A solution of Deligne's Hochschild cohomology
conjecture.
James E. McClure and Jeffrey H. Smith*
Department of Mathematics, Purdue University, West Lafayette IN 479071395
February 1, 2001
ABSTRACT: Deligne asked in 1993 whether the Hochschild cochain complex
of an associative ring has a natural action by the singular chains of the *
*little
2cubes operad. In this paper we give an affirmative answer to this questi*
*on. We
also show that the topological Hochschild cohomology spectrum of an associ*
*ative
ring spectrum has an action by an operad that is equivalent to the little *
*2cubes
operad.
1 Introduction.
Let us first recall some facts about the Hochschild cochain complex C*(R) of an*
* associative
ring R. An element of Cp(R) is a map of abelian groups
x : R p ! R.
Hochschild [14] observed that there is a cup product in C*(R); if x 2 Cp(R) and*
* y 2 Cq(R)
then x ` y is the (p + q)cochain defined by
(x ` y)(r1 . . .rp+q) = x(r1 . . .rp) . y(rp+1 . . .rp+q)
The cup product is clearly associative but not commutative. In 1962 Gerstenhabe*
*r [9] showed
that it is chainhomotopy commutative; the chain homotopy between the multiplic*
*ation and
its twist is a sum of operations
Ok : Cp(R) Cq(R) ! Cp+q1(R)
that take x y to the (p + q  1)cochain x Ok y defined by
(x Ok y)(r1 . . .rp+q1) = x(r1 . . .rk1 y(rk . . .rk+q1) . . .r*
*p+q1)
________________________________
*The authors were partially supported by NSF grants, and by SFB 343 at the U*
*niversity of Bielefeld.
1
Gerstenhaber also used the Ok operations to construct a Lie bracket
[ , ] : Cp(R) Cq(R) ! Cp+q1(R)
The operations ` and [ , ] induce corresponding operations on the Hochschild co*
*homology
H*(R) which satisfy the relations making H*(R) a Gerstenhaber algebra.
Another example of a Gerstenhaber algebra is the homology of a 2fold loop s*
*pace:
H*( 2A) (the fact that this is a Gerstenhaber algebra is a consequence of the w*
*ork of Fred
Cohen [5]). In this case, the Gerstenhaber algebra structure in homology is a c*
*onsequence
of the action of the little 2cubes operad C2 on 2A. In 1993, Deligne [6] ask*
*ed whether
there was a closer relation between these two examples: specifically, he asked *
*whether the
Gerstenhaber algebra structure of H*(R) is induced by an action on C*(R) of a c*
*hain operad
quasiisomorphic to the singular chain operad of C2. This is usually known as *
*Deligne's
conjecture, although in the original letter it was expressed as a desire or pre*
*ference ("I would
like the complex computing Hochschild cohomology to be an algebra over [the sin*
*gular chain
operad of the little 2cubes] or a suitable version of it"). In this paper we g*
*ive an affirmative
answer to this question, except that we have to replace C*(R) by the normalized*
* Hochschild
cochains ~C*(R) (a cochain x is ön rmalized" if x(r1 . . .rn) = 0 whenever some*
* ri= 1; see
[18, Section 10.3]), and in order to make the signs work correctly it is necess*
*ary to work with
the desuspension of ~C*(see [9, page 279]). Other affirmative answers to Delign*
*e's question
for differential graded algebras in characteristic 0 have been found by Tamarki*
*n [25] and
[26], Kontsevich [16], and Voronov [27]. Tamarkin also has a more recent approa*
*ch which is
similar to ours. The approach in this paper answers Deligne's question for diff*
*erential graded
algebras in any characteristic as well as for associative ring spectra.
We show that the singular chain operad of C2 is quasiisomorphic (over the i*
*ntegers) to
a suboperad of the endomorphism operad of the desuspended reduced Hochschild co*
*chain
functor 1C~*. Explicitly, for each n 0 let O(n) be the graded abelian group*
* of natural
transformations
: ( 1C~*) n ! 1C~*
(where the grading of is the amount by which it lowers degrees). We give O(n*
*) the
differential
@( ) = @ O  O @.
Then O is an operad in the category of chain complexes and it is the endomorphi*
*sm operad
of 1C~*. Next we consider certain specific elements in O. We have already men*
*tioned the
element `2 O(2). Let
e : Z ! ~C*
denote the element of O(0) which takes 1 to the unit element of the ring R. For*
* each n 2
there is a brace operation
(C~*) n ! ~C*
2
which takes x1 . . .xn to the cochain
x1{x2, . .,.xn}
defined by
X
x1{x2, . .,.xn} = (1)"x1(id, . .,.id, x2, id, . .,.id, xn, id, . *
*.,.id),
where id is the identity map of R, the summation runs over all possible substit*
*utions of x2,
. .,.xn (in that order) into x1 and " is
Xn
xjij,
j=2
(here   denotes the degree in the desuspension and ij is the total number of *
*inputs in front
of xj). Note that if deg(x)< n then the sum is empty and the brace is zero. (*
*The brace
operation for n = 2 was defined by Gerstenhaber in [9], and the higher braces w*
*ere defined
by Kadeishvili [15] and Getzler [11].)
Let H be the suboperad of O generated by e,`, and the brace operations { }n *
*2 O(n)
for n 2. Our first main result is
Theorem 1.1. The singular chain operad of C2 is quasiisomorphic as a chain o*
*perad to H.
We can give a more explicit description of H by considering the relations sa*
*tisfied by e,
` and the { }n ([10, Section 1]). First we have
e ` x = x ` e = x (*
*1)
and
(x ` y) ` z = x ` (y ` z). (*
*2)
Because we are using normalized cochains, we have
x1{x2, . .,.xn} = 0 whenever some xi= 1. (*
*3)
There is a relation between {} and `:
Xn
(x1 . x2){y1, . .,.yn} = (1)"x1{y1, . .,.yk} . x2{yk+1, . .,.yn*
*},(4)
k=0
P k
where this time " = x2 j=1yj. We can write this relation more compactly b*
*y using the
standard notation Ok 0 for the composition in the operad O which inserts 0 *
*in the kth
input position of ; then equation (4) becomes
X
{ }n+1O1 `= ({ }k+1 ` { }nk) O ø (*
*5)
3
(where ø is the permutation that shuffles the second input to the k + 2 positio*
*n). If we
compose brace operations, we have
P h "
x{x1, . .,.xm }{y1, . .,.yn} = (1) x{y1, . .,.yi1, x1{yi1+1, . .,.yj1},*
*(yj1+1,6)
*
* i
. .,.yim, xm {yim+1, . .,.yjm}, yjm+1, . .*
*,.ym } ,
wherePthe sum isPtaken over all sequences 0 i1 j1 i2. . . im jm n *
*and
" = mp=1xp ipq=1yq; this may be written more compactly as
X
{ }n+1 O1 { }m+1 = { } (id, . .,.id, { }, id, . .,.id, { }, id,(*
*.7.,.id))
where the sum is over all ways of interpreting the righthand side. Since [ is *
*a chain map
we have
@(`) = 0 (*
*8)
and the differential on a brace operation is given by
_ !
X
@{ }n = (` O2{ }n1) O ø + ({ }n1Oi `)  (` O1{ }n1) (*
*9)
i
where ø is the transposition that switches the first and second entries. If we *
*consider H as
an operad in the category of graded abelian groups (that is, if we neglect the *
*differential)
then H is the quotient of the free operad generated by e, ` and the brace opera*
*tions by the
relations (1), (2), (3), (5) and (7), and the differential of H is determined b*
*y (8) and (9).
Our method for proving Theorem 1.1 is to construct a topological operad C wh*
*ose struc
ture is based on that of H and to show that the singular chain operad of C is q*
*uasiisomorphic
to H (this is Corollary 7.3) and that C is weakly equivalent (as an operad) to *
*C2 (this is
Theorem 8.1).
The operad C has another significant property. If Xo is a cosimplicial space*
* (or spectrum)
which has cup products and Ok operations which satisfy the same relations as th*
*ose in the
Hochschild cochain complex, then C acts on Tot(Xo). We give a careful statement*
* of this fact
in Theorem 3.3. In particular, this implies that the öt pological Hochschild co*
