from Proposition_4.7
and Proposition 4.8. __
THE HOMOTOPY THEORY OF E1 ALGEBRAS 13
Let ssEc denote the full subcategory of ssE consisting of the cell Ealgebras*
*. Let
E [Q1] denote the category obtained from the category of Ealgebras by formally
inverting the quasiisomorphisms. The canonical functor fl :E ! E [Q1] clearly
factors through ssE , and so we have a functor fl:ssEc ! E [Q1]. As an immedia*
*te
consequence of the Whitehead theorem, we have the following corollary, which is
the analogue of [7, Theorem 11'].
Corollary 4.10.The functor fl:ssEc ! E [Q1] is an equivalence.
5. Categories of E1 Algebras and Adjoint Functors
We prove Theorem 1.9 by following the proof of the analogous statement for
closed model categories given in [1, 9.7]. Following this proof, we need three *
*ad
ditional observations beyond the work of the previous section. The first is The*
*o
rem 1.10.
Proof of Theorem 1.10.This is proved just as [1, 9.8]. The only thing we still *
*need
to observe is that the acyclic surjections and the surjections of Ealgebras ca*
*n be
characterized as the maps that have the right lifting property with respect to *
*the
collection of relative cell inclusions between cell Ealgebras and the collecti*
*on of
acyclic relative cell inclusions between cell Ealgebras, respectively. These *
*char
acterizations follow from the fact that the acyclic surjections and the surject*
*ions
of differential graded kmodules can be characterized as the maps that have the
right lifting property with respect to the inclusions k[n] ! Ck[n] and k[n] ! I*
*[n],
respectively, for all n. Here [n] is the shift functor, and I is the "unit int*
*erval_
differential graded kmodule" [4, p. 58]. __
The second additional observation is the lemma of K. Brown [1, 9.9] and its d*
*ual.
Proposition 5.1.Let L and R be as in 1.9.(i). Then L converts all quasiisomor
phisms between cell Ealgebras to weak equivalences and R converts all weak equ*
*iv
alences between fibrant objects to quasiisomorphisms.
Proof.The argument for [1, 9.9] and its dual applies. __*
*_
Finally, we need to be able to convert "right homotopies" into homotopies. The
proof is the same as the analogous statement for model categories, except that *
*we
do not know that "right cylinder objects" always exist.
Proposition 5.2.Let A be a cell Ealgebra. Let B, BI be Ealgebras, s: B ! BI
a quasiisomorphism and d0 x d1: BI ! B x B a surjection such that diO s is the
identity for i = 0; 1. Let f0, f1 be maps A ! B. There exists a map h: A ! BI
that makes the following diagram commute if and only if f0 is homotopic to f1.
pBI77
h pppp 
ppp d0xd1
ppppp fflfflfflffl
A ___f0xf1_//_B x B
14 MICHAEL A. MANDELL
Proof.The "if" part is proved as follows. Let f :IA ! B be a homotopy from f0
to f1, and consider the following diagram.
sOf0
Afflffl______//BI

@0~ 
fflffl fflfflfflffl
IA __(f0Ooe)xf//_B x B
Choose a lift g and let h = g O @1. Then h makes the required diagram commute.
For the "only if" part, note that the maps d0 and d1 induce isomorphisms
ss(A; BI) ~= ss(A; B) by Proposition 4.7. Since each di is a retract of the map
s, s induces an isomorphism ss(A; B) ! ss(A; BI) and we see that d0 and d1 indu*
*ce
the same isomorphism. Thus, f0 = d0 O h and f1 = d1 O h are the same element of*
* __
ss(A; B) and it follows from Proposition 4.5 that they are homotopic. *
*__
Proof of Theorem 1.9.The argument for the analogous theorem for closed model__
categories in [1, 9.7] now applies. __
The following theorem is the dual version of Theorem 1.9. The stronger hypoth*
*e
ses are needed because we lack the lift property of Proposition 4.2 for arbitra*
*ry
Ealgebras.
Theorem 5.3. Let L: M ! E and R: E ! M be left and right adjoints between
a closed model category M and the category E of algebras over an E1 operad E.
(i)If R converts surjections to fibrations and quasiisomorphisms to weak equ*
*iv
alences, then the left derived functor of L and the right derived functor *
*of R
exist and are adjoint.
(ii)Suppose that (i) holds and in addition for any cofibrant object X of M and
any Ealgebra B, a map X ! RB is a weak equivalence if and only if the
adjoint LX ! B is a quasiisomorphism. Then the left derived functor of L
and the right derived functor of R are inverse equivalences.
Proof.By Proposition 2.4, L converts cofibrations into retracts of cell inclusi*
*ons. By
Proposition 4.2, L converts acyclic cofibrations between cofibrant objects to r*
*etracts
of acyclic relative cell inclusions between retracts of cell objects. The proof*
* of [1,
9.8] applies to show that L converts any weak equivalence between cofibrant obj*
*ects
to a quasiisomorphism between retracts of cell Ealgebras. It follows that the*
* left
derived functor L of L exists and can be constructed by cofibrant approximation
followed by application of L. The right derived functor R of R clearly exists *
*by
the universal property of the map E ! E [Q1] and is just a factorization of the
functor R through this map.
Let X be a cofibrant object of M and let B be a cell object of E. A left
homotopy in M between maps X ! RB gives a homotopy in E between the
adjoint maps LX ! B. On the other hand, factoring the diagonal map B !
B x B by Proposition 2.5, we obtain a cell Ealgebra BI and maps s: B ! BI
and d0 x d1: BI ! B x B, which by Proposition 4.2 satisfy the hypotheses of
Proposition 5.2. Proposition 5.2 then implies that for homotopic maps LX ! B
in E , the adjoint maps X ! RB are right homotopic in M . It follows that we
have an isomorphism [X; RB] ~=ss(LX; B), natural in cofibrant objects X and cell
objects B. Since R is naturally isomorphic to R O flO "Q, where fland "Qare the
equivalences of Corollary 4.10, it follows that L and R are adjoint.
