Operads and cosimplicial objects: an introduction.
James E. McClure and Jeffrey H. Smith*
Department of Mathematics, Purdue University
150 N. University Street
West Lafayette, IN 479072067
1 Introduction.
This paper is an introduction to the series of papers [26, 28, 29, 30], in whic*
*h we develop
a combinatorial theory of certain important operads and their actions.1 The op*
*erads we
consider are A1 operads, E1 operads, the little ncubes operad and the framed l*
*ittle disks
operad. Sections 2, 6 and 9, which can be read independently, are an introducti*
*on to the
theory of operads.
The reader is also referred to the very interesting papers of Batanin ([3, 4*
*]), which treat
similar questions from a categorical point of view.
Here is an outline of the paper.
In Section 2 we motivate the concept of nonsymmetric operad. We give the de*
*finition
of A1 space and state the characterization (up to weak equivalence) of loop spa*
*ces: a space
is weakly equivalent to a loop space if and only if it is a grouplike A1 space.
In Section 3 we introduce the total space construction Tot for cosimplicial *
*spaces and the
related conormalization construction for cosimplicial abelian groups.
In Section 4 we address the question: when is Tot of a cosimplicial space an*
* A1 space?
We obtain a useful sufficient condition for this to happen.
In Section 5 we reformulate the main result of Section 4 in a way which is c*
*onvenient for
generalization.
In Section 6 we motivate the concept of operad. We give the definition of E1*
* space and
state the characterization (up to weak equivalence) of infinite loop spaces: a *
*space is weakly
equivalent to an infinite loop space if and only if it is a grouplike E1 space.
Section 7 contains motivation for the main result of Section 8.
In Section 8 we give a sufficient condition for Tot of a cosimplicial space *
*to be an E1
space.
In Section 9 we introduce the little ncubes operad Cn. Operads weakly equiv*
*alent to Cn
are called En operads. We give the characterization (up to weak equivalence) of*
* nfold loop
________________________________
*Both authors were supported by NSF grants. This paper is based in part on l*
*ectures given by the first
author at the Newton Institute.
1The relation between these papers and [27] is explained in Remarks 12.5 and*
* 15.5.
1
spaces: a space is weakly equivalent to an nfold loop space if and only if it *
*has a grouplike
action of an En operad.
In Section 10 we give a sufficient condition for Tot of a cosimplicial space*
* to be an En
space.
In Section 11 we describe some category theory which is used in the proof of*
* the main
theorem of Section 10.
In Section 12 we outline the proof of the main theorem of Section 10. As a b*
*yproduct
we get a new, combinatorial, description (up to weak equivalence) of Cn.
In Section 13 we describe applications of the main result in Section 10 to a*
* certain space
of knots and to topological Hochschild cohomology.
In Section 14 we develop a combinatorial description of the framed little di*
*sks operad.
In Section 15 we observe that the theory of Sections 4, 5, 8, 10, 12 and 14 *
*remains valid
with spaces replaced by chain complexes. In particular this leads to concrete a*
*nd explicit
chain models for Cn and for the framed little disks operad.
In Secion 16 we give some applications of the theory developed in Section 15*
*; in particular
we discuss Deligne's Hochschild cohomology conjecture.
2 Loop spaces and the little intervals operad.
Historically, the first use of operads was to give a precise meaning to the ide*
*a that loop
spaces are monoids up to higher homotopy. In this section we recall how this wo*
*rks.
The first step is to reformulate the concept of monoid in a way that is amen*
*able to
generalization.
Proposition 2.1. A monoid structure on a set S determines and is determined by *
*a family
of maps
M(k) : Sk ! S
for k 0 (where Sk denotes the kfold Cartesian product) such that
(a) M(1) is the identity map, and
(b) the set {M(k)}k 0 is closed under multivariable composition.
Proof. If S is a monoid with multiplication M : S2 ! S and unit e : S0 ! S we d*
*efine M(0)
to be e, M(1) to be the identity map, and M(k) to be the iterated multiplicatio*
*n for k 2.
The monoid axioms show that the set {M(k)}k 0 is closed under multivariable com*
*position.
Conversely, if S is a set with maps M(k) satisfying (a) and (b) then S is a *
*monoid with
*
* __
multiplication M(2) and unit M(0). *
* __
Next let Z be a based space with basepoint denoted by *. We consider the spa*
*ce Z of
based loops on Z. For each r 2 (0, 1) there is a multiplication
Mr : ( Z)2 ! Z
which takes a pair of loops (ff, fi) to the loop Mr(ff, fi) which is ff (suitab*
*ly rescaled) on the
interval [0, r] and fi (suitably rescaled) on the interval [r, 1]. We represent*
* the loop Mr(ff, fi)
by the picture
2
ff fi
___________________________________
r
We write * 2 Z for the constant loop at the basepoint, which we represent by t*
*he picture
*
___________________________________
and we write
e : ( Z)0 ! Z
for the map whose image is *.
Motivated by Proposition 2.1, we consider the space M(k) of all maps
( Z)k ! Z
that can be obtained by multivariable composition from the maps Mr and e. A ty*
*pical
example is the map in M(4) which takes a 4tuple (ff1, ff2, ff3, ff4) to the lo*
*op
___________________________________*ff1ff2*ff3*ff4
In general, a map in M(k) is determined by k closed intervals in [0, 1] with di*
*sjoint interiors
(notice that, as in the example just given, the union of these intervals doesn'*
*t have to be all
of [0, 1]). This motivates the following definition:
Definition 2.2. Let A(k) be the set in which an element is a set of k closed in*
*tervals in [0, 1]
with disjoint interiors (in particular, A(0) is a point). Give A(k) the topolog*
*y induced by the
following imbedding of A(k) in R2k: given k closed intervals in [0, 1], list th*
*e 2k endpoints
of the intervals in increasing order.
What we have shown so far is that M(k) is homeomorphic to A(k).2 Moreover, i*
*t is easy
to see that each A(k) is contractible, so to sum up we have
Proposition 2.3. If Y = Z for some Z then there is a family of subspaces M(k)
Map (Y k, Y ) such that
(a) M(1) contains the identity map,
(b) the family M = {Mk}k 0 is closed under multivariable composition, and
(c) each M(k) is contractible.
The crucial fact about this situation is that Proposition 2.3 has a converse*
* up to weak
equivalence: if Y is any connected space which has a family of contractible sub*
*spaces M(k)
Map (Y k, Y ) satisfying (a), (b) and (c) then Y is weakly equivalent to Z for*
* some Z (this
________________________________
2In making this statement we must exclude the special case where the pathco*
*mponent of the basepoint
in Z is a single point. On the other hand, the proposition which follows remain*
*s true in this case.
3
is a special case of Theorem 2.12 below). This gives us a way of recognizing th*
*at a space is
a loop space (up to weak equivalence) without knowing in advance that it is a l*
*oop space.
Motivated by Propositions 2.1 and 2.3 we make a first attempt at the definit*
*ion of non
symmetric3 operad.
Provisional Definition 2.4. A nonsymmetric operad O is a collection of subspac*
*es
O(k) Map (Y k, Y ) k 0
(for some space Y ) such that
(a) O(1) contains the identity map and
(b) the collection O is closed under multivariable composition.
Critique of Provisional Definition 2.4. This definition is formally analogous *
*to the
nineteenthcentury definition of a group as a family of bijections of a set S, *
*closed under
composition and inverses. The advantage of such a definition is its concretenes*
*s and the ease
with which our minds assimilate it. The disadvantage (in the case of groups) is*
* that the set
S is really external to the group. The resolution of this difficulty (in the ca*
*se of groups) was
to split the original definition into two concepts: the concept of (abstract) g*
*roup and the
concept of group action.
Motivated by the Critique, we will split the Provisional Definition into two*
* concepts: the
concept of (abstract) nonsymmetric operad and the concept of operad action.
First observe that (in the situation of 2.4) the multivariable composition o*
*perations in
{Map (Y k, Y )}k 0 restrict to give maps
fl : O(k) x O(j1) x . .x.O(jk) ! O(j1 + . .+.jk)
for each choice of k, j1, . .,.jk 0. The associativity property of multivaria*
*ble composition
implies that the following diagram commutes for all choices of k, j1, . .,.jk, *
*{imn}m k, n jm.
Yk i jmY j k
1xfl Y
(2.1) O(k) x O(jm ) x O(imn)_______//O(k) x O(im1 + . .+.imjm)
m=1 n=1 m=1 
= 
fflffl 
k j
i Y Y fl
O(k) x O(jm ) x O(imn) 
m=1 m,n 