*homologyö f
an A1 ring spectrum (see Example 3.4 for the definition) is a C2ring spectrum.
In a sequel to this paper we will construct similar models for the little n*
*cubes operads
Cn when n > 2.
The organization of the paper is as follows. Section 2 is a warmup in which*
* we give
a sufficient condition for Tot(Xo) to have an A1 structure. We also introduce*
* our basic
technical tool, prismatic subdivision. The results in this section are closely *
*related to those
in section 5 of Batanin's paper [3]; our proofs are simpler, but less general. *
*In section 3 we
give the background needed to state Theorem 3.3 and some examples to which the *
*theorem
applies. In sections 46 we give the definition of the operad C and prove that*
* it acts on
Tot(Xo) for suitable Xo. We begin in sections 4 and 5 with a simplified model *
*C0 which
captures most but not all of the structure of C2, and in section 6 we complete *
*the definition
4
of C. In section 7 we prove that the singular chain operad of C is quasiisomor*
*phic to H. In
sections 8 and 9 we prove that C is weakly equivalent to C2, using a method of *
*Fiedorowicz.
We would like to thank Lucho Avramov, Clemens Berger, Michael Brinkmeier, Dan
Grayson Rainer Vogt, and Zig Fiedorowicz for helpful conversations, and Peter M*
*ay, Jim
Stasheff, and Sasha Voronov for their continued interest and encouragement duri*
*ng the long
gestation of this project.
5
2 Maps from Tot (Xo ) x Tot (Y o) to Tot (Zo ).
Suppose that Xo, Y oand Zo are cosimplicial spaces. In this section we show how*
* to construct
a map
Tot(Xo) x Tot(Y o) ! Tot(Zo).
from cosimplicial data.
Of course, the simplest way to do this is to begin with a cosimplicial map
Xo x Y o! Zo
and then apply Tot (using the fact that Tot commutes with products) but we will*
* give a
construction more general than this.
In order to explain how a more general construction can be useful, let us co*
*nsider the
cobar construction Xo on a based space A: here Xn is the Cartesian product Axn,*
* the coface
maps insert basepoints and the codegeneracy maps are projections. In this case *
*Tot(Xo) is
homeomorphic to A and thus there is a multiplication map
Tot(Xo) x Tot(Xo) ! Tot(Xo);
in fact there is one such map for each partition of the unit interval into two *
*parts. On the
other hand, there is clearly no sensible way to map Axn x Axn to Axn, so there *
*is in general
no cosimplicial map Xo x Xo ! Xo that can induce the multiplication.
The following definition describes the structure at the cosimplicial level t*
*hat our con
struction accepts as input; it is closely related to Batanin's definition of th*
*e tensor product
of cosimplicial spaces [2].
Definition 2.1. (i) A cuppairing OE : (Xo, Y o) ! Zo is a family of maps
OEp,q: Xp x Y q! Zp+q
satisfying
æ
OEp+1,q(dix, y)if i p
(a) diOEp,q(x, y) = ip
OEp,q+1(x, d y)if i > p
(b) OEp+1,q(dp+1x, y) = OEp,q+1(x, d0y)
æ
OEp1,q(six, y)if i p  1
(c) siOEp,q(x, y) = ip
OEp,q1(x, s y)if i p
(ii) A morphism of cup pairings, from OE : (Xo, Y o) ! Zo to OE0: (X0 o, Y 0*
*o) ! Z0o is a
triple of cosimplicial maps
~1 : X ! X0, ~2 : Y ! Y 0, ~3 : Z ! Z0
such that
~3 O OE0p,q= OEp,qO (~1 x ~2)
6
for all p, q.
(iii) Given a cuppairing OE : (Xo, Xo) ! Xo, a unit for OE is a sequence of*
* points en 2 Xn
such that the set { e
n } is closed under all cofaces and codegeneracies and OE0*
*,p(e0, x) =
OEp,0(x, e0) = x for all x (i.e., the en determine a map from the trivial cosim*
*plicial space to
Xo, and e0 is a unit in the usual sense).
From now on we shall usually drop the subscripts and just write OE(x, y) for*
* OEp,q(x, y).
Remark 2.2. (i) The cobar construction on A has the cuppairing
OE((a1, . .,.ap), (b1, . .,.bq)) = (a1, . .,.ap, b1, . .,.bq)
(ii) The definition of cuppairing is modeled on the properties of the cup p*
*roduct in the
Hochschild cohomology complex. It is easy to check that the function OE(x, y) =*
* x ` y is a
cuppairing in our sense if the Hochschild complex is given the usual cosimplic*
*ial structure
in which siinserts a unit in the i + 1st position and diis defined by
8
< r1x(r2, . .,.rp+1)if i = 0
(dix)(r1, . .,.rp+1) = x(. .,.riri+1, . .).if 0 < i < p + 1
: x(r
1, . .,.rp)rp+1if i = p + 1
(iii) A cosimplicial map ~ : Xo x Y o! Zo induces a cup pairing by
OE(x, y) = ~(dp+qdp+q1. .d.p+1x, d0d0. .d.0y)
(that is, the last coface map is applied q times to x, and the zeroth coface ma*
*p is applied
p times to y, where p is the degree of x and q is the degree of y; then ~ is ap*
*plied to the
resulting pair).
Theorem 2.3. (i) A cuppairing
OE : Xo x Y o! Zo
induces a map
~OEu: Tot(Xo) x Tot(Y o) ! Tot(Zo).
for each u with 0 < u < 1.
(ii) A morphism of cuppairings induces a commutative diagram
~ffiu
Tot(Xo) x Tot(Y o) ! Tot(Zo)
# #
f~fi0u o
Tot(X0 o) x Tot(Y 0o)! Tot(Z0 )
(iii) If the cup pairing comes from a cosimplicial map ~ then each ~OEuis ho*
*motopic to
the usual map induced by ~.
7
Remark. Part (i) of this theorem is implicit in Batanin's paper [3], particular*
*ly in the proof
of [3, Theorem 5.2], which situates this result in a more general context.
Our next result refers to Tot0, which is a construction related to Tot in th*
*e same way
that the Moore loop space is related to the ordinary loop space; we will give t*
*he definition
below.
Theorem 2.4. A strictly associative cuppairing on Xo with a unit induces a s*
*trictly asso
ciative multiplication on Tot0(Xo) and an action of the little 1cubes operad o*
*n Tot(Xo).
Remark 2.5. (i) Batanin [3, Theorems 5.1 and 5.2] uses trees to construct an *
*A1 operad
which acts on Tot(Xo) when Xo has a strictly associative cuppairing. The 0th*
* space of
Batanin's operad is not a point, so that an action of his operad on a space or *
*spectrum
provides a multiplication with a homotopy unit rather than a strict unit.
(ii) If Xo is the cobar construction on a space A then the action of the lit*
*tle 1cubes on
Tot(Xo) is homeomorphic to the usual action of the little 1cubes on (A). Also*
*, there is a
continuous bijection from Tot0(Xo) to the Moore loop space of A which takes the*
* multipli
cation on Tot0(Xo) to the usual multiplication on the Moore loop space.
Before giving the proof of Theorems 2.3 and 2.4, we will describe a way of s*
*ubdividing
a simplex which we call the "prismatic subdivision." This is a homeomorphism di*
*scovered
independently and at various times by Lisica and Mardesic [17], Grayson [13], a*
*nd ourselves.
(There is a related "edgewise subdivision" discovered by Quillen, Segal, Bökste*
*dt and Good
willie; the edgewise subdivision of a simplex is a subdivision of the prismatic*
* subdivision.)
It will be convenient from now on to let s, t, etc. stand for a point of a s*
*implex and let
s0, s1, . .(.respectively, t0, t1, . .,.etc.) be its coordinates.
Define a space Dn for each n 0 by
an
Dn = ( p x np)= ~,
p=0
where ~ is defined by (dp+1s, t) ~ (s, d0t) if s 2 p and t 2 np1. For each*
* u with
0 u 1, there is a map
oen(u) : Dn ! n
whose restriction to p x np takes
(s0, . .,.sp), (t0, . .,.tnp)
to
(us0, . .,.usp1, usp + (1  u)t0, (1  u)t1, . .,.(1  u)tnp).
It is easy to check that this map is welldefined, continuous and onto, and whe*
*n 0 < u < 1
it is also onetoone. Thus we have:
Proposition 2.6. oen(u) is a homeomorphism if 0 < u < 1.
8
This homeomorphism is the prismatic subdivision of the simplex n.
Here are the prismatic subdivisions of the 1simplex and the 2simplex (with*
* u = 1_2):
__________________