THE HOMOTOPY THEORY OF E1 ALGEBRAS 15
Finally, assume the hypothesis of (ii). When X is cofibrant in M , the derived
unit map X ! RLX is represented by the unit map X ! RLX whose adjoint is the
identity map on RX and therefore a quasiisomorphism. For a Ealgebra B, choose*
* a
cofibrant approximation X ! RB. Then the counit map LRB ! B is represented
by the map LX ! B, whose adjoint, the map X ! RB, is by assumption a
weak equivalence. It follows that the unit and counit of the derived adjunction*
*_are
isomorphisms. Thus, the functors L and R are inverse equivalences. _*
*_
6.The Proofs of Lemma 4.1 and Theorems 1.4 and 1.5
The proofs of Lemma 4.1 and Theorems 1.4 and 1.5 are of a similar nature and
are based on applying the lifting properties of Section 2 to the various natura*
*l trans
formations between the operad "push forward" and related functors of [4, p. 136*
*].
We begin by developing notation for these functors and natural transformations.
Let O and P be E1 operads and let S be the E1 operad O P. Let O, P, and
S be the free O, P, and Salgebra functors. The augmentations of O and P induce
maps of operads S ! O and S ! P and therefore maps of the associated monads.
In this situation, embedingwe can form a "twosided monadic bar construction"
[4, xII.4]: For a map of monads A ! B and an Aalgebra X, the twosided bar
construction Bo(B; A; X) is the simplicial Balgebra that is given in simplicia*
*l degree
n by
Bn(B; A; X) = B A_._.A.z__"X;
n factors
natural in A ! B and X. Consider the functors
Oo = Bo(O; S; ); Po = Bo(P; S; ); and So = Bo(S; S; )
from Salgebras to O, P, and Salgebras respectively. Normalization makes these
into functors O, P , and S from Salgebras to O, P, and Salgebras.
The maps of monads S ! O and S ! P induce natural transformations o: S ! O
and p: S ! P . Since for any differential graded kmodule X, SX ! OX and SX !
PX are quasiisomorphisms, the maps SoA ! OoA and SoA ! PoA are degreewise
quasiisomorphisms for every Salgebra A, and so the natural transformations o
and p are quasiisomorphisms for all Salgebras. As in [4, II.4.2], we have a n*
*atural
transformation oe :S ! Id which is a quasiisomorphism for all Salgebras. The
maps of operads S ! O and S ! P allow us to regard Oalgebras and Palgebras
as Salgebras; we typically omit notation for these functors, but when importan*
*t,
we denote them as UO and UP . The natural transformations
Bo(O; S; UO ) ! Bo(O; O; ) ! IdO and Bo(P; S; UP ) ! Bo(P; P; ) ! IdP
induce natural transformations ! :OUO ! Id and ss :P UP ! Id such that the
following diagrams commute.
p
S______o______//@@O" S ____________//_@@P"
@@ """ @@@ """
oe@@OO@"""!"" oe @OO@"""ss""
Id Id
It follows that the maps ! and ss are quasiisomorphisms for all Oalgebras and*
* all
Palgebras respectively.
The proofs of the results in this section are based on the following lifting *
*lemma.
16 MICHAEL A. MANDELL
Lemma 6.1. Let A, B, C be cell Palgebras and let A ! B, A ! C be relative
cell inclusions. Suppose given a commutative diagram of Oalgebras
B0 oo___oA0o__//__//C0
  
~ ~ ~
fflfflfflfflfflfflfflfflfflfflfflffl
OB oo___OA _____//OC;
with A0, B0, C0 cell Oalgebras. Then there are quasiisomorphisms of Palgebras
A ! P A0, B ! P B0, C ! P C0 such that the diagram
B oo_____A _______//C
~ ~ ~
fflffl fflffl fflffl
P B0 oo___P A0_____//P C0
commutes and the composite B qA C ! P (B0qA0C0) ! P O(B qA C) is a quasi
isomorphism.
Lemma 4.1 is now an easy consequence.
Proof of Lemma 4.1.Let O = C and P = E, A = E(0), and C = ECk[n]. Construct
A0, B0, C0by Proposition 1.7. Applying Lemma 6.1, we obtain quasiisomorphisms
B ! P B0 and ECk[n] ! C0 that make the following diagram commute.
B _____________//P B0____________//_P OB
  
  
fflffl fflffl fflffl
B q ECk[n]_____//P (B0qA0C0)____//P O(B q ECk[n])
The top row consists of quasiisomorphisms and the composite map in the bottom
row is a quasiisomorphism by the lemma. The middle vertical map is a quasi
isomorphism by the work of Section 3. It follows that the map B ! B q ECk[n]_is
a quasiisomorphism. __
For the proof of Theorems 1.4 and 1.5, recall that for maps of Ealgebras A !*
* B
and A ! C, TorA*(B; C) is defined as TorV*A(V B; V C), where V is the functor of
[4, V.1.7]. Choose cell Calgebras A0, B0, and C0by applying Proposition 1.7 to*
* the
diagram V B V A ! V C, i.e. A0! B0, A0! C0 are relative cell inclusions, and
A0! V A, B0! V B, C0! V C are acyclic surjections. The work of Section 3 then
implies that we can define TorA*(B; C) as H*(B0qA0C0). Theorems 1.4 and 1.5 are
then consequences of the following more specific theorem.
Theorem 6.2. Let A, B, C, A0, B0, C0 be as above. If A is a cell Ealgebra and
the maps A ! B, A ! C are relative cell inclusions, then the canonical map
B0qA0C0! V (B qA C) is a quasiisomorphism.
Proof.Let G be C E. The functor V is the functor N*(Bo(C; G; UE)) from
Ealgebras to Calgebras described in the previous section; denote by W the cor
responding functor N*(Bo(E; G; UC)) from Calgebras to Ealgebras. Applying
Lemma 6.1 with O = C and P = E, we obtain quasiisomorphisms A ! W A0,
B ! W B0, C ! W C0, and a map B qA C ! W (B0qA0C0) such that the compos
ite map B qA C ! W V (B qA C) is a quasiisomorphism. In particular, the map
B qA C ! W (B0qA0C0) induces an injection on homology.