 
flx1 
fflffl fl fflffl
O(j1 + . .+.jk) x O(i11) x . .x.O(ikjk)______//O(i11+ . .+.ikjk)
Definition 2.5. A nonsymmetric operad O is a collection of spaces {O(k)}k 0 to*
*gether with
an element 1 2 O(1) and maps
fl : O(k) x O(j1) x . .x.O(jk) ! O(j1 + . .+.jk)
________________________________
3The word ön nsymmetric" refers to the fact that we haven't yet used, or ne*
*eded, the action of the
symmetric group on Y k; see Section 6.
4
(for each choice of k, j1, . .,.jk 0) such that
(a) for each k and each s 2 O(k), fl(1, s) = s and fl(s, 1, . .,.1) = s, and
(b) Diagram (2.1)commutes for all choices of k, j1, . .,.jk, i11, . .,.ikjk.
Remark 2.6. Any collection O which satisfies Provisional Definition 2.4 will al*
*so satisfy
Definition 2.5.
Remark 2.7. Here are some examples which will be important later.
(a) The collection A = {A(k)}k 0 defined in Definition 2.2 is a nonsymmetri*
*c operad
(by Remark 2.6); it is called the little intervals nonsymmetric operad. It is *
*instructive to
work out the explicit description of the maps fl in this case (cf. Section 9).
(b) If Y is any space the collection
{Map (Y k, Y )}k 0,
with its usual multivariable composition, is called the endomorphism operad of *
*Y and de
noted EY.
(c) More generally, if U is any topological category with a monoidal product*
* (see [22,
Section VII.1]) and D is an object of U then the collection of spaces
{Hom U (D k, D)}k 0
with the evident multivariable composition is a nonsymmetric operad. The proof*
* is left as
an exercise for the reader; see Section 11 for a hint.
(d) With U as above, note that is also a monoidal product for Uop. Applyin*
*g part (c)
to Uop gives a nonsymmetric operad whose kth space is
Hom Uop(D k, D) = Hom U(D, D k)
Next we formulate the concept of operad action. First observe that in the si*
*tuation of
Provisional Definition 2.4 the evaluation maps
Map (Y k, Y ) x Y k! Y
restrict to maps
` : O(k) x Y k! Y
and that the diagram
iYk j Yk
(2.2) O(k) x O(jm ) x Y j1+...+jk=//_O(k) x O(jm ) x Y jm
m=1 m=1
  Q
 1x `
 fflffl

