Geometry & Topology Monographs 16 (2009) 283330 283

The construction of E
1
ring spaces
from bipermutative categories
J P MAY
The construction of E
o
ring spaces and thus E
o
ring spectra from bipermutative
categories gives the most highly structured way of obtaining the Ktheory commu-
tative ring spectra. The original construction dates from around 1980 and has never
been superseded, but the original details are difcult, obscure, and slightly wrong.
We rework the construction in a much more elementary fashion.
18C20, 18D10, 18D50, 19D23, 55P48
Introduction
Bipermutative categories give the most important input into multiplicative innite loop
space theory. The classifying space of a permutative category is an E
o
space. We
would like to say that the classifying space of a bipermutative category is equivalent to
an E
o
ring space. That is a deeper statement, but it is also true.
My rst purported proof of this passage, in [11], was incorrect. It was based on a
nonexistent E
o
operad pair. I wrote the quite difcult paper [16] to correct this.
Although the correction is basically correct, there are two rather minor errors of detail
in [16] and the paper is quite hard to read. Fixes for the errors were in place in the
early 1990s, but were never published.
1
While writing the prequel [17], I rethought
the technical details and saw that the easier x leads to quite elementary ideas that
make the harder x unnecessary. I will give the details here, since they substantially
simplify [16]. In a sense the change is trivial. The minor errors referred to above only
concern considerations of basepoints, and I will redo the theory in a way that allows
the basepoints to take care of themselves, following [17, 1.4]. This changes the ground
categories of our monads to ones made up of unbased spaces, and the change trivializes
the combinatorial descriptions of the relevant monads.
1
The more substantial x is purely combinatorial and was given to me by Uwe Hommel in the early
1980s. That correction was submitted to JPAA, where [16] appeared, in 1986. The editors declined to
publish it since the correction was relatively minor and was unreadable in isolation. The introduction of
[4] exaggerated the errors in [16], which helped spur this simplied reworking.
Published: 25 June 2009 DOI: 10.2140/gtm.2009.16.283
284 J P May
Since the treatment of basepoints is so crucial, we state our conventions right away.
We consider both based and unbased spaces in Sections1 and 2. We work solely with
unbased spaces in Sections 313; we alert the reader to a relevant change of notations
that is explained at the start of Section 3. We also x the convention that when we say
that a map is an equivalence, we mean that it is a weak homotopy equivalence.
In fact, the mistakes had nothing to do with bipermutative categories. As I will recall,
my work [9; 16] and that of Woolfson [23] includes two different and entirely correct
ways of constructing (F
R
F)spaces from bipermutative categories. This is quite
standard and, by now, quite elementary category theory. By pullback, (F
R
F)spaces
are (

G
R

C)spaces, where

G
R

C is the category of ring operators associated to an E
o
operad pair (C. G). The minor errors concerned the construction of (C. G)spaces,
that is, E
o
ring spaces, from (

G
R

C)spaces. With the details here, that construction
is now also mainly elementary category theory.
The diagram in Figure 1 will serve as a guide to the revised theory. It expands the top
two lines of the diagram from [12] that we focused on in the prequel [17, 0.1].
PERM CATS

BIPERM CATS

F CATS
B

(F
R
F) CATS
B

F SPACES

(F
R
F) SPACES

C SPACES

G
R

C) SPACES

C.

G) SPACES

C SPACES

(C. G) SPACES

E
o
SPACES E
o
RING SPACES
Figure 1: Guiding diagram
Geometry & Topology Monographs, Volume 16 (2009)
The construction of E
o
ring spaces from bipermutative categories 285
The intermediate pairs of downwards pointing arrows are accompanied by upwards
pointing arrows, as we will explain, but our focus is on the downwards arrows, whose
bottom targets are the inputs of the additive and multiplicative black box of the prequel
[17]. We shall work mainly from the bottom of the diagram upwards, and the following
list of sections may help the reader follow the logic.
1. Operads, categories of operators, and Fspaces
2. Monads associated to categories of operators
3. The comparison between Cspaces and

Cspaces
4. Pairs of operads and pairs of categories of operators
5. Categories of ring operators and their actions
6. The denition of (

C.

G)spaces
7. The monad

J associated to the category J =

G
R

C
8. The comparison of (

C.