THE HOMOTOPY THEORY OF E1 ALGEBRAS 17
Factor the map A ! W A0as A ! A ! W A0as in Proposition 2.4, and factor
the maps BqA A ! W B0and C qA A ! W C0through B and C by Proposition 2.4.
By Proposition 4.3, the maps B ! B and C ! C are quasiisomorphisms. We have
the following commutative diagram.
BfflfflooAo_o_//__//fflfflCfflffl
~ ~ ~
fflffl fflffl fflffl
B oo_____oAo_//____//C
  
~ ~ ~
fflfflfflfflfflfflfflfflfflfflfflffl
W B0oo___ W A0_____//W C0
Applying Lemma 6.1 with O = E and P = C, we obtain quasiisomorphisms A0!
V A, B0 ! V B, C0 ! V C, and a map B0qA0 C0 ! V (C qA B ) such that the
composite B0qA0C0! V W (B0qA0C0) is a quasiisomorphism. Since V preserves
homology, it follows that the map B qA C ! W (B0qA0C0) induces a surjection on
homology.
The map BqA C ! W (B0qA0C0) factors as the composite of the map BqA C !
B qA C and the map B qA C ! W (B0qA0C0). By Proposition 4.3, the map B qA
C ! BqA C is a quasiisomorphism. The map BqA C ! W (B0qA0C0) is then both
an injection and a surjection on homology and is therefore a quasiisomorphism.
The map W (B0qA0C0) ! W V (B qA C) is therefore a quasiisomorphism. Since
W preserves homology, it follows that the map B0 qA0 C0 ! V (B qA C) is_a_
quasiisomorphism. __
Before beginning the proof of Lemma 6.1, we need the following observation.
Proposition 6.3.The functors O, P , and S preserve surjections. The natural
transformations o, !, p, ss, and oe are surjections.
Proof.The first statement follows from the fact that the functors O, P, S, and
normalization preserve surjections. The map oe is a split surjection of the und*
*erlying
differential graded kmodule. The second statement for the other maps follows
from the fact that for any differential graded kmodule X, the maps SX ! OX
and SX ! PX are surjections. This can be seen from the fact that for each n_the
augmentations O(n) ! k and P(n) ! k are surjections. __
Proof of Lemma 6.1.By Proposition 6.3, the natural quasiisomorphism P SA ! A
is a surjection. Thus, we can choose a lift A ! P SA in the following diagram
P(0)_____//P;SA;_
fflffl_________
 _______~__
fflffl___fflfflfflffl__
A ___id__//_A
and lifts B ! P SB and C ! P SC in the following diagrams.
Af__//_flfflP/SA/_P7SB7____ A __//_P SA_//P7SC7____
 __________ fflffl__________
 _________ ~  _________ ~
fflffl_______fflfflfflffl_ fflffl_______fflfflfflffl_
B _____id____//_B C _____id____//_C
18 MICHAEL A. MANDELL
The universal property of the pushout gives a map B qA C ! P S(B qA C).
Composing with the natural transformation P S ! P O, we obtain the following
commutative diagram.
B qA CN
gmmmmmm  NNNNidN
mm  NNN
vvmmmm fflffl NN''
P O(B qA C) oo~__P S(B qA C) ~___//_B qA C
It follows that the map g is a quasiisomorphism. Since the functor P preserves
quasiisomorphisms and surjections, the maps P A0 ! P OA, P B0 ! P OB, and
P C0 ! P OC are acyclic surjections. Thus we can choose diagonal lifts in the
following diagrams.
P(0) _____//_P A0 A _____//_P B0 A _____//_P C0
fflffl;;__________ fflffl<<_________ fflffl<<_________
 _______~__  ______~___  _______~__
fflffl___fflfflfflffl_fflffl_fflfflfflffl_fflffl__fflfflfflffl__*
*___
A ___~__//P OA B __~__//P OB C ___~_//P OC
The bottom horizontal quasiisomorphisms are the composites of the maps A !
P SA, B ! P SB, C ! P SC chosen above and the natural transformations P S !
P O. Note that the lifts A ! P OA, B ! P OB, C ! P OC must be quasi
isomorphisms. The universal property of the pushout gives a map h: B qA C !
P (B0qA0C0) such that the following diagram commutes
B qA C QQ
g ~ QQQhQQQQ

fflffl QQ((Q
P O(B qA C) oPfo_P (B0qA0C0);
where f is the canonical map B0qA0C0! O(B qA C). ___
7. The Proof of Theorem 1.6
Theorem 1.6 proceeds by a comparison of the bar construction fiE in the categ*
*ory
of Ealgebras with the bar construction fiC in the category of Calgebras. We b*
*egin
with the proof of Theorem 1.6 in the case of the operad C.
Proposition 7.1.Let A0 be a cell Calgebra, and let A0 ! B0, A0 ! C0 be rela
tive cell inclusions. Then the natural map fiC(B0; A0; C0) ! B0qA0 C0 is a quas*
*i
isomorphism.
Proof.It follows from the definition of A0 that the functor B0 A0 () com
mutes with normalization, and so fiC(B0; A0; C0) ~=B0A0fiC(A0; A0; C0). The map
fiC(A0; A0; C0) ! C0is a quasiisomorphism since it is a chain homotopy equival*
*ence
of the underlying differential graded kmodules. Since A0 ! B0 is a relative c*
*ell
inclusion, B0 is A0flat, and the composite
fiC(B0; A0; C0) ~=B0A0fiC(A0; A0; C0) ! B0A0C0= B0qA0C0
is therefore a quasiisomorphism. ___
THE HOMOTOPY THEORY OF E1 ALGEBRAS 19
Let V denote the functor from Ealgebras to Calgebras of [4, V.1.7]. Given E
algebras A, B, and C as in Theorem 1.6, find cell Calgebras A0, B0, C0 to make
the following commutative diagram.