flx1 O(k) x Y k

 
 `
fflffl fflffl
O(j1 + . .+.jk) x Y j1+...+jk____`_______//_Y
commutes for all k, j1, . .,.jk 0.
5
Definition 2.8. Let O be a nonsymmetric operad and let Y be a space. An action*
* of O on
Y consists of a map
` : O(k) x Y k! Y
for each k 0 such that
(a) `(1, x) = x for all x 2 Y , and
(b) Diagram (2.2)commutes for all k, j1, . .,.jk 0.
Example 2.9. (a) The nonsymmetric operad A mentioned in Remark 2.7(a) acts on *
* Z
for any space Z.
(b) The endomorphism operad EY acts on Y .
We conclude this section by stating the most general converse to Proposition*
* 2.3. First
we need two definitions.
Definition 2.10. An A1 operad is a nonsymmetric operad O for which each space *
*O(k) is
weakly equivalent to a point.
For example, the nonsymmetric operad A of Remark 2.7(a) is an A1 operad.
Notice the relationship between this definition and Proposition 2.1: the one*
*point sets
{M(k)} in Proposition 2.1 are replaced by contractible spaces in Definition 2.1*
*0.
Now let Y be a space with an action of an A1 operad O. Because O(2) and O(0*
*) are
connected, the maps
` : O(2) x Y 2! Y
and
` : O(0) x Y 0! Y
induce a monoid structure on ß0Y .
Definition 2.11. The action of O on Y is grouplike if the monoid ß0Y is a group.
For example, the action in Example 2.9(a) is grouplike. Also, if Y is connec*
*ted then all
actions are grouplike.
Theorem 2.12. Y is weakly equivalent to Z for some space Z () Y has a groupl*
*ike
action of an A1 operad.
Remark 2.13. This theorem developed gradually during the period from 1960 to 19*
*74. In the
(= direction, the first version was proved by Stasheff [34], assuming that Y is*
* connected and
using a particular nonsymmetric operad, now called the Stasheff operad (but th*
*e concept of
operad hadn't yet been defined at that time). Boardman and Vogt proved the (= d*
*irection
for general A1 operads (except that they used PROP's instead of operads), but s*
*till assuming
Y connected, in [5, 6]. May defined the concept of operad in [24] and proved th*
*e (= direction
for connected Y ; he proved the general version (for groupcomplete actions) in*
* [25]. The =)
direction (for PROP's, which implies the result for operads) is due to Boardman*
* and Vogt
[5, 6].
6
3 Cosimplicial objects and totalization.
Theorem 2.12 leads to the question of how we can tell when a space Y has an act*
*ion of an
A1 operad. In the next section we will give an answer to this question in the*
* important
special case where Y is the total space of a cosimplicial space Xo. In this sec*
*tion we pause
for some background about cosimplicial objects.
Throughout this paper we will use the following conventions for cosimplicial*
* objects.
Definition 3.1. (a) Define to be the category of nonempty finite totally orde*
*red sets (this
is equivalent to the category usually called ). Define [m] to be the finite to*
*tally ordered
set {0, . .,.m}. Define
di: [m] ! [m + 1] 0 i m + 1
to be the unique ordered injection whose image does not contain i, and define
si: [m] ! [m  1] 0 i m  1
to be the unique ordered surjection for which the inverse image of i contains t*
*wo points.
(b) Given a category C, a cosimplicial object Xo in C is a functor from to*
* C. If S is
a nonempty finite totally ordered set then XS will denote the value of Xo at S,*
* except that
we write Xm instead of X[m]. The maps
Xm ! Xm+1
induced by the diare called coface maps and the maps
Xm ! Xm1
induced by the siare called codegeneracy maps.
Note that every object in has a unique isomorphism to an object of the for*
*m [m], so
we can specify a cosimplicial object by giving its value on the objects [m] (to*
*gether with the
coface and codegeneracy maps). For example:
Definition 3.2. o is the cosimplicial space whose value at [m] is the simplex *
* m , with the
usual coface and codegeneracy maps.
Next we define the cosimplicial analog of geometric realization. First recal*
*l (for example,
from [15, Example 2.4(3)]) that the geometric realization of a simplicial space*
* Uo is a tensor
product over (also called a coend):
Uo = Uo o
When we change the variance from simplicial to cosimplicial it is natural to re*
*place by
Hom , which leads us to the following definition.
Definition 3.3. Let Xo be a cosimplicial space. The total space of Xo, denoted *
*Tot(Xo), is
the space of cosimplicial maps Hom ( o, Xo).
7
Here's a more explicit description: a point in Tot(Xo) is a sequence
ff0 : 0 ! X0, ff1 : 1 ! X1, ff2 : 2 ! X2, . . .
which is consistent, i.e.,
diO ffn = ffn+1 O di
and
siO ffn = ffn1 O si
for all i.
Example 3.4. Given a based space Z with basepoint *, we define a cosimplicial s*
*pace F oZ
whose total space is Z (F oZ is called the geometric cobar construction on Z).*
* The mth
space F mZ is the Cartesian product Zm . The coface
di: F mZ ! F m+1Z
is defined by
8
< (*, z1, . .,.zm ) if i = 0
di(z1, . .,.zm ) = (z1, . .,.zi, zi, . .,.zmi)f 1 i m
: (z
1, . .,.zm , *) if i = m + 1.
and the codegeneracy si: F mZ ! F m1Z deletes the (i  1)st coordinate. The p*
*roof that
Tot(F oZ) is homeomorphic to Z is left as an exercise for the reader. (Hint: i*
*f m > 1 then
the map
m1Y m1Y
si: F mZ ! F m1Z
i=0 i=0
is a monomorphism).
We will also consider cosimplicial abelian groups.
Example 3.5. Let W be a space and define SoW to be the cosimplicial abelian gro*
*up
Map (SoW, Z), where SoW is the usual simplicial set associated to W ([9, Exampl*
*e 1.28])
and Map means maps of sets.
Next we define the analog of Tot in this context. Let msimpbe the standard *
*simplicial
model of m ([9, Example 1.4]).
Definition 3.6. Let o*denote the cosimplicial chain complex which in degree m *
*is the
normalized chain complex ([38, pages 265266]) of msimp.
8
Definition 3.7. Let Ao be a cosimplicial abelian group. The conormalization4 of*
* Ao, denoted
C(Ao), is the cochain complex
Y1
Hom ( o*, Ao) Hom ( m*, Am ).
m=0
Here Hom is Hom in the category of cosimplicial graded abelian groups (with A*
*m concen
trated in dimension 0), and the differential is induced by the differentials of*
* the m*.
Remark 3.8. Here are two concrete descriptions of C(Ao); they are dual to the t*
*wo standard
ways of describing the normalization of a simplicial abelian group ([38, pages *
*265266]). The
proof that they agree with Definition 3.7 is left to the reader.
(a) Let C0(Ao) be the cochain complex whose mth group is thePintersection o*
*f the kernels
of the codegeneracies si : Am ! Am1 and whose differential is (1)idi. Then *
*C(Ao) is
isomorphic to C0(Ao).
(b) Let C00(Ao) be the cochain complex whose mth group is the cokernel of
M M
di: Am1 ! Am
i>0 i>0
and whose differential is induced by d0. Then C(Ao) is isomorphic to C00(Ao)
Example 3.9. The conormalization of SoW (Example 3.5) is the complex of singul*
*ar
cochains that vanish on all degenerate singular chains. This is what is usuall*
*y called the
normalized singular cochain complex of W ; we will denote it by s*W .
4 A sufficient condition for Tot (Xo ) to be an A1 space.
Definition 4.1. An A1 space is a space with an action of an A1 operad.
Let Z be a based space and let F oZ be the cosimplicial space defined in Exa*
*mple 3.4.
Then Tot(F oZ) is homeomorphic to Z and in particular (as we have seen in Sect*
*ion 2) it
is an A1 space. This leads us to the question:
Question 4.2. For what other cosimplicial spaces is Tot an A1 space?
As we have seen in Section 3, Tot is analogous to conormalization, so we can*
* gain insight
into Question 4.2 by examining a cosimplicial abelian group whose conormalizati*
*on has a
multiplicative structure, namely SoW (see Example 3.5). The conormalization of*
* SoW is
s*W (see Example 3.9), and s*W has an associative multiplication, the cup produ*
*ct, given
by the usual AlexanderWhitney formula
(x ` y)(oe) = x(oe(0, . .,.p)) . y(oe(p, . .,.p + q));
________________________________
4This functor is usually called normalization, but it seems desirable to hav*
*e separate names for the
cosimplicial and simplicial versions of normalization, analogous to the usual d*
*istinction between Tot and
geometric realization.
9
here x has degree p, y has degree q, oe is in Sp+qW , . is multiplication in Z,*
* and oe(0, . .,.p)
(resp., oe(p, . .,.p + q)) is the restriction of oe to the subsimplex of p+q s*
*panned by the
vertices 0, . .,.p (resp., p, . .,.p+q). The key point for our purpose is that *
*the same formula
defines a map
(4.1) `: SpW x SqW ! Sp+qW
and we can examine the relation between ` and the coface and codegeneracy maps *
*of SoW .
This relation is given by the following formulas:
æ
dix ` y if i p
(4.2) di(x ` y) = ip
x ` d y if i > p
(4.3) dp+1x ` y = x ` d0y
æ
six ` y if i p  1
(4.4) si(x ` y) = ip
x ` s y if i p
Next we observe that the cosimplicial space F oZ has the same kind of struct*
*ure as SoW :
if we define
`: F pZ x F qZ ! F p+qZ
to be the obvious juxtaposition map
Zp x Zq ! Zp+q
then ` satisfies (4.2), (4.3)and (4.4). Moreover, it is associative:
(x ` y) ` z = z ` (y ` z),
and unital: there is an element e 2 F 0Z (namely the basepoint) such that
x ` e = e ` x = x
for all x. This suggests that, as a way of answering Question 4.2, we consider *
*cosimplicial
spaces having the same kind of structure as SoW or F oZ:
Theorem 4.3. If Xo is a cosimplicial space with a cup product
`: Xp x Xq ! Xp+q
which is associative and unital and satisfies (4.2), (4.3)and (4.4)then Tot(Xo)*
* is an A1
space.
Remark 4.4. (a) This result is due to Batanin [2, Theorems 5.1 and 5.2] with a *
*simplified
proof by us [28, Section 3].
(b) Theorem 4.3 gives a sufficient but not a necessary condition for Tot(Xo)*
* to be an
A1 space. However, we expect that any A1 space is weakly equivalent to one prod*
*uced by
Theorem 4.3 (in fact it is likely that Tot induces a Quillen equivalence betwee*
*n cosimplicial
spaces satisfying the hypothesis of Theorem 4.3 and A1 spaces).
10
The remainder of this section gives an outline of the proof of Theorem 4.3; *
*for details see
[28, Section 3].
The first step in the proof is:
Proposition 4.5. The category of cosimplicial spaces has a monoidal structure *
* with the
property that Xo satisfies the hypothesis of Theorem 4.3 if and only if it is a*
* monoid.
This is due to Batanin [1]. The definition of is modeled on equations (4.2*
*), (4.3)and
(4.4): Xo Y ois the cosimplicial space whose mth space is
_ !
a
Xp x Y q = ~
p+q=m
where ~ is the equivalence relation generated by (x, d0y) ~ (dx+1x, y). The *
*coface maps
are defined by
æ
(dix, y) if i x
di(x, y) = ix
(x, d y) if i > x
and the codegeneracy maps by
æ
(six, y) if i x  1
si(x, y) = ix
(x, s y) if i x
Next we apply Remark 2.7(d) with D = o to get a nonsymmetric operad B. The*
* space
B(k) is
Hom ( o, ( o) k)
The composition maps fl are defined as follows: if f 2 B(k) and gi 2 B(ji) for *
*1 i k
then fl(f, g1, . .,.gk) 2 B(j1 + . .+.jk) is the composite
f o k g1 ... gk o (j1+...+j )
o !( ) ! ( ) k
Now let Xo be a monoid. We define an action of B on Tot(Xo) by letting
` : B(k) x (Tot(Xo))k ! Tot(Xo)
take (f, ø1, . .,.øk) to the composite
f o k fi1 ... fiko k ~ o
o !( ) ! (X ) !X
where ~ is the monoidal structure map of Xo.
To complete the proof of Theorem 4.3 it only remains to show that each B(k) *
*is con
tractible. This is an easy consequence of the fact (due to Grayson [14]) that *
*( o) k is
isomorphic as a cosimplicial space to o. See [28, Section 3] for details.
11
5 A reformulation.
Our next goal is to generalize Theorem 4.3. However, it turns out that the anal*
*ogs of equa
tions (4.2), (4.3)and (4.4)for the situations we will be considering are rather*
* complicated
and inconvenient, so we pause to reformulate Theorem 4.3 in a way that is more *
*amenable
to generalization.
Let us return to the motivating example SoW . Define
t : SpW SqW ! Sp+q+1W
by
(x t y)(oe) = x(oe(0, . .,.p)) . y(oe(p + 1, . .,.p + q + 1))
for oe 2 Sp+q+1W . Note that, in contrast to the cup product, the vertex p is n*
*ot repeated in
the formula for t.
This operation is related to the coface and codegeneracy operations in SoW *
*by the
following equations:
æ
dix t y if i p + 1
(5.1) di(x t y) = ip2
x t d y if i > p + 1
æ
six t y if i < p
(5.2) si(x t y) = ip1
x t s y if i > p
Note that there is no analog for t of equation (4.3). t is associative:
x t (y t z) = (x t y) t z
and unital: there exists e 2 X0 with
(5.3) sp(x t e) = s0(e t x) = x.
The operations ` and t determine each other:
(5.4) x t y = (dp+1x) ` y = x ` d0y
(5.5) x ` y = sp(x t y)