G)spaces and Jspaces
9. Some comparisons of monads
10. The comparison of (C. G)spaces and (

C.

G)spaces
11. Permutative categories in innite loop space theory
12. What precisely are bipermutative categories?
13. The construction of (F
R
F)categories from bipermutative categories
14. Appendix A. Generalities on monads
15. Appendix B. Monads and distributivity
We review the input of additive innite loop space theory in Sections 1-3, which largely
follow May and Thomason [20]. The central concept is that of the category of operators

C constructed from an operad C . This gives a conceptual intermediary between

Segals Fspaces, or Ispaces, and E
o
spaces. We recall this notion in Section 1,
and we discuss monads associated to categories of operators in Section 2. A key point
is to compare monads on the categories of based and unbased spaces. We give based
and unbased versions of the parallel pair of arrows relating Fspaces and

Cspaces
in Section 1 and Section 2. Departing from [20], we give an unbased version of the
parallel pair of arrows relating Cspaces and

Cspaces in Section 3. The comparison
uses the two-sided monadic bar construction that was advertised in [17, Section 8] and
used in [20], but with simplifying changes of ground categories as compared with those
used in [20].
We then give a parallel review of the input of multiplicative innite loop space theory,
largely following [16]. Here we have three pairs of parallel arrows, rather than just
two, and we need the intermediate category of (

C.

G)spaces that is displayed in
Figure 1. This category has two equivalent conceptual descriptions, one suitable for
the comparison given by the middle right pair of parallel arrows and the other suitable
Geometry & Topology Monographs, Volume 16 (2009)
286 J P May
for the comparison given by the bottom right pair of parallel arrows. The equivalence
of the two descriptions is perhaps the lynchpin of the theory.
We recall the precise denition of an action of one operad on another and of one
category of operators on another in Section 4. We introduce categories of ring operators
and show that there is a category of ring operators J =

G
R

C associated to an operad
pair (C. G) in Section 5. Actions of J specify the (

G
R

G)spaces of Figure 1. We
also elaborate the comparison of Fspaces with

Cspaces given in Section 1 to a
comparison of (F
R
F)spaces with Jspaces in Section 5. That gives the top right
pair of parallel arrows in Figure 1.
We dene (

C.

G)spaces in Section 6. They are intermediate between Jspaces and
(C. G)spaces, being less general than the former and more general than the latter. To
compare these three notions, we work out the structure of a monad

J whose algebras
are the Jspaces in Section 7. This is where the theory diverges most fundamentally
from that of [16]. We dene

J on a ground category that uses only unbased spaces,
thus eliminating the need for all of the hard work in [16]. This change also leads to
considerable clarication of the conceptual structure of the theory.
We use this analysis to construct the middle right pair of parallel arrows of Figure 1,
comparing (

C.

G)spaces to Jspaces, in Section 8. We use it to compare monads on
the ground category for Jspaces and on the ground category for (

C.

G)spaces in
Section 9. This comparison implies the promised equivalence of our two descriptions of
the category of (

C.

G)spaces. Using our second description, we construct the bottom
right pair of parallel arrows of Figure 1, comparing (C. G)spaces to (

C.

G)spaces,
in Section 10. This comparison is just a multiplicative elaboration of the comparison
of Cspaces and