B0oo___oA0o__//__//_C0
  
~ ~ ~
fflfflfflfflfflfflfflfflfflfflfflffl
V B oo___V A ____//_V C
The maps B0q (A0q . .q.A0) q C ! V (B q (A q . .q.A) q C) induce a map of
simplicial Calgebras
fiCo(B0; A0; C0) ! V fiEo(B; A; C)
By Theorem 6.2, this map is a degreewise quasiisomorphism. The map of simplici*
*al
Ealgebras from fiEo(B; A; C) to the constant simplicial Ealgebra B qA C induc*
*es
a map of simplicial Calgebras
V (fiEo(B; A; C)) ! V (B qA C)
that makes the following diagram commute.
fiC(B0; A0; C0)_____//_B0qA0C0
 
 
fflffl fflffl
N*(V (fiEo(B; A; C)))_//_V (B qA C)
Here N*( .) denotes normalization. Since the normalization of a degreewise quas*
*i
isomorphism is a quasiisomorphism, the left vertical arrow is a quasiisomorph*
*ism.
The top horizontal arrow is a quasiisomorphism by Proposition 7.1, and the rig*
*ht
vertical arrow is a quasiisomorphism by Theorem 6.2. It follows that the bottom
horizontal map is a quasiisomorphism.
Now let S denote the functor S() = N*(Bo(S; S; )) as in the previous sectio*
*n,
where S is the operad C E. As above, we have natural quasiisomorphisms of S
algebras S() ! V () and S() ! Id. From the following commutative diagram
N*(V (fiEo(B; A; C)))~oN*(S(fiEo(B;oA;_C)))~_//fiE(B; A; C)
~  
fflffl fflffl fflffl
V (B qA C)oo___~______S(B qA C) ____~_____//B qA C;
we see that the map fiE(B; A; C) ! B qA C is a quasiisomorphism. This completes
the proof of Theorem 1.6.
8. The Proof of Lemma 3.4
This section reduces Lemma 3.4 to a statement purely in terms of the operad C,
Lemma 8.2 below. The proof of Lemma 8.2 occupies the next two sections.
Let A be a unital C(1)module and let Z be a differential graded kmodule. The
unital C(1)module CZ A has a direct sum decomposition
M (m)
CZ A = A (C(m) k[m ]Z ) B A :
m>0
20 MICHAEL A. MANDELL
Here Z(m) denotes the mth tensor power Z . . .Z. By [4, p. 106], there is a
natural map, canonical up to chain homotopy,
(C(m) k[m ]Z(m)) A ! (C(m) k[m ]Z(m)) B A:
Lemma 3.4 reduces to showing that this map is a quasiisomorphism: the assump
tion on Z in Lemma 3.4 forces the tensor product on the left to have homology t*
*he
differential torsion product [4, xIII.4]. Thus, Lemma 3.4 is a consequence of *
*the
following proposition.
Proposition 8.1.If A is a unital C(1)module and Z is a differential graded k
module, then the map
(C(m) k[m ]Z(m)) A ! C(m) k[m ]Z(m) B A
is a chain homotopy equivalence of differential graded kmodules.
By [4, V.9.2] and the definition of B [4, V.2.1], the Bproduct on the right *
*above
is given by the following pushout diagram.
C(m + 1) k[m ]C(1)(Z(m) k) _____//C(m + 1) k[m ]C(1)(Z(m) A)
 
fflffl fflffl
C(m) k[m ]Z(m) ______________//_(C(m) k[m ]Z(m)) B A
On the other hand, by inspection, the tensor product on the left above is given*
* by
the following pushout diagram.
(C(m) C(1)) k[m ]C(1)(Z(m) k) ____//(C(m) C(1)) k[m ]C(1)(Z(m) A)
" 
fflffl fflffl
C(m) k[m ]Z(m) __________________//(C(m) k[m ]Z(m)) A
In the top diagram, the map labeled is induced by the last operad degeneracy
map C(m + 1) ! C(m); in the bottom diagram, the map labeled " is induced by
the degeneracy map C(1) ! k. The top and bottom diagram differ only in the
replacement of the right (k[m ] C(1))module C(m + 1) in the top diagram with
the right (k[m ] C(1))module C(m) C(1) in the bottom diagram.
The natural map in Proposition 8.1 is induced by the map of right (k[m ]C(1))
modules
: C(m) C(1) ~= (C(m) C(1)) ! C(2) (C(m) C(1)) ! C(m + 1);
where the first map is induced by the inclusion of a certain zero cycle t in C(*
*2) and
the second map is the operad multiplication [4, p. 106]. Since the vertical map*
*s in
the bottom pushout diagram above are isomorphisms, clearly we can describe the
map in Proposition 8.1 in terms of a map of the pushout diagrams above. The map
of the lower left entries is induced by the map
:C(m) ~= (C(m) C(0)) ! C(2) (C(m) C(0)) ! C(m);
THE HOMOTOPY THEORY OF E1 ALGEBRAS 21
since this is the unique map of right k[m ]modules that makes the diagram
C(m) C(1) ____//_C(m + 1)
 
 
fflffl fflffl
C(m) _________//_C(m)
commute. We prove the following lemma in the next two sections.
Lemma 8.2. The map is a chain homotopy equivalence of right (k[m ] C(1))
modules. The homotopy inverse :C(m + 1) ! C(m) C(1) can be chosen so that
the composite
C(m + 1) ! C(m) C(1) ! C(m)
is the last degeneracy map. The chain homotopies G: C(m) C(1) ! C(m) C(1)
and H :C(m + 1) ! C(m + 1) can be chosen such that there are right m module
chain homotopies J; K :C(m) ! C(m) from the identity to that make the following
diagrams commute.
C(m) C(1) __G__//C(m) C(1) C(m + 1)__H__//C(m + 1)
   
   
fflffl fflffl fflffl fflffl
C(m) _____J_____//C(m) C(m) ___K___//_C(m)
Here the vertical maps are induced by the operad degeneracy maps C(1) ! k and
C(m + 1) ! C(m).