Now let Xo be a cosimplicial space. If Xo has an operation
t : Xp x Xq ! Xp+q+1
which is associative and unital (in the sense of equation (5.3)) and satisfies *
*(5.1)and (5.2)
then the operation ` defined by equation (5.5)satisfies the hypothesis of Theor*
*em 4.3
(the verification is left to the reader). Conversely, if Xo has a cup product *
*satisfying the
hypothesis of Theorem 4.3 then the t product defined by (5.4)is associative, un*
*ital and
satisfies (5.1)and (5.2). To sum up:
12
Proposition 5.1. Xo satisfies the hypothesis of Theorem 4.3 if and only it has *
*a product
t : Xp x Xq ! Xp+q+1
which is associative and unital and satisfies (5.1)and (5.2).
Corollary 5.2. If Xo is a cosimplicial space with a product
t : Xp x Xq ! Xp+q+1
which is associative and unital and satisfies (5.1)and (5.2)then Tot(Xo) is an *
*A1 space.
6 Operads.
As we have seen in Section 2, the ä ssociativity up to higher homotopyö f the *
*multiplication
on Z can be formulated rigorously as the action of an A1 operad on Z. We wo*
*uld
now like to give an analogous formulation of öc mmutativity up to higher homoto*
*py." A
multiplication is commutative in the ordinary sense if it is invariant under pe*
*rmutations of
the factors; this suggests that we add symmetricgroup actions to Definition 2.*
*5. We begin
with a provisional form of the definition.
Provisional Definition 6.1. An operad O is a collection of subspaces
O(k) Map (Y k, Y ) k 0
(for some space Y ) such that
(a) O(1) contains the identity map,
(b) the collection O is closed under multivariable composition, and
(c) each O(k) is closed under the permutation action of the symmetric group *
* k.
As in Section 2 we split the provisional definition into the concept of (abs*
*tract) operad
and the concept of operad action.
To formulate the definition of operad, we need to know the relation between *
*the action
of k and the multivariable composition maps fl in Provisional Definition 6.1. *
*This is left as
an exercise for the reader; the answer in given in [24, Definition 1.1(c)].
Definition 6.2. An operad is a nonsymmetric operad O together with, for each k*
*, a right
action of k satisfying the formulas of [24, Definition 1.1(c)].
Remark 6.3. (a) Let Y be a space. The endomorphism operad EY (Remark 2.7(b)) *
*is an
operad, where the kth space Map (Y k, Y ) is given the obvious right action of*
* k.
(b) If U is a topological category with a symmetric monoidal product (see *
*[22, Section
VII.7]) then the nonsymmetric operads in Remarks 2.7(c) and (d) are operads, w*
*ith the
obvious k actions on Hom U(D k, D) and Hom U(D, D k).
(c) If O is a nonsymmetric operad we can define an operad O0by
O0(k) = O(k) x k
with the obvious right k action; the definition of the composition maps fl for*
* O0is left as
an exercise. We call O0the operad generated by O.
13
Definition 6.4. Let O be an operad and let Y be a space. An action of O on Y is*
* an action
of the underlying nonsymmetric operad with the property that each map
` : O(k) x Y k! Y
factors through O(k) x k Y k.
Remark 6.5. If Y is a space then EY acts on Y .
Definition 6.6. An E1 operad is an operad O for which each space O(k) is weakly*
* equivalent
to a point.5
A space with an action of an E1 operad should be thought of as öc mmutative *
*up to all
higher homotopies."
The analog of Theorem 2.12 in this setting is a statement about "infinite lo*
*op spaces."
Recall that an infinite loop space is a space X for which there exists a sequen*
*ce X1, X2, . . .
with X homeomorphic to X1 and Xi homeomorphic to Xi+1for all i (thus an infin*
*ite
loop space is the zeroth space of a spectrum).
Theorem 6.7. Y is weakly equivalent to an infinite loop space () Y has a groupl*
*ike action
of an E1 operad.
The =) direction, and the (= direction for connected Y , are due to Boardman*
* and Vogt
[5, 6]. May gave a simpler proof of the (= direction for connected Y in [24] an*
*d proved the
general case in [25].
7 A family of cochain operations.
We want to give an E1 analog of Corollary 5.2. In this section we prepare the w*
*ay by return
ing to the motivating example, SoW , and defining a family of operations that g*
*eneralizes t.
The idea is that the definition of t is based on the partition of the set {0, .*
* .,.p + q + 1}
into {0, . .,.p} and {p + 1, . .,.p + q + 1}; we can produce more operations by*
* using other
partitions.
First we need some notation. Recall that we have defined to be the catego*
*ry of
nonempty finite totally ordered sets T . For T 2 we define T to be the conve*
*x hull of
T (in particular, [m]is the usual m ). We define STW to be the set of all co*
*ntinuous
maps T ! W (in particular, S[m]W is what we have been calling Sm W ) and STW t*
*o be
Map (STW, Z) (so S[m]W is the same as Sm W ).
Definition 7.1. Given a map oe : m ! W and a subset U of T , let oe(U) be the *
*restriction
of oe to U.
Now observe that a partition of T into two pieces is the same thing as a sur*
*jective function
f : T ! {1, 2}.
________________________________
5For technical reasons, it is usual to require in addition that the action o*
*f each k should be free. The
operad D that we construct in Section 8 does have this property.
14
Definition 7.2. Given a surjection f : T ! {1, 2}, define a natural transformat*
*ion
1(1) f1(2) T
: Sf W x S W ! S W
by the equation
(x, y)(oe) = x(oe(f1(1))) . y(oe(f1(2)))
for oe 2 STW ; here . is multiplication in Z.
Remark 7.3. If f is the function {0, . .,.p + q + 1} ! {1, 2} that takes {0, . *
*.,.p} to 1 and
{p + 1, . .,.p + q + 1} to 2 then is t.
Next we describe the relation between the operations and the cosimplicia*
*l structure
maps of SoW .
Proposition 7.4. Let
ffi
T ______________//EETy0
EEE yyyy
f EE""E __ygyy
{1, 2}
be a commutative diagram, where OE is a map in (i.e., an orderpreserving map*
*). For
i = 1, 2 let
OEi: f1(i) ! g1(i)
be the restriction of OE.
Then the diagram
1(1) f1(2) _//_T
Sf W x S W S W
(ffi1)*x(ffi2)* ffi*
fflffl fflffl
1(1) g1(2) _//_T0
Sg W x S W S W
commutes.
The proof is an immediate consequence of the definitions. In the special cas*
*e of Remark
7.3 we recover equations (5.1)and (5.2).
Next we formulate the commutativity, associativity and unitality properties *
*of the
operations. Commutativity is easy:
Proposition 7.5. The diagram
1(1) f1(2) _//_T
Sf W x S W S W
fi =
fflffl fflffl
1(2) f1(1) _//_T
Sf W x S W S W
commutes, where ø is the switch map and t is the transposition of {1, 2}.
15
For the associativity condition we need some notation. Define
ff : {1, 2, 3} ! {1, 2}
by ff(1) = 1, ff(2) = 1, ff(3) = 2 and
fi : {1, 2, 3} ! {1, 2}
by fi(1) = 1, fi(2) = 2, fi(3) = 2. Given a surjection g : T ! {1, 2, 3} let g1*
* be the restriction
of g to g1{1, 2} and let g2 be the restriction of g to g1{2, 3}.
Proposition 7.6. With the notation above, the diagram
1(1) g1(2) g1(3) x1//_g1{1,2}g1(3)
Sg W x S W x S W S W x S W
1x 
fflffl fflffl
1(1) g1{2,3} __________//T
Sg W x S W S W
commutes for every choice of T and g.
Again, the proof is immediate from the definitions.
For the unital property we need to extend SoW to the category of all finite *
*totally ordered
sets, including the empty set.
Definition 7.7. (a) Define + to be the category of finite totally ordered sets.
(b) Given a category C, an augmented cosimplicial object in C is a functor f*
*rom + to C.
(c) Extend o to a functor on + by defining ; = ;.
(d) Define SoW as a functor from op+to sets by STW = Map ( T, W ); in parti*
*cular
S;W is a point.
(e) Define SoW as a functor from + to abelian groups by STW = Map (STW, Z);*
* in
particular S;W is isomorphic to Z.
With these conventions, Definition 7.2 makes sense when f is not surjective,*
* and Propo
sitions 7.4 (with replaced by +), 7.5 and 7.6 are still valid in this slight*
*ly more general
context.
Now let " 2 S;W be the element corresponding to 1 2 Z.
Proposition 7.8. If f : T ! {1, 2} takes all of T to 1 then (x, ") = x for a*
*ll x and if f
takes all of T to 2 then (", x) = x for all x.
8 A sufficient condition for Tot (Xo ) to be an E1 space.
Definition 8.1. An E1 space is a space with an action of an E1 operad.
In order to state the analog of Corollary 5.2 we need to use augmented cosim*
*plicial spaces.
Definition 8.2. Let Xo be an augmented cosimplicial space. Define Tot(Xo) to be
Hom +( o, Xo).
16
This can be described more simply: Tot(Xo) is the total space (in the sense *
*of Definition
3.3) of the restriction of Xo to .
Theorem 8.3. Let Xo be an augmented cosimplicial space with a map
1(1) f1(2) T
: Xf x X ! X
for each f : T ! {1, 2}. Suppose that the maps satisfy the analogs of Propo*
*sitions 7.4,
7.5, and 7.6, and that there is an element " 2 X; satisfying the analog of Prop*
*osition 7.8.
Then Tot(Xo) is an E1 space.
Remark 8.4. We expect that Tot induces a Quillen equivalence between augmented *
*cosim
plicial spaces satisfying the hypothesis of Theorem 8.3 and E1 spaces.
The remainder of this section gives an outline of the proof of Theorem 8.3; *
*for details see
[28].
The proof follows the same pattern as the proof of Theorem 4.3. The first st*
*ep is
Proposition 8.5. The category of augmented cosimplicial spaces has a symmetric *
*monoidal
structure with the property that Xo satisfies the hypothesis of Theorem 8.3 i*
*f and only if
it is a commutative monoid.
The basic idea in defining Xo Y ois that we build it from formal symbols <*
*f>(x, y).
In order to get a cosimplicial object we have to build in the cosimplicial oper*
*ators, so we
consider symbols of the form
OE*((x, y))
where f : T ! {1, 2} and OE : T ! S is an orderpreserving map: such a symbol w*
*ill represent
a point in the Sth space (Xo Y o)S. We require these symbols to satisfy the *
*relation in
Proposition 7.4.
Our next two definitions make this precise.
Definition 8.6. Given S 2 +, let IS be the category whose objects are diagrams
f ffi
(8.1) {1, 2}oo__T ____//S
where T is a finite totally ordered set and OE is orderpreserving, and whose m*
*orphisms are
commutative diagrams
f ffi
(8.2) {1, 2}oo__T_____//_S
=  _ = 
fflfflffflfflfflffl0ffi0
{1, 2}oo__T 0___//_S
with _ orderpreserving.
17
We will denote an object (8.1)of IS by (f, OE). Given augmented cosimplicial*
* spaces Xo
and Y owe consider the functor from IS to spaces which takes (f, OE) to
1(1) f1(2)
Xf x Y
and a morphism (8.2)to the map
(_1)* x (_2)*
where _i: f1(i) ! (f0)1(i) is the restriction of _.
Definition 8.7. Define Xo Y oby
1(1) f1(2)
(Xo Y o)S = colim Xf x Y
(f,ffi)2IS
for S 2 +.
The verification that is a symmetric monoidal product is given in [28, Sec*
*tion 6].
Remark 8.8. Readers familiar with Kan extensions will recognize that Xo Y ois*
* one; see
[28, Section 6].
Next we apply Remark 6.3(b) with D = o to get an operad D whose kth space *
*is
Hom ( o, ( o) k)
If Xo is a commutative monoid we define an action of D on Tot(Xo) by letting
` : D(k) x (Tot(Xo))k ! Tot(Xo)
be the map that takes (h, ø1, . .,.øk) to the composite
fi1 ... fiko k ~ o
o h!( o) k ! (X ) !X
where ~ is the monoidal structure map of Xo.
To complete the proof of Theorem 8.3 it only remains to show that each D(k) *
*is con
tractible; see [28, Section 10] for the proof of this.
Remark 8.9. One can give a construction analogous to for the category of ordi*
*nary (unaug
mented) cosimplicial spaces by requiring f to be a surjection in Definition 8.7*
*. This gives a
product which is coherently associative and commutative but not unital.
9 The little ncubes operad.
We have seen in Section 2 that Z is an A1 space. For n 2 the space nZ has*
* a
commutativity property intermediate between A1 and E1 ; moreover, nZ has stro*
*nger
commutativity than m Z if n > m. In this section we see how to make this preci*
*se.
Fix n 1. Let I denote the interval [0, 1].
18
Definition 9.1. A TDmap In ! In is a composite T O D, where T is a translation*
* and D
is a dilation (i.e., multiplication by a scalar).
A TDmap takes (t1, . .,.tn) to (a1 + bt1, . .,.an + btn), where a1, . .,.an*
* and b are con
stants with ai 0, b > 0 and ai+ b < 1. The image of a TDmap is called a "litt*
*le ncube."
A TDmap is completely determined by its image.
Definition 9.2. (a) For k 0, let Cn(k) be the space in which a point is a k*
*tuple
(~1, . .,.~k) of TDmaps In ! In such that the images of the ~ihave disjoint in*
*teriors.
(b) Let Cn be the collection of spaces {Cn(k)}k 0.
In the special case n = 2, the elements of C2(k) can be represented by pictu*
*res in the
plane. For example, the picture
__________________
 ______ 
   