@
@
 @
 @
_____@_
 @
 @
  @
  @
_______________@_
We can now prove part (i) of Theorem 2.3 Recall that a point in Tot(Xo) is a*
* sequence
a0 2 X0, a1 : 1 ! X1, a2 : 2 ! X2, . . .
which is consistent, i.e.,
diO an = an+1 O di
and
siO an = an1 O si
Thus what is required is: given consistent sequences an : n ! Xn and bn : n !*
* Y n, to
construct a consistent sequence cn : n ! Zn.
First observe that, by part (b) of Definition 2.1(i), the maps
OE O (ap x bnp) : p x np ! Zn
fit together to give a welldefined map
Dn ! Zn
(where Dn is the space defined before Proposition 2.6). We define cn for each n*
* to be the
composite of this map with (oen(u))1. The fact that the cn commute with coface*
* and code
generacy maps is immediate from the formula for oen(u) and parts (a) and (c) of*
* Definition
2.1(i). This concludes the proof of part (i).
Here are pictures of c1 and c2 (with u = 1_2):
a0 ` b1 a1 ` b0
__________________________
9

@
 @
 @@
 @
a2 ` b0 @
 @
___________@_
 @
  @
 a1 ` b1  @
  @
 a0 ` b@2
  @
_______________________@_
Part (ii) of Theorem 2.3 is immediate from the definitions.
We now turn to the proof of part (iii). By part (ii), it suffices to conside*
*r the case where
~ is the identity map of Xo x Y o. Let OE : (Xo, Y o) ! Xo x Y odenote the pair*
*ing induced
by the identity map of Xo x Y o(as in Remark 2.22(iii)): thus
OE(x, y) = (dp+qdp+q1. .d.p+1x, d0d0. .d.0y)
Let
F : Tot(Xo) x Tot(Y o) ! Tot(Xo) x Tot(Y o)
denote ~OE1=2; it suffices to show that F is homotopic to the identity. Let ß1 *
*and ß2 be the
projections of Tot(Xo)xTot(Y o) on its first and second factors; it suffices to*
* show that ßiOF
is homotopic to ßifor i = 1, 2. We consider the case i = 1; the other case is s*
*imilar. Let ~Dn
denote the space
a 1
( p x ( np ^ [__, 1]))= ~,
p 2
where the basepoint of [1_2, 1] is taken to be 1 and ~ is defined by (dp+1s, t *
*^ u) ~ (s, d0t ^ u)
if s 2 p and t 2 np1. The map ø : ~Dn! n x [1_2, 1] given by
ø(s, t ^ u) = (oen(u)(s, t), u)
is a homeomorphism, since it is 11, onto, and continuous. For each n, the maps
1 o o n
p x ( np ^ [__, 1]) x Tot(X ) x Tot(Y ) ! X
2
which take (s, t^u, {an}, {bn}) to an(dndn1. .d.p+1s) fit together to give a w*
*elldefined map
~Dnx Tot(Xo) x Tot(Y o) ! Xn.
Composing this with (øn)1 gives a map
1 o o n
n x [__, 1] x Tot(X ) x Tot(Y ) ! X ,
2
10
whose adjoint is a map
H 1_ o o n n
n : [2 , 1] x Tot(X ) x Tot(Y ) ! M ap( , X ).
Taken together, the Hn give a map
1 o o o
H : [__, 1] x Tot(X ) x Tot(Y ) ! Tot(X )
2
which is equal to ß1O F when u = 1_2and to ß1 when u = 1. Thus ß1O F is homotop*
*ic to ß1
*
* __
as required. *
* __
Before proving Theorem 2.4 we define Tot0. First we define np, for each p *
* 0, to be the
set
{ (s0, . .,.sn)  si 0 for all i ands0 + . .+.sn = p }
(Note that opis a cosimplicial space for each p, and that o0is is the trivial*
* cosimplicial
space consisting of a point in each degree.) Then we define Tot0(Xo) to consis*
*t of pairs
(p, {an}), where p is 0 and {an} is a cosimplicial map op! Xo. In order to *
*describe
the topology of Tot0we first observe that for p > 0 the cosimplicial spaces op*
*and o are
isomorphic, and thus there is a bijection between Tot0(Xo) and the disjoint uni*
*on
Const(Xo) [ (Tot(X) x R>0),
where Const(Xo) denotes the space of cosimplicial maps from o0to Xo and R>0 de*
*notes the
positive reals. This in turn implies that there is a bijection between T 0(Xo) *
*and the pushout
Const(Xo) x R>0 ! Const(Xo) x R 0
#
Tot(Xo) x R>0
We give Tot0(Xo) the topology which makes this bijection a homeomorphism. (We *
*could
have simply defined Tot0(Xo) to be this pushout, but this would make the formul*
*as in the rest
of the paper more complicated). Note that there is a continuous projection æ : *
*Tot0(Xo) !
Tot(Xo) which makes Tot(Xo) a deformation retract of Tot0(Xo).
Proof of Theorem 2.4. We need a slight generalization of the idea of prismatic *
*subdivision:
define a space Dnp,qfor each n 0 and each p, q 0 by
an
Dnp,q= ( kpx nkq)= ~,
k=0
where ~ is defined by (dk+1s, t) ~ (s, d0t) if s 2 kpand t 2 nk1r. Define
oenp,q: Dnp,q! np+q
by
((s0, . .,.sk), (t0, . .,.tnk)) = (s0, . .,.sk1, sk + t0, t1, . .*
*,.tnk).
11
Then oenp,qis a homeomorphism for every choice of p and q.
Now we can define the multiplication on Tot0. Given a pair of points (p, {a*
*n}) and
(q, {b 0
n}) in Tot then, by part (b) of Definition 2.1(i), the maps
OE O (ak x bnk) : kpx nkq! Xn
fit together to give a welldefined map
Dnp,q! Xn
Composing this map with (oenp,q)1 gives a map cn : np+q! Xn for each n. The c*
*n commute
with coface and codegeneracy maps by parts (a) and (c) of Definition 2.1(i), an*
*d we define the
product of the points (p, {an}) and (q, {bn}) to be the point (p+q, {cn}). This*
* multiplication
is clearly associative and unital, and we leave it to the reader to check that *
*it is continuous.
It remains to give the action of the little 1cubes operad C1 on Tot(Xo). So*
* given a point z
of C1(k) and k points a1, . .,.ak of Tot(Xo), we are required to define a point*
* fl(z, a1, . .,.ak)
in Tot(Xo). The idea is to imitate the way that C1 acts on a loop space, that i*
*s, we scale
the a's to the lengths of the corresponding little intervals and fill in the bl*
*ank spaces with
appropriately scaled copies of the unit element.
First observe that the point z of C1(k) can be written as a sequence of 2k +*
* 1 numbers
(p1, p2, . .,.p2k+1), where the even numbered p's are the lengths of the little*
* intervals and
the odd numbered p's are the lengths of the empty spaces.
Next we introduce some notation. Given a function f : n ! Xn and a number p*
* > 0,
let us write p . f for the function np! Xn which takes s to f(1_p. s). Given a*
* point a = {an}
in Tot(Xo), let us write p . a for the point (p, {p . an}) in Tot0(Xo). Recall*
* that the cup
product has a unit e, which is a map from o0to Xo, and for each p 0 write ep*
* for the
composite
op! o0e!Xo
Let * denote the multiplication on Tot0(Xo). Then we define fl(z, a1, . .,.ak) *
*to be
æ(ep1* (p2 . a1) * ep3* (p4 . a2) * . .*.ep2k+1),
*
* __
where æ is the projection from Tot0(Xo) to Tot(Xo). *
* __
12
3 Operads with multiplication and their associated cosim
plicial objects.
In this section we give a formal statement of the fact that if Xo is a cosimpli*
*cial space or
spectrum with ` and Ok operations that satisfy the same relations as those in t*
*he Hochschild
cochain complex then our operad C acts on Tot(Xo). For this we need some backgr*
*ound.
First recall that if O is an operad with structure maps
fl : O(n) x O(j1) x . .x.O(jn) ! O(j1 + . .+.jn)
and identity element id2 O(1), it is customary to write Oifor the map
O(n) x O(j) ! O(n + j  1)
which takes (o1, o2) to fl(o1, id, . .,.o2, . .,.id) (with i  1 id's before th*
*e o2).
Next let R be a ring and let OR denote the endomorphism operad of the underl*
*ying
abelian group of R; that is, OR(n) is the set of homomorphisms of abelian group*
*s from
R n to R, with the evident operad structure. There are special elements ~ 2 OR*
*(2) (the
multiplication in the ring R) and e 2 OR(0) (the unit element of R). Of course,*
* OR(n) is
the same as the group of ncochains in the Hochschild complex of R, and the cos*
*implicial
structure of the Hochschild complex can be recovered from the operad structure *
*of OR and
the elements ~ and e:
8
< ~ O2 x if i = 0
(dix) = x Oi~ if 0 < i < p + 1 (1*
*0)
: ~ O
1 x if i = p + 1
si(x) = x Oi+1e (1*
*1)
When the di and si are defined in this way the cosimplicial identities follow f*
*ormally from
the identities
~ O1 ~ = ~ O2 ~ (1*
*2)
and
~ O1 e = ~ O2 e = id. (1*
*3)
This example motivates the following definition, which is due to Gerstenhabe*
*r and
Voronov [10]. Recall ([19, Definition 3.12]) that a non operad is a structur*
*e which has
all the properties of an operad except those having to do with the actions of t*
*he symmetric
groups. Also recall that the definition of operad makes sense in any symmetric*
* monoidal
category.
Definition 3.1. Let S be a symmetric monoidal category with unit object S. An o*
*perad
with multiplication in S is a non operad O in S together with maps e : S ! O(*
*0) and
~ : S ! O(2) satisfying (12) and (13). The associated cosimplicial object Oo co*
*nsists of the
objects O(n) with coface and codegeneracy maps given by (10) and (11).
13
The reader will perhaps be relieved to know that the only symmetric monoidal*
* categories
we will be concerned with in this paper are the category of spaces, the categor*
*y of Smodules
[7, Chapter II], and (briefly) the category of sets.
Remark 3.2. (i) If we let Ass be the non operad for which Ass(n) is the obj*
*ect S for
each n (so that an algebra over Ass is an associative monoid in S) then a more *
*economical
(but equivalent) way to define an operad with multiplication is: a non operad*
* together
with a non operad morphism from Ass.
(ii) If O is an operad with multiplication then the associated cosimplicial *
*object has a
cuppairing in the sense of Section 2: we define OEp,q(x, y) to be (~ O1 x) Op+*
*1y.
We can now state our main result. Recall that a weak equivalence of operads *
*is an operad
morphism which is a weak equivalence on each object. We say that two operads ar*
*e weakly
equivalent if there is a third operad which maps to each of them by a weak equi*
*valence.
Theorem 3.3. There is an operad C in the category of spaces with the followin*
*g properties:
(i) C is weakly equivalent to the little 2cubes operad C2.
(ii) If O is an operad with multiplication in the category of spaces or spec*
*tra then C acts
on Tot(Oo).
Here are some examples to which Theorem 3.3 can be applied. In the examples *
*that refer
to the category of Smodules we will always write ^ for ^S.
Example 3.4. (The Hochschild cohomology complex of a ring spectrum.) Let R
be an Salgebra in the sense of [7, Section II.3] and let O be the endomorphism*
* operad of
R:
O(n) = FS(R^n, R)
(see [7, Section II.1] for the definition of FS). Let e : S ! R be the unit map*
* of R and let
~ : S ! O(2) be adjoint to the multiplication map R ^ R ! R. Then O is an opera*
*d with
multiplication and we define the Hochschild cohomology complex of R to be the c*
*osimplicial
Smodule associated to O.
Example 3.5 (The loop space of a topological monoid.). (We would like to than*
*k Zig
Fiedorowicz for pointing out this example to us.) Let A be a topological monoid*
*, with the
unit of A chosen as basepoint. We will define an operad with multiplication O w*
*hose asso
ciated cosimplicial space is the cobar construction on A. Of course, we let O(n*
*) = Axn. In
order to define the operad structure we observe (as in [20, p. 6]) that it suff*
*ices to specify
the operations
Oi: O(n) x O(j) ! O(n + j  1),
and we define these by the equation
(a1, . .,.an) Oi(b1, . .,.bj) = (a1, . .,.ai1, aib1, . .,.aibj, ai+*
*1,(.1.a.n)4)
Finally, we choose e 2 O(0) and ~ 2 O(2) to be the basepoints. (Observe that t*
*he cup
pairing determined by ~ is the same as that defined in Remark 2.2(i).)
14
In preparation for our next two examples it is helpful to reformulate Exampl*
*e 3.5 in a
fancier way. Let Bo denote the simplicial circle 1=@ 1. Recall that a nonbase*
*point simplex
in B
n is a sequence (b0, b1, . .,.bn) of n + 1 zeroes and ones, with the zeroes*
* coming before
the ones. We can put the nonbasepoint simplices of Bn in onetoone correspond*
*ence with
the numbers 1 to n by letting the number j correspond to the simplex with j zer*
*oes, and this
gives a homeomorphism ff between O(n) and the space of based maps from Bn to A;*
* under
ff the coface and codegeneracy maps of O(n) agree with the maps induced by the *
*faces and
degeneracies of Bn. If x 2 O(n) and b 2 Bn then we write x(b) for the element f*
*f(x)(b) of
A. Now equation (14) is equivalent to the following: if x 2 O(n), y 2 O(j) and *
*b 2 Bn then
(x Oiy)(b) = x(b0ij) . y(b00ij), (1*
*5)
where . denotes the product in A, b0ijis the simplex
(b0, b1, . .,.bi1, bi+j1, . .,.bn+j1)
and b00ijis the simplex
(bi1, . .,.bi+j1).
We can generalize the definitions of b0ijand b00ijto any simplicial set Bo a*
*s follows. If s t
we write @(s, t) for the composite of face maps
@s@s+1. .@.t
Now if n 1, j 0, 1 i n, and b 2 Bn+j1, we define
b0ij= @(i, i + j  2)b
and
b00ij= @(0, i  2)@(i + j, n + j  1)b.
These definitions will be familiar to many readers because they are related to *