The map from Lemma 8.2 induces a map of the pushout diagrams above from
the top diagram to the bottom diagram and therefore a map of differential graded
kmodules
(C(m) k[m ]Z(m)) B A ! (C(m) k[m ]Z(m)) A:
The homotopies G, J and H, K induce chain homotopies on the diagrams above
that give chain homotopies on
(C(m) k[m ]Z(m)) A and (C(m) k[m ]Z(m)) B A:
from the composite maps to the identity. This proves Proposition 8.1.
9. The Linear Isometries Operad
The proof of Lemma 8.2 requires Lemmas 9.1 and 9.2 below regarding the linear
isometries operad. Lemma 9.1 is the equivariant version of [4, V.9.3]. Lemma 9.2
provides the additional compatibility with the degeneracy maps that we need for
our application to Lemma 8.2. These two lemmas together provide a topological
version of Lemma 8.2. In the next section, we show how to convert the topology
into algebra.
Let U = R1 = ColimRn regarded as an inner product space. Recall that the
linear isometries operad, L, is the operad whose nth space is the space of lin*
*ear
isometries from Un = U . . .U to U with product fl given by composition.
See [3, xI.3,xXI] for details. The space L(m + 1) has a right action of the mon*
*oid
m x L(1) by letting L(1) act on the last copy of U in Um+1 and letting m act of
the first m copies. If ff is an element of L(m + 1), we can ignore the first m *
*copies
of U in Um+1 and associate to ff an element of L(1). Similarly, we can ignore *
*the
22 MICHAEL A. MANDELL
last copy of U and associate to ff an element of L(m). This association defines*
* a
m x L(1)equivariant map of spaces
h: L(m + 1) ! L(m) x L(1):
If we choose an isometric isomorphism t: U2 ! U, then for every element (ffi; f*
*fl) in
L(m) x L(1), the multiplication fl(t; ffi; ffl) = t O (ffi ffl) gives an eleme*
*nt of L(m + 1).
Again, it is easy to see that this defines a m x L(1)equivariant map of spaces
g :L(m) x L(1) ! L(m + 1):
Lemma 9.1. The maps g and h are inverse m xL(1)equivariant homotopy equiv
alences.
Proof.We use 1 to denote the identity map on U regarded as an element of L(1).
Write i1 for the isometric embedding of U in Um+1 as the last copy and im for t*
*he
isometric of Um in Um+1 as the first mcopies. By ignoring the second copy of*
* U
in U2, we can associate to t an element ^t1in L(1), and by ignoring the first c*
*opy,
we obtain an element ^t2in L(1). In this notation, h is the map
h: ff 7! (ff O im ; ff O i1);
and h O g is the map that takes an element (ffi; ffl) in L(m) x L(1) to (^t1O f*
*fi; ^t2O ffl).
Since L(1) is contractible, we can choose paths OEj in L(1) from 1 to ^tjfor j *
*= 1; 2.
Composition on the left on L(m) x L(1) with the path (OE1; OE2) in L(1) x L(1) *
*gives
a m x L(1)equivariant homotopy from the identity on L(m) x L(1) to h O g.
Consider the space A of isometric automorphisms of Um Um and denote by
Am the subspace of the m equivariant maps, where we give Um Um the diagonal
m action. It is straightforward to write explicitly a path in Am from the ide*
*ntity
to the map that switches the copies of Um . Choose to be such a path. We use
the path to construct the homotopy for L(m + 1).
The map g O h acts on an element ff of L(m + 1) as follows:
g O h: ff 7! t O (ff O im ff O i1):
Consider tO(ffOim ff). This is a linear isometry from Um Um U to U. We can
compose on the right with 1 to obtain a path of linear isometries Um Um U
to U. Writing i for the inclusion of Um U in Um Um U as the last m + 1
copies of U,
t O (ff O im ff) O ( 1) O i
gives a path in L(m + 1) from ^t2O ff to g(h(ff)). Composition on the left with*
* the
path OE2 provides a path from ff to ^t2Off. These paths assemble to a homotopy *
*from
the identity to g O h. This is a m x L(1)equivariant homotopy since ( _1)_O i*
* is
a path through m x L(1)equivariant maps. __
Write ^t1as in the proof above for the element of L(1) obtained from t by ign*
*oring
the second copy of U in U2. Let g:L(m) ! L(m) be the map that takes the element
ffi of L(m) to the element ^t1O ffi. Then the following diagrams commute.
L(m) x L(1) _g__//_L(m + 1) L(m + 1) __h_//_L(m) x L(1)
   
   
fflffl fflffl fflffl fflffl
L(m) ____g____//_L(m) L(m) ____id___//_L(m)
THE HOMOTOPY THEORY OF E1 ALGEBRAS 23
Here the vertical maps are the projection L(m)xL(1) ! L(m) and the last operad
degeneracy map L(m + 1) ! L(m). The following lemma explains the relationship
of these diagrams to the homotopies above.
Lemma 9.2. There are m equivariant homotopies J and K from the identity of
L(m) to g that make the following diagrams commute
L(m) x L(1) x I_G__//_L(m) x L(1) L(m + 1) x I_H___//L(m + 1)
   
   
fflffl fflffl fflffl fflffl
L(m) x I ____J_____//_L(m) L(m) x I___K____//L(m);
where G and H are the homotopies constructed in the proof of Lemma 9.1.
Proof.Let OE1, OE2, and be the paths in the proof of Lemma 9.1. The homotopy
J is given by composition on the left with the path OE1. Write for the inclusi*
*on of
Um in Um Um as the last m compies of U. The homotopy K takes the element *
* __
fi in L(m) along the path OE2 O fi followed by the path t O (fi fi) O O . *
* __
10.The Proof of Lemma 8.2
Recall from [4, xV.9] that the operad C is built out of the singular chain co*
*mplex
of the linear isometries operad. We have that C(n) = C*L(n), and the multiplica*
*tion
C(n) (C(j1) . . .C(jn)) ! C(j)
is the composite of the shuffle map [2, I.5.3]
C(n) (C(j1) . . .C(jn)) ! C*(L(n) x (L(j1) x . .x.L(jn))) ! C(j1+ . .+.jn)
and the map induced by the multiplication of L. We remind the reader that the
shuffle map is an associative and commutative natural transformation [2, I.5.10*
*].