   
  2  
 _____ 
____ 
 
 3  _____ 
___   
  1 
 ____ 
 
__________________
represents an element of C2(3).
Next we define an operad structure for Cn.
Definition 9.3. (a) Let 1 2 Cn(1) be the identity map of In.
(b) Give Cn(k) the right k action that permutes the ~i.
(c) Define
fl : Cn(k) x Cn(j1) x . .x.Cn(jk) ! Cn(j1 + . .+.jk)
as follows: if
c = (~1, . .,.~k)
is a point of Cn(k), and
di= (~i1, . .,.~iji)
is a point of Cn(ji) for 1 i k, then fl(c, d1, . .,.dk) is the point ( 11, *
*. .,. kjk), where
il= ~iO ~il.
For example, if n = 2 and c 2 C2(2), d1 2 C2(3) and d2 2 C2(2) are represent*
*ed by
__________________ __________________ __________________
 _______  ______   _______ 
          
          
  1    2     1  
    _____     
 ______ _____   ______  
_______      _______ 
    1  _____    
   ____      
 2     3    2 
    ____    
______      ______ 
__________________ __________________ __________________
19
respectively, then fl(c, d1, d2) 2 C2(5) is represented by
__________________
 ___ 
 2 
 ___  
 _1_ 
 _3
 
 
___ 
 4  
____ 
   
 _5_ 
__________________
Remark 9.4. (a) The definition of Cn and its composition maps is due to Boardma*
*n and Vogt
[5, 6]. They were working in a somewhat different context (PROP's instead of op*
*erads).
(b) C1 is the operad generated by the nonsymmetric operad A defined in Sect*
*ion 2 (see
Remark 6.3(c)).
The reason for defining the operad Cn is that it acts on nZ. To describe t*
*his action
we think of an element of nZ as a map In ! Z which takes the boundary of In to*
* the
basepoint * of Z. Then
` : Cn(k) x ( nZ)k ! nZ
is defined as follows: if
c = (~1, . .,.~k)
is a point of Cn(k) and ffi 2 nZ for 1 i k then `(c, ff1, . .,.ffk) is the*
* map In ! Z
which is ffiO (~i)1 on the image of ~iand * for points which are not in the im*
*age of any ~i.
For example, if c is the element of C2(3) represented by
__________________
 ______ 
   