*Steenrod's
original definition of the `1 product [24]: if , 2 Cn(Bo; Z=2) and j 2 Cj(Bo; Z*
*=2) are mod2
simplicial cochains of Bo and b 2 Bn+j1 then
X
(, `1 j)(b) = ,(b0ij)j(b00ij)
i
Example 3.6 ( k of a space when k 2.). Let A be any based space and fix k *
*2.
We can define a "higher" cobar construction Xo, with Tot(Xo) homeomorphic to k*
*A, as
follows: let Bo be the simplicial ksphere k=@ k and let Xn be the space of ba*
*sed maps
from Bn to A, with coface and codegeneracy maps induced by the face and degener*
*acy maps
of Bo. Next we define an operad with multiplication O whose associated cosimpli*
*cial space
is Xo. Of course, we let O(n) = Xn. We define the Oi operations by equation (*
*15), but
note that the symbol . now requires interpretation, since we are not assuming t*
*hat A is a
monoid. We are saved by the fact that (since k 2) either b0ijor b00ijis alway*
*s the basepoint,
and thus either x(b0ij) or y(b00ij) is always the basepoint * of A. It therefor*
*e suffices to define
a . * = * . a = a for all a 2 A to make equation (15) meaningful in this contex*
*t. Finally, we
choose e 2 O(0) and ~ 2 O(2) to be the basepoints.
15
As background for our final example we recall that, if R is a commutative ri*
*ng and m 2,
Pirashvili [21] has defined a higher Hochschild homology complex Yo(R) by letti*
*ng Yn(R) be
the tensor product
O
R
Bn
(where Bo is the simplicial set from example 3.6), with face and degeneracy map*
*s induced by
those of Bo; thus Yo(R) is the tensor product Bo R in the category of simplicia*
*l commutative
rings, as defined by Quillen [22].
We also recall [7, Proposition VII.1.6] that ^ is the coproduct in the categ*
*ory of commu
tative Salgebras. It follows that if R is an Salgebra, A and B any sets and f*
* : A ! B any
map then f induces a map
^ ^
f* : R ! R
A B
of Salgebras.
Example 3.7. (The Pirashvili cohomology complex of a commutative ring spec
trum.) Fix k 2. Let R be a commutative Salgebra, and define the kth Piras*
*hvili
homology complex Yo(R) by letting
^
Yn(R) = R
Bn
(where Bo is the simplicial set from example 3.6) with face and degeneracy maps*
* induced by
those of Bo. The inclusion of the basepoint in Bn induces a map R ! Yn(R) of co*
*mmutative
Salgebras which makes Yn(R) an Rmodule. We define the kth Pirashvili cohomo*
*logy
complex of R to be the cosimplicial Rmodule Xo(R) with
Xn(R) = FR(Yn(R), R)
(see [7, Section III.6] for the definition of FR), with coface and codegeneracy*
* maps induced by
the face and degeneracy maps of Yo. Next we want to define an operad with multi*
*plication O
whose associated cosimplicial object is Xo. Let O(n) = Xn. In order to define t*
*he operation
Oiwe first observe that, since k 2, the map
Bn+j1! Bn x Bj
which takes b to (b0ij, b00ij) factors through the wedge to give a map
Bn+j1! Bn _ Bj
and this in turn induces a map
^ ^
fi: Yn+j1(R) = R ! R
Bn+j1 Bn_Bj
16
We next observe that there is a natural isomorphism
^ R ~=(^ R) ^R (^ R) = Yn(R) ^R Yj(R)
Bn_Bj Bn Bj
We can therefore define Oito be the composite
^R
Xn ^ Xj = FR(Yn(R), R) ^ FR(Yj(R), R) ! FR(Yn(R) ^R Yj(R), R ^R R)
^ f*
~= FR( R, R) i!FR(Yn+j1(R), R) = Xn+j1
Bn_Bj
It remains to specify the maps e : S ! O(0) and ~ : S ! O(2). We note that O(0)*
* is just
R, so we can let e be the unit map of R. If k > 2 then O(2) is also R, and we c*
*an let ~ be
the unit map. If k = 2 then O(2) is isomorphic to FS(R, R), and we let ~ be the*
* adjoint of
the identity map.
17
4 The spaces of the operad C0.
Recall that C2 denotes the operad of little 2cubes. The zeroth space of C2 is *
*a point. We
can define a suboperad C02of C2 by letting C02(n) = C2(n) for all n > 0 and let*
*ting C02(0) be
the empty set. All of the information about C2 and its operad structure is cont*
*ained in C02
except for the socalled "degeneracy" maps
Ok : C2(n) x C2(0) ! C2(n  1)
(the corresponding maps for C02have the empty set as their domain).
In this section and the next we define an operad C0 which will turn out to b*
*e equivalent
to C02. In section 6 we will define the operad C of theorem 3.3 by adding one m*
*ore ingredient
to the definition of C0.
4.1 The spaces P(n) and F(n).
The nth space C0(n) is empty for n = 0 and for n > 0 it is the Cartesian produ*
*ct of a space
F(n) and a contractible space P(n). (The idea here is that the spaces F(n) by t*
*hemselves
have a structure like that of an operad but with a composition operation fl whi*
*ch is only
A1 associative. By combining the F(n) with the P(n) we get a true operad).
The space P(n) is easy to define: it is the empty set for n = 0 and for n > *
*0 it is the set
of ntuples (p1, . .,.pn) of positive real numbers that add up to 1.
The structure of the space F(n) is related to that of the algebraic operad H*
* defined in
the introduction. We will describe it in several stages: first we define the se*
*t that indexes
the cells of F(n), then we define the cells themselves, and finally we define t*
*he attaching
maps.
4.2 The set of öf rmulas of type n."
The cells of F(n) will be indexed by what we call öf rmulas of type n." Specif*
*ically, a
formula of type n is a symbol constructed from the numbers 1, 2, . .n.(with no *
*repetitions)
by formal cup products and formal operad compositions. (The formulas of type n*
* corre
spond to certain multilinear operations in the Hochschild complex. The formal c*
*up products
correspond to cup products in the Hochschild complex, and the formal operad com*
*positions
correspond to brace operations.) For ease of notation we write the formal opera*
*d composition
fl(x0; x1, . .,.xn) as x0(x1, . .,.xn).
Some examples of formulas of type 5 are: 2(1(3, 5), 4), 1(2 ` 4, 3(5)), 4(2,*
* 3) ` 1(5).
Here is a more precise definition. Define a sequence of sets {Sk} for k 0*
* as follows:
Sk is the set of positive integers when k 6= 2 and S2 is the set of positive in*
*tegers with the
symbol ~ adjoined. Take the free operad generated by the sequence of sets Sk an*
*d impose
the relation (12) from Section 3. Let the resulting operad be denoted by J 0(th*
*e prime refers
to the fact that we are engaged in defining C0). Then the set of formulas of ty*
*pe n is precisely
the set of elements in J 0(0) which contain each of the symbols 1, . .,.n exact*
*ly once.
Remark 4.1. For clarity we should point out that, in this way of defining the*
* formulas of
type n, the cup products are denoted by ~ instead of `, so that for example the*
* formula
18
1(2 ` 4, 3(5)) would be written 1(~(2, 4), 3(5)) and the formula 4(2, 3) ` 1(5)*
* would be
written ~(4(2, 3), 1(5)).
For our later definitions it will be convenient to define the valence of an *
*integer in Sk to
be the number k. Thus in a formula of type n the valence of a symbol 1, . .,.n *
*is the number
of formal inputs for that symbol. For example, in the formula
3(2, 4(5, 6), 1, 7) ` 8(9)
the valence of the symbol 3 is 4, the valence of the symbol 4 is 2, the valence*
* of the symbol
8 is 1 and the valences of 1, 2, 5, 6, 7, 9 are each 0.
We will write v(i) for the valence of i.
4.3 The cells of F(n).
To each formula f of type n we associate the product of simplices
Yn
v(i)
i=1
where v(i) is the valence of the symbol i. This product of simplices is a cell *
*of F(n). We
will frequently denote the cell associated to a formula f by Ff.
Here are some examples of cells of F(5):
F2(1(3,5),4)= 2 x 2 x 0 x 0 x 0,
F1(2`4,3(5))= 2 x 0 x 1 x 0 x 0,
and
F4(2,3)`1(5)= 1 x 0 x 0 x 2 x 0.
4.4 Identifications Along The Boundary.
The boundary of each cell Ff is a union of other cells which are selected by a *
*rule modeled
on the relation (9) given in the introduction. Before spelling this out we give*
* some examples
of what the rule will say.
Example. The cell F1(2)is an interval whose initial point is identified with F1*
*`2 and whose
terminal point is identified with F2`1. Similarly, the cell F2(1)is an interva*
*l whose initial
point is identified with F2`1 and whose terminal point is identified with F1`2.*
* Since the
only cells of F(2) are F1(2), F2(1), F1`2, and F2`1, we conclude that the space*
* F(2) is
homeomorphic to a circle.
Example. F1(2,3)is a triangle whose three faces are identified with F2`1(3), F*
*1(2`3)and
F1(2)`3.
19
Example. F1(2(3))is a square whose four faces are identified with F2(3)`1, F1`2*
*(3), F1(3`2),
and F1(2`3).
Next we give a formal description of the rule which underlies these examples*
*. We will
use the operad J 0described in subsection 4.1.
Let f be a formula of type n and let i be an integer with 1 i n. Let k b*
*e the valence
of i in f and let j be an integer with 0 j k. If k 1 we define @ijf to be*
* the formula
obtained from f by replacing the symbol i by
8
< ~ O2 ik1if j = 0
ik1Oj~ if 0 < j < k
: ~ O
1 ik1if j = k
Here we are writing ik for the copy of i in Sk; also see Remark 4.1. Of course,*
* this definition
is motivated by equation (10). Here are some examples:
@101(2) = 2 ` 1 and @111(2) = 1 ` 2
@101(2, 3) = 2 ` 1(3), @111(2, 3) = 1(2 ` 3), and @121(2, 3) = 1(2) *
*` 3
@101(2(3)) = 2(3) ` 1, @111(2(3)) = 1 ` 2(3)
@201(2(3)) = 1(3 ` 2), @211(2(3)) = 1(2 ` 3)
Now let In be the set of formulas of type n. We can define a partial orderi*
*ng on In
as follows: given a formula f of type n, the maximal elements in the set { g  *
*g < f } are
{ @ijf  1 i n, 0 j v(i) }.
Let us now think of the partially ordered set In as a category in the usual *
*way. We
can make the function F defined in the previous subsection into a functor from *
*In to the
category of topological spaces by taking the arrow @ijf ! f to the map
i1Y nY Yn
1 x . .x.djx . .x.1 : v(k)x v(i)1x v(k)! v(k)
k=1 k=i+1 k=1
where dj is the jth face map
v(i)1! v(i)
Now we can give a formal definition of the space F(n):
Definition 4.2. F(n) = colimInF
This definition implies that the boundary of a cell Ff is the union of the c*
*ells F@ijf.
20
5 The operad structure of C0 and the action of C0 on
Tot (Xo ).
Our next goal is to do two things: to define the operad structure of C0 and to *
*define the
action maps
`n : C0(n) x (Tot(Xo))n ! Tot(Xo)
when Xo is a cosimplicial space arising from an operad with multiplication (the*
* case where Xo
is a cosimplicial Smodule requires only routine changes and we will not discus*
*s it separately).
It turns out that the second of these tasks is easier and provides helpful back*
*ground for the
first, so we begin with it and defer the definition of the operad structure to *
*the end of this
section.
We will construct `n as a map of sets and leave it to the reader to verify c*
*ontinuity.
Recall that C0(n) = P(n) x F(n). In order to construct `n it suffices to co*
*nstruct, for
each choice of p = (p1, . .,.pn) 2 P (n) and of a cell Ff of F(n), a map
`p,f: Ff x (Tot(Xo))n ! Tot(Xo)
Next recall that
Tot(Xo) = Hom ( o, Xo),
where Hom denotes morphisms of cosimplicial spaces, so in order to construct `*
*n it suffices
to construct, for each k 0, a suitable map
~`p,f,k: k x Ff x (Tot(Xo))n ! Xk.
We do the case k = 0 first, since this illustrates the general idea in a simple*
* situation.
Example 5.1 (The case k = 0.). Given a formula f of type n and elements xi2 Xv*
*(i)for
1 i n, we can define an element ~f(x1, . .,.xn) 2 X0 by replacing each symb*
*ol i in f
by xi and then interpreting the formal compositions and cup products in f to be*
* genuine
compositions and cup products in Xo. For example, if f = 3(1(2 ` 4), 6(5)) then
f~(x1, . .,.x6) = x3(x1(~(x2, x4)), x6(x5)).
This process gives a map
Yn
~f: Xv(i)! X0.
i=1
Now let s 2 Ff, and ai 2 Tot(Xo) for 1 i n; thus s is an ntuple s1, . .,.s*
*n with
si2 v(i)and each aiis a sequence { aim : m ! Xm }. We define
~`p,f,0(s, a1, . .,.an) = ~f(a1v(1)(s1), . .,.anv(n)(sn)).
(Note that p plays no role in this definition, but it will have a role for k > *
*0).
21
In order to extend this idea to kQ> 0 we need a way of decomposing the produ*
*ct kx Ff
as a union of products of the form ni=1 ki. Here is an example which should *
*help the
reader to follow the general description. It may also be helpful to compare thi*
*s example to
the relation (7) for iterated brace products given in the introduction.
Example 5.2. The following picture shows how to define the map
`~p,f,k: k x Ff x (Tot(Xo))n ! Xk
for the case in which n is2,f is 1(2), k is 1, and p = (1=2,1=2).
__________________________________
 