By associativity, it follows that the map described in Section 8 is the comp*
*osite
of the shuffle map C(m) C(1) ! C*(L(m) x L(1)) and the map induced by the
map g described in Section 9. We define the map for Lemma 8.2 as the composite
of the map induced by the map h of Section 9 with the AlexanderWhitney map
C(m + 1) C*h!C*(L(m) x L(1)) ! C(m) C(1):
We obtain the homotopies G, H, J, and K for Lemma 8.2 from the corresponding
homotopies in Lemmas 9.1 and 9.2 as follows.
Since the composite of the shuffle map followed by the AlexanderWhitney map
is the identity, the composite O is C*(h O g) and we take the chain homotopy G
for Lemma 8.2 from the identity to O to be the chain homotopy induced by the
homotopy G of Lemma 9.1. Likewise, we can take the chain homotopies J and K
for Lemma 8.2 to be the chain homotopies obtained from the homotopies J and K
of Lemma 9.2.
The composite O is not C*(g O h) but rather the composite
C*g O oe O ae O C*h;
where ae denotes the AlexanderWhitney map and oe denotes the shuffle map. We
obtain the homotopy H in Lemma 8.2 using the chain homotopy of [2, II.2]
from the identity to oe O ae and the chain homotopy induced by the homotopy H of
Lemma 9.1.
24 MICHAEL A. MANDELL
Both the AlexanderWhitney map and the shuffle map are isomorphisms when
one of the tensor factors comes from a constant simplicial kmodule. In additio*
*n,
the homotopy is zero when one of the factors is constant. It follows from the
topological diagrams in Lemma 9.2 that the algebraic diagrams
C(m) C(1) __G__//C(m) C(1) C(m + 1)__H__//C(m + 1)
   
   
fflffl fflffl fflffl fflffl
C(m) _____J_____//C(m) C(m) ___K___//_C(m)
commute. Likewise, it follows from naturality that the map and the chain ho
motopies we have constructed are m equivariant. Lemma 8.2 is then completed
by showing that the map is a map of right C(1)modules and that the chain
homotopies above are chain homotopies of right C(1)modules.
For the chain homotopies G, J, and K, this is relatively straightforward. Den*
*ote
by I the "unit interval differential graded kmodule" [4, p. 58], the free diff*
*erential
graded kmodule with a generator [I] in degree 1, and generators [0] and [1] in
degree zero with d[I] = [1]  [0]. A chain homotopy X ! Y is the same thing as
a map of differential graded kmodules X I ! Y . Let Io denote the simplicial
kmodule free on the simplicial set [1]; so I is the normalization of Io. We de*
*note
normalization of a simplicial kmodule Xo as N*(Xo). Then the chain homotopy
N*(Xo) I ! N*(Yo) induced by a simplicial homotopy Xo x Io ! Yo is just the
composite with the shuffle map and normalization:
N*(Xo) I oe!N*(Xo x Io) ! N*(Yo):
It therefore follows from the associativity of the shuffle map that the chain h*
*omo
topies G, J, and K are chain homotopies of right C(1)modules.
To see that the map is a map of right C(1)modules, we need an associativity
relation between the shuffle map and the AlexanderWhitney map. The following
lemma is proved in the next section.
Lemma 10.1. Let oe denote the shuffle map and let ae denote the AlexanderWhit
ney map. For any simplicial kmodules Ao, Bo, Co, the following diagram com
mutes.
N*(Ao Bo) N*(Co) ___oe____//N*(Ao Bo Co)
aeId ae
fflffl fflffl
N*(Ao) N*(Bo) N*(Co) Idoe_//N*(Ao) N*(Bo Co)
Let us denote by SoX the simplicial kmodule free on the singular simplicial
set of a space X, so that C*X = N*(SoX). Then plugging in Ao = SoL(m),
Bo = SoL(1), Co = SoL(1), and composing with So applied to the multiplication
L(1) x L(1) ! L(1), we see that the AlexanderWhitney map
ae: C*(L(m) L(1)) ! C(m) C(1)
is a map of right C(1)modules, and it follows that is as well.
To see that the chain homotopy H respects the right C(1)module structure,
it suffices to see that the homotopy from the identity to oe O ae does. This i*
*s a
consequence of the following lemma proved in Section 12, again plugging in Ao =
SoL(m), Bo = SoL(1), Co = SoL(1).
THE HOMOTOPY THEORY OF E1 ALGEBRAS 25
Lemma 10.2. Let be the natural chain homotopy of [2, II.2] from the identity *
*to
oe O ae. For any simplicial kmodules Ao, Bo, Co, the following diagram commute*
*s.
N*(Ao Bo) N*(Co) __oe_//N*(Ao (Bo Co))
Id  
fflffl fflffl
N*(Ao Bo) N*(Co) __oe_//N*(Ao (Bo Co))
This completes the proof of Lemma 8.2.
11. The Proof of Lemma 10.1
Write f = aeOoe, g = (Idoe)O(aeId). We show that f = g by direct calculation.
For a homogeneous element of N*(Ao Bo) N*(Co) of the form am bm cn,
(where the subscript denotes the degree), the formulas for oe and ae yield the *
*following
formulas for f and g.
m+nX X
f(am bm cn) = (1)o()@"m+nis a @i0s b @i0s c
i=0 ((;)man;n)shuffle
mX X
g(am bm cn) = (1)o(ff)"@mia sfi@i0b sffc
i=0(m(ff;fi)ani;n)shuffle
In the above formulas, "@denotes the last face map, @j for an element in degree*
* j,
so for example
"@mix = @i+1O . .O.@m x
when x has degree m. The notation s denotes the composite of degeneracies s1 ,
s2 , etc, i.e. for an (m; n)shuffle (; ),
s = sm O . .O.s1 s = sn O . .O.s1 :
When i > m in the formula for f, "@m+ni consists of fewer than n face maps
whereas s consists of n degeneracy maps; it follows that in this case "@m+nis*
* a is
zero in the normalized complex N*( .). Since
( m+ni1
@"m+nisj = @" when j i
sj"@m+ni when j < i
it follows that "@m+nis a is zero unless j = j  1 for 1 j i. In the remaini*
*ng
cases when "@m+nis a is nonzero, it is equal to "@mia, and
@i0sn . .s.0b = sni . .s.1i@i0b; @i0sm . .s.1c = sm i . .s.i+1ic:
Matching up the terms, we conclude that f = g.