   
  2  
 _____ 
____ 
 
 1  _____ 
___   
  3 
 ____ 
 
__________________
then `(c, ff1, ff2, ff3) is the map I2 ! Z represented by the picture
__________________
 ______ 
   
  ff  
  2 
 _____ 
____ 
 
 ff1 _____ 
___   
 *  ff3 
 ____ 
 
__________________
(where the ff's in the picture are appropriately scaled).
As one would expect, there is an analog of Theorems 2.12 and 6.7: if Y has a*
* grouplike
Cn action then Y is weakly equivalent to nZ for some Z. In fact something a *
*bit more
20
general is true; we pause to give the relevant definitions, which will also be *
*used in Section
10.
Definition 9.5. Let O, O0 be operads. An operad morphism i : O ! O0 is a sequen*
*ce of
maps
ik : O(k) ! O0(k)
such that
(a) i1 takes the unit element in O(1) to that in O0(1),
(b) each ik is k equivariant, and
(c) the diagram
``kx``j1x...
O(k) x O(j1) x . .x.O(jk)________//O0(k) x O0(j1) x . .x.O0(jk)
fl fl
fflffl ``j1+...+j fflffl
O(j1 + . .+.jk)________________k_//O0(j1 + . .+.jk)
commutes for all k, j1, . .,.jk 0.
Definition 9.6. A morphism i : O ! O0is a weak equivalence if each ik is a weak*
* equivalence
of spaces. Two operads O and O0are weakly equivalent if there is a diagram of o*
*perads and
weak equivalences of operads
O . .!.O0
Definition 9.7. An operad is an En operad if it is weakly equivalent to Cn.
Now the analog of Theorems 2.12 and 6.7 is
Theorem 9.8. Y is weakly equivalent to nZ for some space Z () Y has a groupl*
*ike
action of an En operad.
The =) direction, and the (= direction for connected Y , are due to Boardman*
* and
Vogt [5, 6]. A simpler proof of the (= direction for connected Y was given by M*
*ay in [24].
The general case of the (= direction is due to May [25].
Theorem 9.8 is aesthetically pleasing but has not often been applied because*
* it is usually
hard to show that a space is an En space (for 1 < n < 1) without knowing in adv*
*ance that
it is an nfold loop space. In Section 10 we will address this difficulty by gi*
*ving a sufficient
condition for Tot of a cosimplicial space to be an En space.
Since this section is intended as an introduction to Cn, we should mention t*
*hat the most
important uses of Cn in algebraic topology come from the ä pproximation theorem*
*" [24,
Theorem 2.7]. This theorem gives a model for n nZ, built from Z and Cn. Usin*
*g this
model, Fred Cohen has given a complete description of the homology of n nZ [7]*
*. Another
basic fact is that the model splits stably as a wedge of pieces of the form
Cn(k)+ ^ k Z^k
(where + means add a disjoint basepoint and ^ is the smash product); this is ca*
*lled the
Snaith splitting [33, 8].
21
10 A sufficient condition for Tot (Xo ) to be an En space.
In this section we give the analog of Theorem 8.3 for En actions. The hypothesi*
*s of Theorem
8.3 refers to operations where f ranges through functions T ! {1, 2}. For o*
*ur current
purpose we need to consider functions f : T ! {1, . .,.k} for all k 2 but we *
*will only use
those of öc mplexity n" (see Definitions 10.3 and 10.4).
Let us return to the motivating example SoW . The extension of the definitio*
*n of to
functions f : T ! {1, . .,.k} is routine:
Definition 10.1. Given f : T ! {1, . .,.k} define a natural transformation
1(1) f1(k) T
: Sf W x . .x.S W ! S W
by the equation
(x1, . .,.xk)(oe) = x1(oe(f1(1))) . . ...xk(oe(f1(k)))
for oe 2 STW ; here . is multiplication in Z.
These operations satisfy properties analogous to Propositions 7.4, 7.5, 7.6 *
*and 7.8; the
precise formulation is left to the reader (see [28, Definitions 9.39.7] for a *
*hint).
Remark 10.2. operations with k > 2 are composites of those with k = 2; that*
* is why we
were able to restrict to k = 2 in Sections 7 and 8. However, it is not true tha*
*t an operation
with complexity n (see Definitions 10.3 and 10.4) can be decomposed into *
*operations
with k = 2 and complexity n, which is why we cannot restrict to k = 2 in this*
* section.
As we have seen in Section 9, an En operad encodes commutativity which is in*
*termediate
between A1 (no commutativity) and E1 (full commutativity). We therefore want, f*
*or each
n, a family of operations which interpolates between t and the family of all operations.
Notice that, in general, the ordered sets f1(1), . .,.f1(k) are mixed togethe*
*r in T , but
when f corresponds to an iterate of t they are not mixed. We therefore introduc*
*e a way of
measuring the amount of mixing.
We begin with the case k = 2. First observe that a function f from a finite*
* totally
ordered set to {1, 2} is the same thing as a finite sequence of 1's and 2's.
Definition 10.3. (a) The complexity of a finite sequence of 1's and 2's is the *
*number of
times the sequence changes from 1 to 2 or from 2 to 1.
(b) The complexity of a function f : T ! {1, 2} is the complexity of the cor*
*responding
sequence.
For example, if f corresponds to the sequence 11222122112 then the complexit*
*y of f is
5.
Next let k > 2. A function f : T ! {1, . .,.k} corresponds to a finite sequ*
*ence with
values in {1, . .,.k}. We consider the subsequences that have only two values: *
*for example
in the sequence 12313212 we consider the subsequences 121212, 23322 and 13131. *
* As in
Definition 10.3, the complexity of such a subsequence is the number of times it*
* changes its
value.
22
Definition 10.4. (a) The complexity of a sequence with values in {1, . .,.k} is*
* the maximum
of the complexities of the subsequences with only two values.
(b) The complexity of f : T ! {1, . .,.k} is the complexity of the sequence *
*corresponding
to f.
In the example just given, the complexity of 121212 is 5, the complexity of *
*23322 is 2,
and the complexity of 13131 is 4, so the complexity of 12313212 is 5.
Remark 10.5. The definition of complexity is suggested by [32]; the reason we u*
*se it here
is that it is welladapted to the proof of Theorem 10.6 below. There may be oth*
*er ways of
defining complexity that would also lead to Theorem 10.6, although this seems u*
*nlikely.
We can now state the analog of Theorem 8.3.
Theorem 10.6. Fix n. Let Xo be an augmented cosimplicial space with a map
1(1) f1(k) T
: Xf x . .x.X ! X
for each f : T ! {1, . .,.k} with complexity n. Suppose that the maps are*
* consistent
with the cosimplicial operators (in the sense of [28, Definition 9.4]) and are *
*commutative,
associative and unital (in the sense of [28, Definitions 9.5, 9.6 and 9.7]). Th*
*en Tot(Xo) is
an En space.
We expect that Totinduces a Quillen equivalence between augmented cosimplici*
*al spaces
satisfying the hypothesis of Theorem 10.6 and En spaces.
The proof of Theorem 10.6 is similar in outline to the proofs of Theorems 4.*
*3 and 8.3.
However, just as En interpolates between A1 and E1 , we need a way to interpola*
*te between
the concepts of monoidal product and symmetric monoidal product. The next sect*
*ion is
devoted to this.
11 An extension of Remark 6.3(b).
Remark 6.3(b) says that a symmetric monoidal product on a topological categor*
*y U,
together with a choice of an object D 2 U, leads to an operad O. We begin with *
*an outline
of the proof of this fact.
is a binary operation, so the first step is to choose, for each k > 2, a s*
*pecific way of
inserting parentheses6 to get a kfold iterate of which we denote by
k : Uk ! U.
We also define 1 to be the identity functor and 0 to be the unit object of .
Next we define the spaces of the operad O by
O(k) = Hom U(D, k(D, . .,.D))
for k 0.
________________________________
6A different choice gives a naturally isomorphic functor, by MacLane's coher*
*ence theorem [22, Section
VII.7].
23
To define the action of k on O(k), we use MacLane's coherence theorem. This*
* gives, for
each oe 2 k, a natural isomorphism
oe* : k(X1, . .,.Xk) ! k(Xff(1), . .,.Xff(k))
and in particular a selfmap of k(D, . .,.D).
We use the coherence theorem again to get a natural isomorphism
: k( j1, . .,. jk) ! j1+...+jk
which induces the structure map fl of O.
It remains to check that fl has the associativity, unitality and equivarianc*
*e properties
required by the definition of operad; for this we apply the coherence theorem o*
*ne more time
to see that has associativity, unitality and equivariance properties (see [28*
*, Section 4] for
the explicit statements) from which those for fl can be deduced. This completes*
* the proof
of Remark 6.3(b).
The same proof proves something more general. Assume that for each k we are *
*given a
subfunctor of k, that is, a functor Fk with a natural monomorphism to k. Assu*
*me further
that Fk is closed under oe* (that is, oe* takes Fk(X1, . .,.Xk) to Fk(Xff(1), .*
* .,.Xff(k))) and
that the collection {Fk}k 0 is closed under . The argument given above shows:
Proposition 11.1. Under these assumptions the collection {Hom U (D, Fk(D, . .,.*