 
 
 
 a2 O1 b0 
 
 
 
 
 
 
 
 a1 O1 b1 
 
 
 
 
 
 
 
 
 a2 O2 b0 
 
 
 
___________________________________
Here a = {am } and b = {bm } are points of Tot(Xo), and the picture defines a m*
*ap from
1x F1(2)to X1 (note: the F1(2)coordinate is the horizontal direction). This pi*
*cture is part
of a homotopy from b ` a to a ` b, since the left and righthand edges are the*
* projections
of b ` a and a ` b on Hom ( 1, X1).
Notice that each vertical crosssection of the picture in example 5.2 is a 3*
*fold prismatic
subdivision of 1 (except at the endpoints where it degenerates to a 2fold pri*
*smatic subdivi
sion). We therefore think of the picture in example 5.2 as a "fiberwise prismat*
*ic subdivision"
of 1 x F1(2). Our next goal is to define the fiberwise prismatic subdivision o*
*f k x Ff in
general, and for this we first need to define the "thickeningsö f a formula f.
5.1 The thickenings of a formula.
The cells of the fiberwise prismatic subdivision of k x Ff will correspond to *
*a collection
of formulas called the kthickenings of f. To thicken a formula means to add ex*
*tra inputs
(denoted by the symbol id) to the formula (we will give a formal definition in *
*a moment).
The new formula is called a kthickening if there are k copies of the symbol id.
Before giving the formal definition we give an example. The 1thickenings o*
*f 1(2) are
1(id, 2), 1(2, id), 1(2(id)), 1(id` 2), 1(2 ` id), id` 1(2)) and 1(2) ` id, and*
* in Example 5.2
these correspond to the cells which are labeled by a2 O2 b0, a2 O1 b0, a1 O1 b1*
*, and to the two
diagonal and the two horizontal edges, respectively.
22
For the formal definition, we use the operad J 0defined in subsection 4.1. L*
*et us define a
formula of type (n, k) to be an element of J 0(k) which contains each of the sy*
*mbols 1, . .,.n
exactly once (note that such a formula is forced to contain the symbol idexactl*
*y k times).
There is a reduction map
æ : J 0(k) ! J 0(0)
which removes each copy of id, together with the typographical symbols that are*
* immediately
adjacent to it (these may be commas, parentheses, or `'s). For example, æ take*
*s 1(2 `
id, id, 3(4(id))) to 1(2, 3(4)).
Definition 5.3. If f is a formula of type n, the kthickenings of f are the for*
*mulas g of
type (n, k) for which æ(g) = f.
5.2 Fiberwise prismatic subdivision.
Given a formula f of type n, let If,kbe the set of kthickenings of f. We can *
*define a
function F from If,kto the set of topological spaces exactly as in subsection 4*
*.3, that is,
Yn
Fg = v(i)
i=1
where g is in If,kand v(i) denotes the valence of i in g.
For our present purposes we need to generalize this, using simplices of vari*
*able sizes as
in Section 2 : given q = (q1, . .,.qn) 2 (R 0)n and g 2 If,kwe define
Yn
Fg,q= v(i)qi;
i=1
this gives a function F,qfrom If,kto the set of topological spaces for each ch*
*oice of q.
Next observe that we can make If,kinto a partially ordered set, and F,qinto*
* a functor
from If,kto the category of topological spaces, exactly as we did with In in su*
*bsection 4.4.
Definition 5.4. Ff,k,qis the space colimIf,kF,q.
(In example 5.2, F1(2),1,(1,1)is the union of two triangles and a square, wi*
*th the edge
identifications indicated in the picture there.)
The fiberwise prismatic subdivision is a certain map
oe : Ff,k,q! kq1+...+qnx Ff,q
which will turn out to be a homeomorphism. In order to describe it we need some*
* terminol
ogy.
Let g be an element of If,k. Then g is a characterstring consisting of the *
*integers 1, . .,.n
together with k copies of the symbol idand some commas and parentheses. If i 2 *
*{ 1, . .,.n }
and v(i) > 0 then there are v(i)1 commas, a left parenthesis, and a right pare*
*nthesis which
belong to i (to be precise, the characters that belong to i are the left and ri*
*ght parentheses
23
that enclose the inputs of i and the commas that separate them). If i has vale*
*nce 0 then
we say that i belongs to itself. Note that in either case there are v(i) + 1 ch*
*aracters which
belong to i. We refer to the characters of g which belong to some i as the elig*
*ible characters
of g; thus the eligible characters are all of the commas, all of the parenthese*
*s, and the i with
valence 0.
Next let s be an element of Fg,q. Now Fg,qis a product of simplices indexed *
*by 1, . .,.n,
and we write si for the projection of s on the ith factor; thus si 2 v(i)qi. *
* Let us fix i
temporarily. As an element of v(i)qi, sihas v(i) + 1 coordinates, and we denot*
*e these by sij.
Since there are v(i) + 1 characters belonging to i in g, we can match the numbe*
*rs sijwith
the characters that belong to i (going from left to right as j increases), and *
*letting i vary we
get a onetoone correspondence between the eligible characters of g and the nu*
*mbers sij,
1 i n, 0 j v(i). Here is an example, for the formula g = 3(2(1, id), 4,*
* id, id, 5(6)):
3 ( 2 ( 1 , id ) , 4 , id , id , 5 ( 6 ) )
s30 s20 s10 s21 s22 s31 s40 s32 s33 s34 s50 s60 s51 s35
We will refer to this diagram as the tableau of g. The tableau of g has two lin*
*es, with the
first being the string of characters of g and the second the symbols sijin the *
*order we have
prescribed.
Now we are ready to describe the fiberwise prismatic subdivision map
oe : Ff,k,q! kq1+...+qnx Ff,q
We write oe2 for the projection of oe on the second factor:
oe2 : Ff,k,q! Ff,q
Since Ff,qis itself a product, it suffices to define the projection of oe2 on t*
*he ith factor of
Ff,q; we denote this projection by oe2i. Let g 2 If,kand let s = (s1, . .,.sn) *
*2 Fg,q. In order
to apply oe2ito s, we select the coordinates sijfrom the second line of the tab*
*leau of g and
then insert + signs whenever the symbol idoccurs in the first line. (The motiva*
*tion for this
is that f is obtained from g by collapsing out the symbols idthat occur in g.) *
*For example,
in the tableau given above we have
oe21(s)= s10
oe22(s)= (s20, s21+ s22)
oe23(s)= (s30, s31, s32+ s33+ s34, s35)
oe24(s)= s40
oe25(s)= (s50, s51)
oe26(s)= s60
(Note that the restriction of oe2 to Fg,qis a Cartesian product of iterated deg*
*eneracies, with
the idsymbols telling what degeneracies are to be used.)
Next we define the projection of oe on the first factor, which we denote by *
*oe1:
oe1 : Ff,k,q! kq1+...+qn
24
Again let g 2 If,kand let s = (s1, . .,.sn) 2 Fg,q. In order to apply oe1 to s,*
* we insert +
signs in the second line of the tableau wherever the symbol iddoes not occur in*
* the first line.
For example, in the tableau given above we have k = 3 and
oe1(s) = (s30+ s20+ s10+ s21, s22+ s31+ s40+ s32, s33, s34+ s50+ s60+ s51+*
* s35).
When these definitions are applied to example 5.2, the formulas that result *
*are:
o If s 2 F1(id,2),qthen oe1(s) = (s10, s11+ s20+ s12) and oe2(s) = ((s10+ s1*
*1, s12), s20).
o If s 2 F1(2,id),qthen oe1(s) = (s10+ s20+ s11, s12) and oe2(s) = ((s10, s1*
*1+ s12), s20).
o If s 2 F1(2(id)),qthen oe1(s) = (s10+ s20, s21+ s11) and oe2(s) = ((s10, s*
*11), s20+ s21).
Returning to the general situation, it is not difficult to see that the rest*
*riction of oe1 to
each fiber of oe2 is a prismatic subdivision of kq1+...+qn; this accounts for *
*the name "fiberwise
prismatic subdivisionä nd also implies
Proposition 5.5. oe is a homeomorphism.
Finally, it is convenient to introduce a variant of oe. Let p 2 P(n) and let*
* f be a formula
of type n. For each g 2 If,kdefine
.p : Fg ! Fg,p
to be the map which takes (s1, . .,.sn) 2 Fg to (p1s1, . .,.pnsn) 2 Fg,p. Thes*
*e maps fit
together to give a map
.p : Ff,k! Ff,k,p
(where, as the reader may have guessed, Ff,kis an abbreviation for Ff,k,(1,...,*
*1)). Now define
oe(p) : Ff,k! k x Ff
to be the composite
.p ff k 1x(.p)1k
Ff,k! Ff,k,p! x Ff,p! x Ff
Note that the projection of oe(p) on the second factor Ff is the same as the pr*
*ojection of oe
on Ff (that is, it is the map oe2 defined above).
5.3 Definition of the map ~`p,f,k
We are now ready to define the map
~`p,f,k: k x Ff x (Tot(Xo))n ! Xk;
it is the composite
ff(p)1x1 o n `0f,kk
k x Ff x (Tot(Xo))n ! Ff,kx (Tot(X )) ! X ,
25
where oe(p) was defined at the end of subsection 5.2 and `0f,kis a map which we*
* define next.
Recall that Ff,kis colimg2If,kFg. It therefore suffices to define the restri*
*ction of `0f,kto
F
g; we denote this restriction by
`0g: Fg x (Tot(Xo))n ! Xk.
So fix g 2 If,k. As in Example 5.1, if we are given elements xi 2 Xv(i)for *
*1 i n
(where v(i) denotes the valence of i in g) we can define an element ~g(x1, . .,*
*.xn) 2 Xk by
replacing each symbol i in g by xi and then interpreting the formal composition*
*s and cup
products in g to be genuine compositions and cup products in Xo; this process g*
*ives a map
Yn
~g: Xv(i)! Xk.
i=1
Now given s 2 Fg and ai2 Tot(Xo) for 1 i n we define
`0g(s, a1, . .,.an) = ~ga1v(1)(s1), . .,.anv(n)(sn),
where v(i) denotes the valence of i in g.
This completes the definition of the maps ~`p,f,k. It is straightforward to *
*check that these
maps fit together to give the map `n : C0(n) x (Tot(Xo))n ! Tot(Xo) that we set*
* out to
define.
5.4 The operad structure of C0.
The operad structure of C0 is determined by the fact that we want `n to be an a*
*ction of C0.
First we describe the action of the symmetric group n on C0(n) (we depart s*
*lightly from
[19] by having the symmetric group act on the left instead of the right.) Let ø*
* 2 n. The
action of ø on the P(n) factor permutes the coordinates (p1, . .,.pn). Given a *
*formula f of
type n, we define ø(f) to be the formula obtained from f by replacing each symb*
*ol i in f by
ø(i). There is an evident homeomorphism ø : Ff ! Ffi(f)which permutes the coord*
*inates,
and passing to the colimit over f we get a homeomorphism ø : F(n) ! F(n).
Next we will describe the operad composition Ok. We begin by giving the coll*
*ection of
spaces P(n) an operad composition: we define
Ok : P(n) x P(j) ! P(n + j  1)
to be the map that takes the pair
(p1, . .,.pn), (q1, . .,.qj)
to
(p1, . .,.pk1, pkq1, . .,.pkqj, pk+1, . .,.pn).
(If we think of an element of P(n) as a collection of little intervals whose le*
*ngths add up to
1 then P(n) is a suboperad of the little intervals operad C01.)
26
Now we want to define
Ok : (P(n) x F(n)) x (P(j) x F(j)) ! (P(n + j  1) x F(n + j  1))
for 1 i n. The projection of Ok on the P(n + j  1) factor is defined to be*
* the composite
(P(n) x F(n)) x (P(j) x F(j)) ! P(n) x P(j) ! (P(n + j  1),
where the first map is the projection and the second is the Ok operation for th*
*e operad P.
The projection of Ok on the F(n + j  1) factor is defined to be the composite
ci,n,j
(P(n) x F(n)) x (P(j) x F(j)) ! P(j) x F(n) x F(j) ! F(n + j  1),
where the first map is the projection and the map ck,n,jwill be defined next. (*
*This means
that the operad structure of C0 is like a semidirect product of P with F, excep*
*t that F is
not an operad.)
In order to construct ck,n,jit suffices to construct, for each choice of p =*
* (p1, . .,.pn) 2
P (n), each formula f of type n, and each formula f0 of type j, a map
ck,p,f,f0: Ff x Ff0! F(n + j  1)
Let us write F6=kffor
Y
v(i);
i6=k
thus Ff = F6=kfx v(k), where v(k) denotes the valence of k in f. We define ck,*
*p,f,f0to be
the composite
1xff(p)16=k c0k,f,f0
Ff x Ff0= F6=kfx v(k)x Ff0! Ff x Ff0,v(k)!Fn+j1,
where c0k,f,f0remains to be defined.
Of course, it suffices to define the restriction of c0k,f,f0to F6=kfx Fg0whe*
*re g0 is a v(k)
thickening of f0; we denote this restriction by
c0k,f,g0: F6=kfx Fg0! Fn+j1
Next we define a formula f *k g0of type n + j  1 by üs bstituting g0for k,ä*
* s follows:
we replace the symbols k + 1, . .,.n in f by k + j, . .,.n + j  1 respectively*
*, replace the
symbols 1, . .,.j in g0 by k, . .,.k + j  1 respectively, replace the symbols *
*idin g0 by the
entries of k in f, and then replace k by g0. For example, if k = 1, f = 1(2(3),*
* 4 ` 5, 6(7, 8)
and g0= 1(2, id, 3, id, id) then f *k g0= 1(2, 4(5), 3, 6 ` 7, 8(9, 10)).
Observe that F6=kfx Fg0= Ff*kg0. Since f *k g0 is a formula of type n + j  *
*1, we can
define c0k,f,g0to be the composite
F6=kfx Fg0= Ff*kg0 Fn+j1,
27
and this completes the definition of the operations Ok for C0.
The fact that the operations Ok so defined make C0into an operad, and the fa*
*ct that the
maps ` o o
n defined earlier give an action of this operad on Tot(X ) when X is the*
* cosimplicial
space associated to an operad with multiplication, are both easy consequences o*
*f an asso
ciativity property of the fiberwise prismatic subdivision which will be stated *
*and proved in
the next subsection. First we pause to give two examples to illustrate the defi*
*nition of Ok.
Example 5.6. Let f = f0 = 1(2), let p = (p1, p2), and let k = 2. Then v(k) = 0*
*, so there
is only one g0(which is equal to f0) and ck,p,f,f0is the composite
Ff x Ff0= F1(2(3)) F(3)
Example 5.7. Again let f = f0 = 1(2), p = (p1, p2), but now let k = 1. Then v(*
*k) = 1,
and the topdimensional 1thickenings g0of f0 are 1(id, 2), 1(2, id), and 1(2(i*
*d). Now ck,p,f,f0
is themap of Ff x Ff0into F(3) indicatedin the following picture.
 