12. The Proof of Lemma 10.2
Mixing the terminology from Eilenberg and Mac Lane [2] with the modern ter
minology and notation, we define a "monotonic operator" to be a map in op
regarded as an operation on simplicial kmodules. More generally, an "operator"
is any natural homomorphism
(A1)p1 . . .(An)pn ! (A1)q1 . . .(An)qn
26 MICHAEL A. MANDELL
where A1; : :;:An denote simplicial kmodules and the subscripts pm and qm de
note simplicial degrees. Any operator can be written uniquely as a linear combi
nation of tensor products of monotonic operators. If ff is an operator from deg*
*ree
p1; : :;:pn to degree q1; : :;:qn, we define the "derived operator", ff0 from d*
*egree
p1 + 1; : :;:pn + 1 to degree q1 + 1; : :;:qn + 1, by the following prescriptio*
*n. The
derived operator of @i is @i+1; the derived operator of si is si+1; for operato*
*rs ff
and fi,
(ff fi)0= ff0 fi0
and whenever ff O fi or ff + fi is defined,
(ff O fi)0= ff0O fi0 (ff + fi)0= ff0+ fi0:
It is straightforward to verify that this is welldefined. Finally, we say tha*
*t a
monotonic operator is "frontal" if it can be written in the form
sim. .s.i1@jn. .@.j1
where im > . .>.i1 0 and j1 > . .j.n> 0, i.e. if it does not require the zeroth
face operation when written in standard form. We call an operator frontal if it
can be written as the linear combination of tensor products of frontal monotonic
operators. It is easy to check that the composition of frontal operators is fro*
*ntal and
that all derived operators are frontal. Furthermore, if ff is frontal then s0ff*
* = ff0s0.
Departing from the terminology of [2], we say an operator ff from degree p1; : *
*:;:pn
to degree q1; : :;:qn is "degenerate" if q1 = . .=.qn and ff can be written as a
linear combination of operators of the form
(si . . .si) O fi
where fi is some operator from degree p1; : :;:pn to degree q1  1; : :;:qn  1.
Note that since derivation commutes with composition, the derived operator of a
degenerate operator is degenerate.
The advantage of this point of view is that the unnormalized shuffle map oe
restricted to a given degree is a frontal operator. The concept of derived ope*
*ra
tor allows an inductive definition of the shuffle map [2, I.5.7]. We are prima*
*rily
interested in the shuffle map
N*(Ao Bo) N*(Co) ! N*(Ao Bo Co)
so it is convenient to regard this as an inhomogeneous sum of operators oep;p;q*
*from
degree p; p; q to degree p + q; p + q; p + q. In terms of these operators, the *
*inductive
definition is as follows.
8
>>>oe0p1;p1;qO (1 1 s0) + (1)poe0p;p;q1(s0p s0> 1)0; q > 0
0; q = 0
oep;p;q= > p1;p1;00 0
>>:oe0;0;q1(s0 s0 1) p = 0; q > 0
Id p = 0 = q
If we use the convention that oe01;1;q= 0 = oe0p;p;1, this reduces to the si*
*mpler
formula
oep;p;q= oe0p1;p1;qO (1 1 s0) + (1)poe0p;p;q1(s0 s0 1) p + q > 0:
Let h = oe O ae; restricted to any given degree, h determines an operator hp;*
*pfrom
degree p; p to itself. Regarding h as a map
N*(Ao (Bo Co)) ! N*(Ao (Bo Co))
THE HOMOTOPY THEORY OF E1 ALGEBRAS 27
we also obtain an operator hp;p;pfrom degree p; p; p to itself. By Lemma 10.1 a*
*nd
the associativity of the shuffle map, the diagram
N*(Ao Bo) N*(Co) __oe_//N*(Ao (Bo Co))
hId  h
fflffl fflffl
N*(Ao Bo) N*(Co) __oe_//N*(Ao (Bo Co))
commutes. This is equivalent to the observation that
oep;p;qO (hp;p 1)  hp+q;p+q;p+qO oep;p;q
is degenerate. It follows that
(12.1) oe0p;p;qO (h0p;p 1)  h0p+q;p+q;p+qO oe0p;p;q:
is degenerate.
The chain homotopy is defined [2, II.2.13] by 0;0= 0 and
p;p= 0p1;p1+ h0p;pO (s0 s0) p > 0:
When we apply this operator to Ap (Bo Co)p, we can regard it as an operator
Ap Bp Cp ! Ap+1 Bp+1 Cp+1 that we denote as p;p;p. Note that p;pand
p;p;pare frontal operators. In this notation, the assertion of the lemma for th*
*is
chain homotopy is equivalent to the assertion that
oep+1;p+1;qO (p;p 1)  p+q;p+q;p+qO oep;p;q
is degenerate. We prove this by induction on p and induction on q for fixed p.
When p = 0, p;p= 0, so it suffices to show that q;q;qO oe0;0;qis degenerate.
This latter assertion follows from [2, II.2.4] since the image of oe0;0;qapplie*
*d to
Ao Bo Co lies in the image of the shuffle map oe applied to A0 diag(Bo Co)q.