*D))}k 0
is an operad, with k action induced by the maps oe* and fl induced by .
Remark 11.2. In [28, Section 4] we give a more general version of 11.1, using t*
*he concept of
üf nctoroperad.Ä functoroperad is a collection of functors
Fk : Uk ! U
with just enough structure to satisfy the conclusion of Proposition 11.1. This *
*concept has
been discovered independently, in a different context and in a more general for*
*m, by Batanin
[3], who calls them "internal operads."
12 Proof of Theorem 10.6.
Recall that in the proof of Theorem 8.3 we constructed a symmetric monoidal pro*
*duct
on the category of augmented cosimplicial spaces. Our first task is to give a f*
*ormula for the
iterate k.
Given S 2 + and k 0 let IS(k) be the category whose objects are diagrams
f ffi
(12.1) {1, . .,.k}oo_T ____//S
where T is a finite totally ordered set and OE is orderpreserving, and whose m*
*orphisms are
commutative diagrams
f ffi
{1, . .,.k}oo_T_____//_S
=  _ = 
fflfflf0 fflfflfflfflffi0
{1, . .,.k}oo_T 0___//_S
24
with _ orderpreserving.
We will denote an object (12.1)of IS by (f, OE).
Definition 12.1. Let Xo1, . .,.Xokbe augmented cosimplicial spaces. Define k(X*
*o1, . .,.Xok)
to be the cosimplicial space whose value at S 2 + is
1(1) f1(k)
colim Xf1 x . .x.Xk
(f,ffi)2IS(k)
When k = 2 this is the same as the definition of Xo1 Xo2. In general we have
Proposition 12.2. k is naturally isomorphic to k.
For the proof see [28, Section 6] (also cf. Remark 10.2).
Now fix n, and let InS(k) be the full subcategory of IS(k) whose objects are*
* the (f, OE) for
which the complexity of f is n.
Definition 12.3. Let Xo1, . .,.Xokbe augmented cosimplicial spaces. Define nk(*
*Xo1, . .,.Xok)
to be the cosimplicial space whose value at S 2 + is
1(1) f1(k)
colim Xf x . .x.X
(f,ffi)2InS(k)
Proposition 12.4. For each n, the sequence of functors { nk}k 0 satisfies the h*
*ypothesis of
Proposition 11.1.
For the proof see [28, Section 8].
Applying Proposition 11.1 we obtain an operad with kth space
(12.2) Hom +( o, n( o, . .,. o))
We will denote this operad by Dn.
If Xo satisfies the hypothesis of Theorem 10.6 then there are maps
,k : nk(Xo, . .,.Xo) ! Xo
for each k 0. We define an action of Dn on Tot(Xo) by letting
` : Dn(k) x (Tot(Xo))k ! Tot(Xo)
be the map that takes (h, ø1, . .,.øk) to the composite
nk(fi1,...,fik)no o ,k o
o h! nk( o, . .,. o) ! k(X , . .,.X ) ! X
To complete the proof of Theorem 10.6 it remains to show that Dn is weakly e*
*quivalent
to Cn. This is more difficult than the corresponding step in the proofs of Theo*
*rems 4.3 and
8.3 because in those cases it was only necessary to show that certain spaces we*
*re contractible,
whereas here we need to show not just that the spaces Dn(k) and Cn(k) are weakl*
*y equivalent
but that the operad structures are compatible. The proof is given in [28, Secti*
*on 12]; the
basic idea is to show that the operads Cn and Dn can be written as homotopy col*
*imits, over
the same indexing category, of contractible suboperads.
25
Remark 12.5. The operad Dn is of interest in its own right, as an En operad who*
*se structure
is in some ways simpler than that of Cn. In [28, Section 11] it is shown that *
*Dn(k) is
homeomorphic to
Znkx Tot( o)
where Znkis the zeroth space of nk( o, . .,. o). Moreover, the space Znkhas a*
*n explicit
cell decomposition which is wellrelated to the operad structure of Dn. The cel*
*lular chain
complexes of the Znkform a chain operad which is studied in [27] (where it is c*
*alled Sn).
13 Applications.
13.1 The topology of a space of knots.
The space of imbeddings of S1 in Rk is of considerable interest in topology. It*
* turns out that
a closely related space satisfies the hypothesis of Theorem 10.6 with n = 2, an*
*d is therefore
a twofold loop space.
To be specific, let us consider the manifoldwithboundary Rk1 x I. Fix po*
*ints x0 2
Rk1 x {0} and x1 2 Rk1 x {1}, and also fix tangent vectors v0 at x0 and v1 at*
* x1. Let
Emb(I, Rk1 x I) be the space of embeddings of I in Rk1 x I which take 0 and 1*
* to x0
and x1 respectively, with tangent vectors v0 at 0 and v1 at 1. Let Imm(I, Rk1x*
* I) be the
analogous space with immersions instead of imbeddings. Finally, let Fib(I, Rk1*
*x I) be the
fiber of the forgetful map
Emb (I, Rk1x I) ! Imm (I, Rk1x I)
It follows from a theorem of Hirsch and Smale that Imm(I, Rk1x I) is homotopy *
*equivalent
to Sk1, so Fib(I, Rk1x I) contains most of the information in Emb(I, Rk1x I*
*).
Now assume k 4.
Dev Sinha [31] (building on earlier work of Goodwillie and Weiss) has given *
*a cosimplicial
space Xo with Tot(Xo) weakly equivalent to Fib(I, Rk1x I). He has also shown t*
*hat Xo
satisfies the hypothesis of Theorem 10.6 with n = 2. It follows that Fib(I, Rk*
*1 x I) is a
twofold loop space.
Remark 13.1. When a cosimplicial space Xo satisfies the hypothesis of Theorem 1*
*0.6, the
spectral sequence converging to the homology of Tot(Xo) will have extra structu*
*re coming
from the operations. This should be useful for analyzing the spectral sequ*
*ence that
converges to the homology of Fib(I, Rk1x I).
13.2 Topological Hochschild Cohomology.
Theorems 4.3, 8.3 and 10.6 are still true, with essentially the same proofs, fo*
*r cosimplicial
spectra (except that Cartesian products in the category of spaces are replaced *
*by smash
products in the category of spectra).
The definition of Hochschild cohomology for associative rings (which will be*
* recalled in
Section 16) has an analog for associative ring spectra in the sense of [11] or *
*[16]. If R is an
26
associative ring spectrum there is a cosimplicial spectrum T Ho(R) (see [26, Ex*
*ample 3.4]
for the definition) whose total spectrum Tot(T Ho(R)) is called the topological*
* Hochschild
cohomology spectrum of R.
In [26] it is shown that T Ho(R) satisfies the hypothesis of Theorem 10.6 wi*
*th n = 2, and
therefore the topological Hochschild cohomology spectrum of R is an E2 spectrum*
*. This is
a spectrum analog of Deligne's Hochschild cohomology conjecture (see Section 16*
*).
14 The framed little disks operad.
The framed little disks operad was defined by Getzler in [13]; it is a variant *
*of the little
2cubes operad.
Let B denote the closed unit disk in R2.
Definition 14.1. A TDRmap B ! B is a composite T O D O R, where T is a transla*
*tion,
D is a dilation and R is a rotation.
Definition 14.2. (a) For k 0, let F(k) be the space in which a point is a kt*
*uple
(~1, . .,.~k) of TDRmaps B ! B such that the images of the ~ihave disjoint int*
*eriors.
(b) Let F be the collection of spaces {F(k)}k 0.
F is an operad, where the k action on F(k) permutes the ~i and the definiti*
*on of fl is
analogous to Definition 9.3(c).
Remark 14.3. It is instructive to consider the relation between F and C2.
(a) If we require the ~i in Definition 14.2 to be T D maps (that is, composi*
*tes of trans
lations and dilations), we get a suboperad F0 of F. By restricting TD maps B ! *
*B to the
square inscribed in B we get an equivalence of operads F0 ! C2.
(b) The kth space of F is the Cartesian product F0(k) x (S1)k (but note tha*
*t the
projections F(k) ! F0(k) do not give a map of operads).
(c) An action of F on a space X is the same thing as an F0 action together w*
*ith a suitably
compatible S1 action.
Remark 14.4. One reason that F is important is that an F action on a space X in*
*duces a
BatalinVilkovisky structure on H*X (see [13]).
The analog of Theorem 10.6 for F actions has a surprisingly simple form. As *
*motivation
we consider the following situation: let Vo be a cyclic set, that is, a simplic*
*ial set together
with maps
t : Vm ! Vm
for each m 0 satisfying certain relations with the simplicial operators (see *
*[38, Definition
9.6.1]). Define Ao to be Map (Vo, Z). Then Ao is a cocyclic abelian group, th*
*at is, it is a
cosimplicial abelian group together with maps
ø : Am ! Am
for each m 0 which satisfy appropriate relations with the cosimplicial operat*
*ors. We can
define t and the other operations on Ao in analogy with Definition 7.2 (the*
* precise
27
definition is left as an exercise for the reader; the basic idea is to use iter*
*ated face maps
to interpret the symbol oe(U) in this context). The relations between the maps*
* t and the
simplicial operators imply that all operations of complexity 2 are genera*
*ted by t and
ø, subject to the relation
(14.1) øp+1(x t y) = y t x,
where x is in Ap and øp+1 denotes the (p + 1)st iterate of ø.
Theorem 14.5. If Xo is a cocyclic space with a product
t : Xp x Xq ! Xp+q+1
which is associative and unital and satisfies (5.1), (5.2)and (14.1)then Tot(Xo*
*) has an
action of F.
The proof is similar to that of Theorem 10.6; see [30].
15 Cosimplicial chain complexes.