__________________________________
 
 
 
 
 1(2, 3) 
p  p
1  2
 
 
 
 
 
 1(2(3)) 
 
 
 
 
 
 
p2 p1
 
 1(3, 2) 
 
 
 
___________________________________
5.5 An associativity property of the fiberwise prismatic subdivi
sion.
Given k, n, j and l, let f be a formula of type n, f0 a formula of type j, g an*
* lthickening of
f, g0a vf(k) thickening of f0 (where, of course, vf(k) denotes the valence of k*
* in f), and h0
a vg(k)thickening of f0. Note that g *k h0is an lthickening of f *k g0. Let p*
* 2 P(n) and let
p02 P(j).
It is straightforward to check from the definitions that the following diagr*
*am commutes.
28
Fg*kh0 = F6=kgx Fh0
oe(p Ok p0)# # 1 x oe(p0)
lx Ff*kg0 F6=kgx vg(k)x Ff0
= # # =
lx F6=kfx Fg0 Fg x Ff0
1 x 1 x oe(p0)# # oe(p) x 1
lx F6=kfx vf(k)x Ff0 = lx Ff x Ff0
29
6 The operad C and its action on Tot (Xo ).
In this section we modify the definition of C0 to obtain the operad C of theore*
*m 3.3.
As we have already mentioned, the reason we need to modify C0is that it does*
*n't model
the degeneracy maps
Ok : C2(n) x C2(0) ! C2(n  1)
of the little 2cubes operad. Here is another way to describe the difficulty. *
* If Xo is as in
Theorem 3.3, the operad C0 maps to a suboperad of the endomorphism operad of To*
*t(Xo),
but this suboperad is not closed under the degeneracy maps of this endomorphism*
* operad
(these are the maps that insert one or more copies of the basepoint). We will r*
*emedy this
defect by adjoining new points to each C0(n1) which will serve as the degenera*
*cies of points
in C0(n).
Before proceeding we need to specify the basepoint of Tot(Xo). The definitio*
*n of operad
with multiplication provides a special element e 2 X0. The conditions (12) and*
* (13) of
Section 3 imply that, for each n, all nfold iterated cofaces of e are equal, s*
*o there is a
unique cosimplicial map from o to Xo which is constant on each n and takes 0*
* to e.
This cosimplicial map is the basepoint of Tot(Xo); we will denote it by ~efrom *
*now on.
Remark 6.1. For later use we describe ~emore explicitly. The projection of ~e*
*on Hom ( m , Xm )
will be denoted by ~em. It is a constant map whose image is e 2 X0 for m = 0, *
*id 2 X1
for m = 1, ~ 2 X2 for m = 2, and for m 3 its image is the "iterated multiplic*
*ation"
~ O1 (~ O1 (. .~.) . .).in Xm .
In order to define C we follow the general outline of sections 4 and 5. Fir*
*st we need
to define an "indexingö perad J analogous to the operad J 0defined in section *
*4. Define
a sequence of sets {Sm } for m 0 as follows: Sm is the set of positive intege*
*rs with the
symbol ä djoined when m 6= 2 and S2 is the set of positive integers with the s*
*ymbols "
and ~ adjoined. Take the free operad generated by the sequence of sets Sm and i*
*mpose the
relation (12) from Section 3. Let the resulting operad be denoted by J .
Let us write Imnfor the set of elements of J (0) which contain each of the s*
*ymbols 1, . .,.n
exactly once and the symbol " exactly m times.
If f 2 Ikn, we define the valence of each of the symbols 1, . .,.n and of ea*
*ch of the copies
of " in the usual way, as the number of entries of the symbol in question. We d*
*efine Ff to
be
Yn Y
v(i)x v(")
i=1 "2f
where the second product is indexed by the copies of " in f.
We can define a partial ordering on Imnand make F into a functor from Imnto *
*the category
of topological spaces exactly as in subsection 4.4. We will denote the space co*
*limImnF by
F(n, m).
Next we define P(n, m) to be
X X
{ (p1, . .,.pn, q1, . .,.qm )  pi> 0, qi 0, pi+ qi= 1 }
30
Next we define the spaces C(n). We define C(0) to be a point. For n > 0 the *
*space C(n)
that we are seeking to define is a quotient
_ !
[
P(n, m) x F(n, m) = ~
m 0
and our next task is to define the equivalence relation ~. If f 2 Imnand 1 k *
* m we write
fk for the formula obtained from f by "pruning" the kth copy of " in f (counti*
*ng from left
to right). What this means is that if the copy of " has valence j with j > 0 we*
* replace that
" by ~ej(see Remark 6.1). If the copy of " has no entries then we remove it, to*
*gether with
the typographical symbols immediately adjacent to it (these can be commas, pare*
*ntheses,
or `'s). Now let f 2 Imn, and let
x = (p, q), (s, t)
be a point of P(n, m)xFf. If qk = 0 then x will be equivalent to a point of P(n*
*, m1)xFfk.
There are three cases. If the kth copy of " in f has valence 0 and if this " i*
*s the ith entry
of the integer k0then x is equivalent to the point
(p, q1, . .,.bqk, . .,.qm ), (s1, . .,.sisk0, . .,.sn, t1, . .,.bt*
*k, . .,.tm )
of P(n, m  1) x Ffk(where the hats indicate that the coordinates qk and tk are*
* deleted, and
0)
si is the ith degeneracy map of the simplex vf(k). If the kth copy of " in f*
* has valence
0 and this copy of " is the ith entry of the k0th copy of " in f then x is eq*
*uivalent to
(p, q1, . .,.bqk, . .,.qm ), (s, t1, . .,.sitk0, . .,.btk, . .,*
*.tm ).
Otherwise x is equivalent to
(p, q1, . .,.bqk, . .,.qm ), (s, t1, . .,.btk, . .,.tm ).
This completes the definition of the equivalence relation ~ and of the space C(*
*n).
Remark 6.2. For later use we remark that C0(n) is a deformation retract of C(*
*n) by the
homotopy Ht with
` P '
1  t qi
Ht((p, q), (s, t)) = _________Pp, tq, (s, t)
pi
It remains to define the operad structure of C and the action of C on Tot(Xo*
*) when X is
the cosimplicial space associated to an operad with multiplication.
First we define the degeneracy maps of C. If f 2 In,m and 1 k n we write*
* kf for the
formula obtained from f by replacing the symbol k by ", and then replacing k + *
*1, . .,.n
respectively by k, . .,.n  1; thus kf is an element of Im+1n1. The degeneracy*
* map
Ok : C(n) ! C(n  1)
31
takes a point
(p, q), (s, t)
of P(n, m) x Ff to the point
((p1, . .,.bpk, . .,.pn), (q1, . .,.pk, . .,.qn)), ((s1, . .,.bsk, . .,.sn)*
*, (t1, . .,.sk, . .,.tn)),
of P(n  1, m + 1) x Fkf (the hats mean that the coordinates pk and sk are dele*
*ted).
Notice in particular that every point of C(n) is an iterated degeneracy of s*
*ome point
in C0(n). This implies that the operad structure maps of C and its action on To*
*t(Xo) are
determined by the corresponding data for C0, and it is straightforward to verif*
*y that the
conditions for an operad structure and an operad action are satisfied. We are n*
*ow finished
with the definition of C and the proof of part (ii) of Theorem 3.3.
32
7 The chain operad of C.
The (normalized) singular chains functor, applied to a topological operad, give*
*s a chain
operad since the shuffle map is strictly associative and commutative. In this s*
*ection we show
that the singular chain operad of our operad C is quasiisomorphic to the chain*
* operad H
described in the introduction.
We begin with a general result, and for this we need some terminology. We wi*
*ll consider
filtered chain complexes of abelian groups (all of our chain complexes will be *
*nonnegatively
graded, and all of our filtrations will be increasing). We say that a chain co*
*mplex X is
homologically filtered if the homology of FiX=Fi1X is concentrated in dimensio*
*n i. (The
motivating example is when X is the singular chain complex of a CWcomplex A an*
*d FiX
is the singular chain complex of the iskeleton of_A.) If X is a homologically *
*filtered chain
complex we define the condensation of X, denoted X , to be the chain complex wh*
*ose ith
group is Hi(FiX=Fi+1X) and whose ith boundary operator is the boundary operato*
*r of
the triple (FiX, Fi1X, Fi2X). (In the motivating example, the condensation_of*
*_X is the
usual cellular chain complex of A). Note for later use that the homology of X *
*is naturally
isomorphic to the homology of X (this follows from the usual spectral sequence *
*argument).
By a filtered chain operad we mean a chain operad D together with an increas*
*ing filtration
Fiof each D(n) such that for each n, j and k the operation
Ok : D(n) D(j) ! D(n + j  1)
takes FiD(n) x Fi0D(j) to Fi+i0D(n + j  1). We say that D is homologically fil*
*tered if the
filtration of each_D(n) is homological. In this case we can_define_the condensa*
*tion of D to be
the chain operad D whose nth object is the condensation D(n) and whose operad *
*structure
maps are determined by those of D.
Theorem 7.1. Let D be a homologically filtered chain_operad. Then D is quasi*
*isomorphic
in the category of chain operads to its condensation D .
Before proving this we give some applications to our situation. Recall the c*
*hain operad
H defined in the introduction. Let H0be the suboperad of H defined by H0(n) = H*
*(n)Pfor
n > 0 and H0(0) = 0. We can filter C0by declaring P(n) x Ff to be in filtration*
* vf(i). It
is easy to check that this induces a homological filtration of the singular cha*
*ins operad of C0,
and that the condensation of this filtered chain operad is isomorphic to H0. No*
*w Theorem
7.1 implies
Corollary 7.2. The singular chain operad of C0 is quasiisomorphic to H0.
P *
* P
Similarly, we can filter C by declaring P(n, m) x Ff to be in filtration v*
*f(i) + vf("),
and this induces a homological filtration of the singular chain operad of C. N*
*ow the de
formation retraction defined in Remark 6.2 is filtration preserving, and thus t*
*he inclusion
C0(n) ,! C(n) induces an isomorphism (not just a quasiisomorphism) of condensa*
*tions for
all n > 0. Moreover, the definition of the degeneracy maps for C given in Secti*
*on 6 shows
that the degeneracy maps in the condensation of C are zero (because if we begin*
* with a
cell in C(n), apply a degeneracy, and then apply the retraction of Remark 6.2 w*
*e will end
up in a lower filtration than that of the original cell). Together, these facts*