Now consider the case when p > 0. By induction, we can assume the assertion
for degree p  1; q and derive it to conclude that
oe0p;p;qO (0p1;p1 1)  0p+q1;p+q1;p+q1O oe0p1;p1;q
is degenerate. If q > 0, the assertion holds for degree p; q  1 by induction o*
*n q,
and we conclude that
oe0p+1;p+1;q1O (0p;p 1)  0p+q1;p+q1;p+q1O oe0p;p;q1
is degenerate. In the case when q = 0, this last expression is zero by our conv*
*ention
that oe0n;n;1= 0. Therefore, for any q, composing these formulas on the right *
*with
(1 1 s0) and (s0 s0 1) respectively, adding with a sign, and making the
substitution
oep;p;q= oe0p1;p1;qO (1 1 s0) + (1)poe0p;p;q1O (s0 s0 1)
we see that
oe0p;p;q(0p1;p1 s0)  (1)p+1oe0p+1;p+1;q1(0p;p 1)(s0 s0 1)
 0p+q1;p+q1;p+q1oep;p;q
is degenerate. Using the fact that is a frontal operator, we see that
oe0p;p;q(0p1;p1 s0)  (1)p+1oe0p+1;p+1;q1(s0 s0 1)(p;p 1)
 0p+q1;p+q1;p+q1oep;p;q
28 MICHAEL A. MANDELL
is degenerate. Using the definition of and the fact that oe is frontal, we can*
* make
the substitution
0p+q1;p+q1;p+q1oep;p;q= p+q;p+q;p+qoep;p;q h0p+q;p+q;p+q(s0 s0 s0)oep;p;q
= p+q;p+q;p+qoep;p;q h0p+q;p+q;p+qoe0p;p;q(s0 s0 s0):
Since expression (12.1) is degenerate, we have that
h0p+q;p+q;p+qoe0p;p;q(s0 s0 s0) and oe0p;p;q(h0p;p 1)(s0 s0 s0)
differ by a degenerate operator, and we conclude that
oe0p;p;q(0p1;p1 s0)  (1)p+1oe0p+1;p+1;q1(s0 s0 1)(p;p 1)
+ p+q;p+q;p+qoep;p;q oe0p;p;q(h0p;p 1)(s0 s0 s0)
is degenerate. Combining the first and last terms and using the definition of ,*
* we
see that
 oe0p;p;q(p;p s0)  (1)p+1oe0p+1;p+1;q1(s0 s0 1)(p;p 1)
+ p+q;p+q;p+qoep;p;q
is degenerate. Finally, using the inductive description of oe, we see that
oep+1;p+1;q(p;p 1) + p+q;p+q;p+qoep;p;q
is degenerate. This completes the proof.
13. Closed Model Categories of NonSigma Operads
Recall from [4, I.1.2.(i)] the definition of a nonSigma operad. A nonSigma
operad is defined in terms of the same diagrams as an operad but without any
actions of the symmetric groups. For a nonSigma operad G, we obtain an associa*
*ted
operad G, by defining G(n) = G(n) k[n], using block sum of permutations for
the operad multiplication of the symmetric groups part. The monad G associated
to the nonSigma operad G is
M
GX = G(n) X(n);
n0
where X(n)denotes the nth tensor power X . .X., and we understand X(0)= k.
In order to prove Theorem 1.11, it suffices to show that if A is a Galgebra *
*and
Z is a contractible differential graded kmodule, then the map A ! A q GZ is
a homotopy equivalence of the underlying differential graded kmodules. A check
of universal properties shows that the following diagram is a coequalizer in the
category of differential graded kmodules
G(GA Z) ____//_//_G(A _Z)_//_A q GZ:
Here one map on the left is G, where is the action map GA ! A. The other
is the map of Galgebras induced by the maps of differential graded kmodules
GA ! G(A Z) and Z ! G(A Z). Expanding out the definition of G in terms
THE HOMOTOPY THEORY OF E1 ALGEBRAS 29
of G, we see that
M M
G(GA Z) = G(n) (GA)(nm) Z(m)
nm0 (nm)copies
M M
G(A Z) = G(n) A(nm) Z(m) :
nm0 (nm)copies
We have rearranged the tensor factors in order to write a compact formula above,
but the original arrangement must be retained for interpreting the maps in the
coequalizer in terms of the nonSigma operad multiplication.
Both maps in the coequalizer preserve the "power" of Z: they both send direct
summands of G(GA Z) containing Z(m) to direct summands of G(A Z) con
taining the same power Z(m). It follows that the differential graded kmodule of
A q GZ breaks up as a direct sum m0 Ym , where Ym corresponds to the image
of the summands where Z(m) appears. Note that the piece Y0 is the coequalizer of
the diagram
GGA _____////_GA
and so is isomorphic to A. In fact, the splitting of A q GZ as Y0 (m>0 Ym ) is
just the splitting obtained from the retraction
A ! A q GZ ! A
where the section is the inclusion of A into the coproduct and the retraction is
induced by the map Z ! 0. Thus, it suffices to show that Ym is contractible for
m 1. However, Z(m) is a contractible differential graded kmodule. Choosing a
contraction of Z(m) specifies contractions of
M M
G(n) (GA)(nm) Z(m) and G(n) A(nm) Z(m)
nm nm
that commute with both maps in the coequalizer and therefore specify a contract*
*ion
of Ym .
References
[1]W. G. Dwyer, J. Spalinski, "Homotopy Theories and Model Categories," in I. *
*M. James, ed,
Handbook of Algebraic Topology, Elsevier Science B.V., 1995.
[2]S. Eilenberg, S. Mac Lane, "On the Groups H(; n) I, II, III," Annals of Mat*
*h 58 (1953), pp.
55106; 60 (1954), pp. 49139, pp. 513557.
[3]A. D. Elmendorf, I. Kriz, M. A. Mandell, J. P. May, Rings, Modules, and Alg*
*ebras in Stable
Homotopy Theory, Amer. Math. Soc. Surveys & Monographs, vol. 47, 1996.
[4]I. Kriz, J. P. May, Operads, Algebras, Modules, and Motives, Asterisque 233*
*, 1995.
[5]M. A. Mandell, "E1 Algebras and pAdic Homotopy Theory," submitted in conj*
*unction with
this manuscript.
[6]J. P. May, Simplicial Objects in Algebraic Topology, Van Nostrand, 1967.
[7]D. G. Quillen, Homotopical Algebra, Springer Lecture Notes 43, 1967.
Department of Mathematics, University of Chicago, Chicago, IL
Current address: Department of Mathematics, M. I. T., Cambridge, MA
Email address: mandell@math.mit.edu