The theory developed in Sections 4, 5, 8, 10, 12 and 14 has a precise analog wi*
*th spaces
replaced by chain complexes; see [29]. In this section we give a brief discussi*
*on.
First we need the appropriate concept of operad. In fact one can define nons*
*ymmetric
operads in any monoidal category by replacing the Cartesian products in Definit*
*ion 2.5 by
the monoidal product, and one can define operads in any symmetric monoidal cate*
*gory
by analogy with Definition 6.2. The category of chain complexes is a symmetric *
*monoidal
category (the monoidal product is the usual tensor product of chain complexes) *
*and operads
in this category are called chain operads.
Definition 15.1. (a) A chain complex is weakly contractible if its homology is *
*Z in dimension
0 and zero in all other dimensions.
(b) An A1 chain operad is a nonsymmetric chain operad O for which each O(k)*
* is a
weakly contractible chain complex.
(c) An E1 chain operad is a chain operad O for which each O(k) is a weakly c*
*ontractible
chain complex.7
Next we need the analog of Tot. We have already defined the conormalization*
* of a
cosimplicial abelian group (Definition 3.7). We now extend this definition to *
*cosimplicial
chain complexes. Recall the cosimplicial chain complex o*(Definition 3.6).
Definition 15.2. Let Bo*be a cosimplicial chain complex. The conormalization o*
*f Bo*,
denoted C(Bo*), is the cochain complex
Y1
Hom ( o*, Bo*) Hom ( m*, Bm*).
m=0
Here Hom is Hom in the category of cosimplicial graded abelian groups and the*
* differential
is induced by the differentials of o*and Bo*.
________________________________
7For technical reasons, it is usual to require in addition that the action o*
*f each k should be free. The
operad T defined below has this property.
28
In practice it's useful to have an elementary description of C(Bo*). First *
*note that by
fixing the internal degree m we get a cosimplicial abelian group Bomand hence a*
* cochain
complex C(Bom) (see Remark 3.8 for elementary descriptions of this cochain comp*
*lex). The
differential in Bo*induces a differential
C(Bom) ! C(Bom1)
so the cochain complexes C(Bom) assemble into a bicomplex (with differentials l*
*owering degree
in one direction and raising degree in the other). The conormalization of Bo*is*
* the totalization
of this bicomplex:
Y
C(Bo*)p = C(Bom)p+m;
m
in general this is an infinite product.
Next we observe that the definition of in Section 8 has an analog for augm*
*ented
cosimplicial chain complexes, with x replaced by . As a consequence we get a *
*chain
operad T with
T(k) = C(( o*) k)
Theorem 15.3. (a) T is an E1 chain operad.
(b) If Bo*is a cosimplicial chain complex satisfying the hypothesis of Theor*
*em 8.3 then T
acts on C(Bo*).
Remark 15.4. The definition of T looks complicated, but in fact T has a simple *
*explicit
description. For each fixed k 1 and q, r 0 let Uq,r(k) be the free abelian *
*group generated
by the symbols
f ffi
{1, . .,.k}oo_[q]___//[r]
where
(a) f is onto,
(b) the image of OE contains all of {1, . .,.r} (but is allowed to not contain *
*0),
(c) OE is orderpreserving,
(d) OE(i) = OE(i + 1) ) f(i) 6= f(i + 1).
Then the pth group of the chain complex T(k) is
Y
Uq,q+1pk(k)
q
It is not hard to show this from the definitions; see [29]. [29] also gives exp*
*licit formulas for
the differential of T(k) and for the operad structure maps of T.
29
Remark 15.5. T is not the same as the E1 chain operad S defined in [27], but t*
*hey are
related: S can be obtained from T by the condensation process described in [26,*
* Section 7].
We will show in [29] that the structural formulas for S can be deduced from tho*
*se for T; this
is less elementary than the treatment of the structural formulas in [27] but av*
*oids the eight
pages of sign verifications in that paper. Each of S and T has advantages: S is*
* much smaller
but T has useful formal properties (see Section 16.3).
Next we need the definition of weak equivalence for chain operads.
Definition 15.6. A morphism i : O ! O0of chain operads is a weak equivalence if*
* each ik
is a homology isomorphism. Two chain operads O and O0are weakly equivalent if t*
*here is a
diagram of operads and weak equivalences of operads
O . .!.O0
Now fix n 1. Applying the normalized singular chain functor to the little*
* ncubes
operad Cn we obtain a chain operad S*Cn.
Definition 15.7. An En chain operad is a chain operad weakly equivalent to S*Cn.
The definition of nkin Section 12 has an analog for augmented cosimplicial *
*chain com
plexes. As a consequence we get a chain operad Tn with
Tn(k) = C( nk( o*, . .,. o*))
Theorem 15.8. (a) Tn is an En chain operad.
(b) If Bo*is a cosimplicial chain complex satisfying the hypothesis of Theor*
*em 10.6 then
Tn acts on C(Bo*).
Remark 15.9. Tn has an explicit description similar to that in Remark 15.4, exc*
*ept that f
is required to have complexity n; see [29]. Also in [29], we show that the ch*
*ain operad Sn
defined in [27] can be obtained from Tn by condensation.
Remark 15.10. The theory described in Section 14 also has a chain analog. In [*
*30] we
construct a chain operad G which is weakly equivalent to S*F, and we show that *
*if Bo*is a
cosimplicial chain complex satisfying the hypothesis of Theorem 14.5 then G act*
*s on C(Bo*).
16 Applications.
16.1 Deligne's Hochschild cohomology conjecture.
Let R be an associative ring. The Hochschild cochain complex C*(R) is the cocha*
*in complex
which in degree p is
Hom Z(R p, R);
the differential is determined by the formula
(dæ)(r1 . . .rp+1)
Xp
= r1æ(r2 . .).+ (1)iæ(. . .riri+1 . .).+ (1)p+1æ(. . .r*
*p)rp+1
i=1
30
where æ 2 Cp(R). The Hochschild cohomology H*(R) is the cohomology of this comp*
*lex.
Hochschild defined a cup product on C*(R) by
(æ1 ` æ2)(r1 . . .rp+q) = æ1(. . .rp) . æ2(rp+1 . .).
where æ1 2 Cp(R) and æ2 2 Cq(R). This induces a product, also denoted by `, on *
*H*(R).
Gerstenhaber showed in 1963 (see [12]) that H*(R) is what is now known as a *
*Gersten
haber algebra. That is, he showed that the cup product on H*(R) is commutative *
*and that
there is a Lie bracket
[ , ] : Hp(R) Hq(R) ! Hp+q1(R)
such that [x, ] is a derivation with respect to ` for each x 2 H*(R).
About 10 years later, Fred Cohen showed that if X has a C2 action then H*X i*
*s a
Gerstenhaber algebra (but he didn't use this terminology); see [7].
In 1993, Deligne asked in a letter [10] whether these two examples of Gerste*
*nhaber
algebras were related: specifically he asked whether the cup product and Lie b*
*racket on
H*(R) are induced by an action of an E2 chain operad on C*(R).
One reason this conjecture is important is because of its connection with Ko*
*ntsevich's
deformation quantization theorem; see [19].
The conjecture has been proved by several authors using quite different meth*
*ods (see
[35, 36, 26, 37, 19, 20, 17, 18]). In [27, Section 2] we gave an especially si*
*mple proof by
showing that the E2 operad S2 defined in that paper acts on C*(R) by explicit f*
*ormulas. In
[29] we show that the E2 operad T2 (see Section 15) acts on C*(R), also by expl*
*icit formulas;
this argument has the advantage that it avoids the complicated sign verificatio*
*ns needed in
[27].
16.2 Strong Frobenius algebras.
By a strong Frobenius algebra we mean an algebra A over a field such that A is *
*isomorphic to
A* as an Abimodule. In [30] we show that if A is a strong Frobenius algebra th*
*en the chain
operad G (see Remark 15.10) acts on C*(A). This is a strong form of Deligne's H*
*ochschild
cohomology conjecture for these algebras.
16.3 A theorem of Kriz and May.
Let Ab denote the category of cosimplicial abelian groups, and let Ch* 0denote*
* the category
of nonnegatively graded cochain complexes.
The conormalization functor gives an equivalence of categories
(16.1) C : Ab ! Ch* 0
(cf. [38, Section 8.4]).
We mentioned in Remark 8.9 that the category of cosimplicial spaces has a no*
*nunital
symmetric monoidal product ; essentially the same construction (with x replace*
*d by )
gives a nonunital symmetric monoidal product for Ab . It is natural to ask*
* how is
related to the equivalence (16.1), and this question has a simple answer:
31
Theorem 16.1. C(Ao Bo) ~=T(2) T(1) T(1)C(Ao) C(Bo) , where T is the chain operad
defined in Section 15.
See [29] for the proof.
The formula in Theorem 16.1 is precisely analogous to Definition V.1.1 of [2*
*1]. As a
corollary to Theorem 16.1 we recover the results of [21, Section V.3], but with*
* the "lin
ear isometries operad" (actually the singular chains of the usual linear isomet*
*ries operad)
replaced by T. The operad T has the advantage that it is much smaller than the*
* linear
isometries operad and its structure can be described explicitly.
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