* imply that the
condensation of C is isomorphic to H, and Theorem 7.1 implies
33
Corollary 7.3. The singular chain operad of C is quasiisomorphic to H.
We now turn to the proof of Theorem 7.1.
Let Op denote the category of nonnegatively graded chain operads, let N be *
*the non
negative integers, and let ChN be the category whose objects are sequences of c*
*hain com
plexes, indexed by N, and whose morphisms are sequences of chain maps. The for*
*getful
functor
Op ! ChN
has a left adjoint, the free functor, which we will denote by
: ChN ! Op
We will write Sn for the chain complex consisting of a copy of Z in dimensio*
*n n, and Bn
for the chain complex
. .0.! Z id!Z ! 0 . . .
where the copies of Z are in dimensions n and n  1 (of course, the analogy is *
*with the
nsphere and the ndisk). Given a sequence of sets T = {Tm }m 0, we will write *
*Sn(T ) for
the sequence of chain complexes
{ Sn }m 0
Tm
and Bn(T ) for the sequence of chain complexes
{ Bn }m 0
Tm
Now suppose we are given a chain operad D, a sequence of sets T , and a sequ*
*ence of
chain maps
f = { fm : Sn1 ! D(m) }
Tm
Passing to the adjoint gives a map of chain operads
f~: (Sn1(T )) ! D
and we can form a pushout in the category of chain operads:
(Sn1(T )) '! (Bn(T ))
~f# #
D ! D0
where ' is the inclusion Sn1(T ) ,! Bn(T ). We then say that D0 is obtained f*
*rom D by
attaching ncells. If we have a sequence of operads Dn such that D0 = S0(T ) f*
*or some T
and each Dn is obtained from Dn1 by attaching ncells, we say that the colimit*
* colimDn is
*
* n
a CW chain operad.
34
Lemma 7.4. If D is any chain operad, there is a CW chain operad E and a quasi*
*isomorphism
E ! D.
The proof of the lemma is precisely analogous to that of the corresponding f*
*act in the
category of spaces, so we omit it.
We need to define what it means for two morphisms of chain operads to be cha*
*in
homotopic "through operad maps." First note that if C is a differential graded*
* coalgebra
and D is any chain operad we can define a new chain operad Hom (C, D) by lettin*
*g the nth
object be Hom (C, D(n)) and letting the operad structure be determined by that *
*of D. Now
let I be the simplicial chain complex of the standard 1simplex. Let i0 (respec*
*tively i1) be the
maps from S0 to I corresponding to the endpoints of the 1simplex. Then two cha*
*inoperad
morphisms f1, f2 : D ! D0are operadchainhomotopic if there is a chainoperad *
*morphism
H : D ! Hom (I, D0) with i*1O H = f1 and i*2O H = f2.
Next let us observe that if C is a chain complex, then we can give C the fil*
*tration Fi
where Fi(C) is equal to C in dimensions i and is 0 in dimensions > i. We call*
* this the
filtration by degrees. This is a homological filtration, and the associated co*
*ndensation is
isomorphic to the C that we started with.
Proposition 7.5. Let E be a CW chain operad, D a homologically filtered chain o*
*perad,
and f : E ! D a morphism of chain operads. Give E the filtration by degrees. Th*
*en f is
operadchainhomotopic to a morphism f0 of filtered chain operads.
Before proving Proposition 7.5 we use it to prove Theorem 7.1. So let D be a*
* chain operad
with a homological filtration. By Lemma 7.4 there is a CW chain operad E and a*
* quasi
isomorphism of chain operads f : E ! D. By Proposition 7.5, f is operadchainh*
*omotopic
to a morphism f0 of filtered chain operads (where E is given the filtration by *
*degrees.) Now
f0 induces an operad morphism of condensations
__ __ __
f0: E ! D,
__ __
and f0is a quasiisomorphism (because f0has_the same effect in homology as f0, *
*and f0 has
the same effect in homology as f). But E is isomorphic to E_(since E was given *
*the filtration
by degrees), so we conclude that E is quasiisomorphic to D . Since we_also kno*
*w that E is
*
* __
quasiisomorphic to D, we have finally that D is quasiisomorphic to D , as req*
*uired. __
It remains to prove Proposition 7.5. First we need two lemmas.
Lemma 7.6. Let D be a filtered operad, let T be a sequence of sets, and for e*
*ach m let
fm : Sn(Tm ) ! D(m) be a chain map which lands in filtration n. Let f denote th*
*e sequence
{fm }. Then the chainoperad map
f~: (Sn(T )) ! D
is filtration preserving when (Sn(T ) is given the filtration by degrees.
*
* __
Proof. This is immediate from the explicit description of , which may be found*
* in [12]. __
35
Lemma 7.7. Let
D0 ! D1
# #
D2 ! D
be a pushout diagram in the category of chain operads. Suppose that D0 is a fil*
*tered chain
operad and that f : D ! D0 is a morphism of chain operads. Give D1, D2 and D t*
*he
filtration by degrees and suppose that f is filtrationpreserving when restrict*
*ed to D1 and D2.
Then f is filtrationpreserving.
Proof. Suppose first that the pushout diagram is obtained by applying to a p*
*ushout
diagram in the category ChN. In this case the lemma follows from Lemma 7.6 by p*
*assage to
adjoints. But the general case can be reduced to this case by using the fact th*
*at the pushout
of
D0 ! D1
#
D2
*
* __
surjects to D. *
* __
Proof of Proposition 7.5. Let En be a sequence of chain operads, with En obtain*
*ed from
En1 by attaching ncells, and let E = colimEn. Suppose inductively that we ar*
*e given a
n
homotopy Hn1 : En1 ! Hom (I, D) such that i*0O Hn1 = f and i*1O Hn1 is filt*
*ration
preserving. It suffices to show that Hn1 extends to a homotopy Hn : En ! Hom (*
*I, D) such
that i*0O Hn = f and i*1O Hn is filtrationpreserving. Now En is defined by a p*
*ushout diagram
(Sn1(T )) '! (Bn(T ))
~g# # fl
En1 ! En
and it suffices (by Lemma 7.7) to show that Hn1 O ~gextends to a homotopy
H0: (Bn(T )) ! Hom (I, D)
such that i*0O H0= f O fl and i*1O H0 is filtrationpreserving.
Passing to adjoints, we need to find a suitable chain homotopy
s : Bn(T ) ! D,
(where s raises degrees by 1) and we can treat each of the summands Bn in Bn(T *
*) separately,
so we are in the following situation. Choose generators a and b for the two cop*
*ies of Z in Bn
such that @b = a. The homotopy Hn1 determines s(a) and we have
@s(a) = f(a)  x,
where x is in filtration n  1. We need to define s(b) in such a way that
@(s(b)) + s(a) = f(b)  y,
36
where y is in filtration n and @y = x. We know that
@f(b) = f(a),
and this implies that
@(f(b)  s(a)) = x.
The fact that D is homologically filtered now implies that there is an element *
*z in filtration
n with
@z = x.
Since @z = @(f(b)  s(a)) = x, we see that f(b)  s(a)  z is a cycle, so (usin*
*g again the
fact that D is homologically filtered) there is a cycle z0 in filtration n whic*
*h is homologous
to f(b)  s(a)  z; that is, there is an element w in dimension n + 1 with
@w = (f(b)  s(a)  z)  z0
*
* __
We can now define s(b) = w and let y = z + z0. *
* __
37
8 The equivalence between C and the little 2cubes op
erad.
In this section and the next, we will prove
Theorem 8.1. C and C2 are weakly equivalent operads.
For the proof we use a method of Fiedorowicz [8]. Let C1 denote the operad *
*of little
intervals and let øi2 n denote the permutation that transposes i and i + 1.
Proposition 8.2. Suppose we are given an operad D with
(a) a morphism of (non ) operads I : C1 ! D,
(b) for each n, a point cn 2 C1(n) and for each i a path ffi from I(cn) to øiI(*
*Cn).
Suppose moreover that
(c) the universal cover of D(n) is contractible for each n, and
(d) for each n and i the paths
øiøi+1(ffi) . øi(ffi+1) . ffi
and
øi+1øi(ffi+1) . øi+1(ffi) . ffi+1
are path homotopic (where . denotes concatenation of paths).
Then D is weakly equivalent as an operad to C2 (in the sense that there is a th*
*ird operad
which maps to each of them by a morphism which is a weak equivalence on each sp*
*ace.)
Proof of Proposition 8.2 (following Fiedorowicz). For each n let ~D(n) be the u*
*niversal
cover of the space D(n), and let ß : D~(n) ! D(n) be the projection. We assume*
* for
simplicity that I : C1(n) ! D(n) is an inclusion. Since C1(n) is contractible, *
*ß1C1(n) is a
disjoint union of copies of C1(n); we arbitrarily choose one such copy for each*
* n and declare
that the basepoint of D~(n) shall lie in it. Now that we have chosen basepoints*
*, D~has an
induced structure of non operad. Next observe that hypothesis (b) determines*
* a map
~øi: ~D(n) ! ~D(n) for each n and i, and hypothesis (d) implies that the braid *
*relation
~øi~øi+1~øi= ~øi+1~øi~øi+1
is satisfied, so each ~D(n) has an action of the braid group Bn. This makes ~Di*
*nto a braided
operad as defined by Fiedorowicz (this just means that the symmetric groups are*
* replaced
by the braid groups everywhere in the usual definition of operad). Next let C~*
*2be the
corresponding braided operad constructed from C2 (see [8, Example 3.1]). Since *
*C2(n) is a
38
K(Bn, 1) for each n, the spaces ~C2(n) are contractible for each n, so b*
*y hypothesis (c) the
projections
~Dx ~C2! ~D
and
~Dx ~C2! ~C2
are equivalences of braided operads. If we now mod out by the action of *
*the pure braid groups
*
* __
(that is, the kernels of the projections Bn ! n), we get the assertion *
*of the proposition. __.
Now we turn to the proof of Theorem 8.1. We need to show that our ope*
*rad C satisfies
the hypotheses of Proposition 8.2. For (a), let c 2 C1(n). Let (p1, . .,*
*.pn be the lengths of
the little intervals in c and let (q1, . .,.qn+1) be the lengths of the *
*gaps, and define I(c) =
(p, q, e ` 1 ` e ` . .`.n ` e). For (b), we define cn to consist of n in*
*tervals each of length
_1
n , and we define ffito be the 1simplex
{cn} x F1`...`(i1)`i(i+1)`(i+2)`...`n
We defer the verification of (c) to the next section. For (d), assume f*
*irst that n = 3 and
i = 1. In this case the desired path homotopy is given by the following *
*picture:
 
 2 ` 1(3) 
2 ` 1 ` 3 __________________________________2 ` 3 ` 1
 
 
 
 
 1(2, 3) 
 
1(2) ` 3 2(3) ` 1
 
 
 
 
 
1 ` 2 ` 3  1(2(3)) 3 ` 2 ` 1
 
 
 
 
 
 
1 ` 2(3) 3 ` 1(2)
 
 1(3, 2) 
 
 
 
1 ` 3 ` 2 ___________________________________3 ` 1 ` 2
1(3) ` 2
Here the path
ø1ø2(ff1) . ø1(ff2) . ff1
begins at the vertex 1 ` 2 ` 3 and goes clockwise to 3 ` 2 ` 1 while the*
* path
ø2ø1(ff2) . ø2(ff1) . ff2
39
begins at the vertex 1 ` 2 ` 3 and goes counterclockwise to 3 ` 2 ` 1.
The verification of (d) for general n and i is similar: the required homotop*
*y is again the
boundary of a figure consisting of a square and two triangles, but the triangle*
* labeled 1(2, 3)
in the figure above is replaced by
1 ` . .`.(i  1) ` i(i + 1, i + 2) ` (i + 3) ` . .`.n
and so on.
40
9 The homotopy type of the spaces C(n).
In this section we prove
Proposition 9.1. For each n 0 the space C(n) is homotopy equivalent to the sp*
*ace C2(n).
On passage to universal covers this will imply that hypothesis (c) of Propos*
*ition 8.2 is
satisfied for the operad C, and this will complete the proof of Theorem 8.1.
Since C(0) is a point we may assume n > 0.
The homotopy in Remark 6.2 shows that C(n) is homotopy equivalent to F(n), a*
*nd it is
wellknown that C2(n) is homotopy equivalent to the configuration space F (n) o*
*f n ordered
points in R2, so what we really need to show is that F(n) is weakly equivalent *
*to F (n).
The basic idea is to show that F(n) and F (n) are both weakly equivalent to *
*the nerve
of a certain category Tn. In order to motivate the definition of Tn, let us obs*
*erve that if f
is a formula of type n then f induces a total order on the set {1, . .,.n} (nam*
*ely the order
in which these symbols first appear in f as one reads from left to right) and a*
*lso a partial
order (generated by the relation: i < j if j is an entry of i). These two order*
*s are consistent
in the sense that
p t, and if i < j < k in t and i < k in p then i must be < j in p.*
*(16)
(Here p t means that p is contained in t when both are considered as sets of *
*ordered pairs,
i.e., if i < j in p then i < j in t.)
Accordingly, we define Tn to be the following partially ordered set. The obj*
*ects Tn are
pairs (t, p), where t is a total order of the set {1, . .,.n} and p is a partia*
*l order of this set,
subject to the consistency condition (16). In order to define the partial order*
*ing of Tn, let
us first define top to be the ordering which is the reverse of t (that is, i < *
*j in top if and
only if j < i in t). Then there is a morphism in Tn from (t1, p1) to (t2, p2) *
*if both of the
conditions p1 p2 and t2 \ top1 p2 are satisfied. (We don't know a good way t*
*o motivate
this partial order on Tn, except that it is what is needed to make the proof wo*
*rk. But note
that if f < f0 in the partial order of subsection 4.4, then the pair (t, p) ass*
*ociated to f is <
the pair associated to f0 in the partial order of Tn.)
The main step in proving Proposition 9.1 is to show
Proposition 9.2. (a) There is a functor In from Tn to spaces such that
(i)the spaces In(t, p) are an open cover of F (n),
(ii)F (n) is the colimit of In, and
(iii)each In(t, p) is contractible.
(b) There is a functor I0nfrom Tn to spaces such that
(i)the spaces I0n(t, p) are subcomplexes of F(n),
(ii)F(n) is the colimit of I0n, and
(iii)each I0n(t, p) is contractible.
41
Proof of Proposition 9.1. Parts (a)(i) and (a)(ii), together with the proof of *
*[23, Propo
sition 4.1], imply that the natural map
hocolimIn ! colimIn
is a weak equivalence. Part (a)(iii), together with the homotopy invariance of*
* homotopy
colimits ([4, XII.4.2]), implies that hocolim In is weakly equivalent to hocoli*
*m * (where * is
the constant functor that takes every object of Tn to a point.)
Similarly, parts (b)(i) and (b)(ii), together with [1, Proposition 6.9], imp*
*ly that the
natural map
hocolimI0n! colimI0n
is a weak equivalence, and part (b)(iii) implies that hocolim I0nis weakly equi*
*valent to
*
* __
hocolim *. *
* __
We proceed to the proof of Proposition 9.2(a). Let ß1 and ß2 denote the proj*
*ections of R2
on its two factors. Let In(t, p) be the open subspace of F (n) consisting of al*
*l configurations
x = (x1, . .,.xn) such that
if i < j in t and ß1(xi) ß1(xj) then i < j in p
if i < j in t and ß1(xi) = ß1(xj) then i < j in p and ß2(xi) < ß2(xj).
Part (a)(i) is evident. For part (a)(ii), since the sets In(t, p) cover F (*
*n) it suffices to
show that the map colim In ! F (n) is 11, and for this it suffices to show tha*
*t, if x is in
In(p, t) \ In(p0, t0), there is a pair (p00, t00) which is both (p, t) and (p*
*0, t0) and is such that
x 2 In(p00, t00). And this in turn follows from the stronger fact that for each*
* configuration
x there is a unique minimal (t, p) with x 2 In(t, p): define t by i < j in t if*
* and only if
xi< xj in lexicographic order, and define p by i < j in p if and only if ß1(xi)*
* = ß1(xj) and
ß2(xi) < ß2(xj).
The proof of part (a)(iii) is by induction on n. The case n = 1 is obvious,*
* so assume
n > 1, and let (t, p) 2 Tn. To simplify the notation we treat the case where t *
*is the standard
total order 1 < 2 < . .<.n; the general case is precisely similar. Let t0 be t*
*he standard
total order of {1, . .,.n  1}, and let p0be the restriction of p to {1, . .,.n*
*  1}. It suffices
by induction to show that In(t, p) is homotopy equivalent to In1(t0, p0). Let
ff : In(t, p) ! In1(t0, p0)
be the projection map that takes (x1, . .,.xn) to (x1, . .,.xn1). Let
fi : In(t, p) ! In1(t0, p0)
take (x1, . .,.xn1) to (x1, . .,.xn1, fl(x1, . .,.xn1)), where
fl(x1, . .,.xn1) = (1 + max{ß1(xi)}, 1 + max{ß2(xi)}) 2 R2
Then ff O fi is the identity map, and it suffices to show that fi O ff is homot*
*opic to the identity
map of In(t, p). Let H be the homotopy which leaves x1, . .,.xn1 fixed and mo*
*ves xn
42
vertically until it has the desired second coordinate, and then horizontally un*
*til it has the
desired first coordinate: that is,
æ
(x1, . .,.xn1, (1  2t)xn + 2t(ß1(xn), ß2(fl))) if t 1=2
Ht(x) =
(x1, . .,.xn1, (2  2t)(ß1(xn), ß2(fl)) + (2t  1)fl) if 1=*
*2 t
where we have written fl for fl(x1, . .,.xn1). (To see that this homotopy stay*
*s inside In(t, p)
note that there cannot be a point in x directly above xn, and if xi is to the r*
*ight of xn we
must have i < n in p.) This concludes the proof of part (a) of Proposition 9.2.
For part (b), let I0n(t, p) be the following subcomplex of F(n):
[
I0n(t, p) = Ff
{f (tf,pf) (t,p)}
where (tf, pf) is the pair determined by the formula f and Ff is as in subsecti*
*on 4.3.
Part (b)(i) is evident. For part (b)(ii), since the sets I0n(t, p) cover F(*
*n) it suffices to
show that the map colim I0n! F(n) is 11, and for this it suffices to show that*
*, if Ff is
contained in I0n(p, t) \ I0n(p0, t0), there is a pair (p00, t00) which is bot*
*h (p, t) and (p0, t0) and
is such that Ff I0n(p00, t00). And this in turn follows from the stronger fac*
*t that for each
formula f the pair (tf, pf) is the unique minimal pair whose image under In con*
*tains Ff.
We now turn to the proof of (a)(iii), so let us fix a pair (t, p). If i 2 {1*
*, . .,.n}, we write
h(i) (the height of i with respect to p) for the length of the longest chain j1*
* < . .<.i in p,
and we write h(p) (the height of p) for the length of the longest chain in p. W*
*e write w(p)
(the width of p) for the number of elements i 2 {1, . .,.n} with h(i) = h(p). T*
*he proof of
(a)(iii) will be by double induction on h(p) and w(p).
Let us fix an element i with h(i) = h(p)  1. There are two cases:
Case 1 There is only one j with i < j in p.
Case 2 There is more than one j with i < j in p.
We begin with Case 1. In this case, condition (16) implies that j is the im*
*mediate
successor of i in the total order t.
We define three subcomplexes of In(p, t). It will be convenient to write f *
* (t, p) to mean
(tf, pf) (t, p).
S
A = f2S1Ff, where S1 = { f (t, p)  f contains one of the strings i(j), i *
*` j}or.j ` i
S
B = f2S2Ff, where S2 = { f (t, p)  i is to the left of j in f but j is no*
*t an entry}of i in f
S
C = f2S3Ff, where S3 = { f (t, p)  i is to the right of j}in.f
Clearly In(t, p) = A [ B [ C. We will show that In(t, p) is contractible by *
*showing that
A, B, C, A \ B, and A \ C are contractible and that B \ C is empty.
It is easy to see that B = In(t, p  (i, j)) (where p  (i, j) means the par*
*tial order which
is the same as p except that i is no longer < j; this is indeed a partial order*
* since j is the
immediate successor of i in p) so B is contractible by the inductive hypothesis*
*. Similarly,
C = In(~t, p  (i, j)) (where ~tis the same as t except that i and j are switch*
*ed; this is indeed
43
a total order since j is the immediate successor of i in t) so C is contractibl*
*e by the inductive
hypothesis. It is also clear that B \ C is empty.
Next we claim that A is contractible. Let
jt (respectively jp) be the restr*
*iction of t
(respectively, p) to {1, . .,.j  1, j + 1, . .,.n}. If i has valence 0 in g le*
*t us write g *ih to
mean "plug h in for i" (this is a little different from the use of this symbol *
*in subsection
5.4). Then S1 = {g *ih  g (jt,jp) and h i(j)}. This implies that A = In1(*
*jt,jp) x 1,
and so A is contractible by the inductive hypothesis.
Similarly, S1 \ S2 = {g *ih  g (jt,jp) and h = i `}j. This implies that *
*A \ B =
In1(jt,jp) x 0 which is contractible by the inductive hypothesis. Also, S1 \*
* S3 = {g *i
h  g (jt,jp) and h = j `}i. This implies that A \ B = In1(jt,jp) x 0 whic*
*h is con
tractible by the inductive hypothesis. This concludes the proof of Case 1.
For Case 2, let j1, . .,.jm be the successors of i in p. We may assume that*
* j1 < j2 <
. .<.jm in t. Condition (16) then implies that i, j1, . .,.jm is a consecutive *
*sequence in t
(that is, j1 is the immediate successor of i, etc.).
Let t0(respectively p0) denote the restriction of t (respectively p) to {1, *
*. .,.n}{j1, . .,.jm }.
Now define subcomplexes A, B, and C of In(t, p) by
S
A = f2S1Ff, where S1 = {g *ih  g (t0, p0) and h i(j1, .}.,.jm.)
S
B = f2S2Ff, where S2 = { f  j1 is to the left of}i.in f
S
C = f2S3Ff, where S3 = { f  jm is to the right of i in f but is not an entr*
*y}of.i
Then A [ B [ C = In(p, t), because if f is not in S2 or S3 then j1 will be t*
*o the right of i,
jm will be an entry of i, and this will imply that f contains the string i(j1, *
*. .,.jm ) so that
f 2 S1.
We will show that In(t, p) is contractible by showing that A, B, C, A \ B, A*
* \ C, B \ C
and A \ B \ C are all contractible.
First of all, B is contractible by induction because it is equal to In(~t, p*
*  (i, j1)) (where ~t
is the same as t but with i and j1 switched). Similarly, C is contractible beca*
*use it is equal
to In(t, p  (i, jm )), and B \ C is contractible because it is equal to In(~t,*
* p  (i, j1)  (i, js)).
Next, A is contractible because it is homeomorphic to Inm (t0, p0) x m . S*
*imilarly, we
have
o S1\S2 = {g*ih  g (t0, p0) and h j1 ` i(j2, .}.,.jm,)and so A\B Inm*
* (t0, p0)x
m1.
o S1\S3 = {g*ih  g (t0, p0) and h i(j1, . .,.jm1)}`,jmand so A\C In*
*m (t0, p0)x
m1.
o S1\ S2\ S3 = {g *ih  g (t0, p0) and h j1 ` i(j2, . .,.jm1)}`,jmand s*
*o A \ B \
C Inm (t0, p0) x m2.
*
* __
This concludes the proof of Case 2. *
* __
44
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