Cspaces in Section 3.
The theory described so far makes considerable use of general categorical results about
monads and, following Beck [3], about how monads are used to encode distributivity
phenomena. These topics are treated in Appendices A and B.
With this theory in place, we recall what permutative and bipermutative categories
are in Section 11 and Section 12. Some examples of bipermutative categories will be
recalled in the sequel [18]. There are several variants of the denition. We shall focus
on the original precise denition in order to relate bipermutative categories to E
o
ring
spaces most simply, but that is not too important. It is more important that we include
topological bipermutative categories, since some of the nicest applications involve
the comparison of discrete and topological examples. In line with this, all categories
throughout the paper are understood to be topologically enriched and all functors and
natural transformations are understood to be continuous. We sometimes repeat this for
emphasis, but it is always assumed.
Geometry & Topology Monographs, Volume 16 (2009)
The construction of E
o
ring spaces from bipermutative categories 287
We explain how to construct E
o
spaces from permutative categories in Section 11
and how not to construct E
o
ring spaces from bipermutative categories in Section 12.
We recall one of the two correct passages from bipermutative categories to (F
R
F)
categories in Section 13. Applying the classifying space functor B = [N()[, we
obtain (F
R
F)spaces, from which we can construct E
o
ring spaces.
There is a more recent foundational theory analogous to that reworked here, which is
due to Elmendorf and Mandell [4]. As recalled in the prequel [17], their work produces
(naive) E
o
symmetric spectra, and therefore commutative symmetric ring spectra,
from (a weakened version of) bipermutative categories. Most importantly, they show
how to construct algebra and module spectra as well as ring spectra from categorical
data.
2
However, their introduction misstates the relationship between their work and
the 1970s work. The 1970s applications all depend on E
o
ring spaces and not just
on E
o
ring spectra. That is, they depend on the passage from bipermutative categories
to E
o
ring spaces, and from there to E
o
ring spectra. Such applications, some of
which are summarized in the sequel [18], are not accessible to foundations based on
diagram ring spectra. We reiterate that a comparison is needed.
1 Operads, categories of operators, and Fspaces
We review the input data of additive innite loop space theory, since we must build
on that to describe the input data for multiplicative innite loop space theory. We rst
recall the denition of a category of operators D and the construction of a category of
operators

C from an operad C . We then recall the notion of a Dspace for a category
of operators D, and nally we show how to compare categories of Dspaces as D
varies. Aside from the correction of a small but illuminating mistake, this material is
taken from [20], to which we refer the reader for further details.
Recall that F denotes the category of nite based sets n ={0. 1. . . . . n], with 0 as
basepoint, and based functions. The category F is opposite to Segals category I [21],
and Fspaces are just Ispaces by another name. Let F be the subcategory
whose morphisms are the based functions : m n such that [
-1
(j )[ 1 for
1 j n, where [S[ denotes the cardinality of a nite set S. Such maps are
composites of injections (
-1
(0) = 0) and projections ([
-1
(j )[ = 1 for 1 j
n). The permutations are the maps that are both injections and projections. For an
injection : mn, dene

n
to be the subgroup of permutations such that
3
2
In work in progress with Vigleik Angeltveit, we dene algebras and modules on the E
o
space level,
which is completely new, and we elaborate the theory of this paper to give a comparison between E
o
rings, modules, and algebras of spaces and of spectra.
3
This is a slight correction of [20, 1.2], the need for which was observed in [16, page 11].
Geometry & Topology Monographs, Volume 16 (2009)
288 J P May
o(Im) = Im. We shall later make much use of the subcategory whose
morphisms are the projections.
4
Note that 0 is an initial and terminal object of and
of F, giving a map 0 between any two objects, but it is only a terminal object of .
Denition 1.1 A category of operators is a topological category D with objects
n={0. 1. . . . . n], n0, such that the inclusion F factors as the composite of an
inclusion D and a surjection c: DF, both of which are the identity on objects.
We require the maps D(q. m) D(q. n) induced by an injection : mn to be

cobrations. A map v: D E of categories of operators is a continuous functor

v over F and under . It is an equivalence if each map v: D(m. n) E (m. n) is
an equivalence.
Recall that we understand equivalences to mean weak homotopy equivalences. More
details of the following elementary denition are given in [20, 4.1]; see also Notations
4.5 below. The cobration condition of the previous denition is automatically satised
since the maps in question are inclusions of components in disjoint unions. As in [17],
we require the 0th space of an operad to be a point.
Denition 1.2 Let C be an operad. Dene a category

C by letting its objects be the
sets n for n 0 and letting its space of morphisms mn be

C(m. n) =
a
F(m,n)
Y
1jn
C([
-1
(j )[).
When n =0, this is to be interpreted as a point indexed on the unique map m0
in F. Units and composition are induced from the unit id C(1) and the operad
structure maps ; . If the C(j ) are all nonempty,

C is a category of operators. The
inclusion of is obtained by using the points + =C(0) and id C(1). The surjection
to F is induced by the projections C(j ) +.
Remark 1.3 There is a unique operad P such that P(0) and P(1) are each a point
and P(j ) is empty for j >1. The category

P is . There is also a unique operad
N such that N (j ) is a point for all j 0. Its algebras are the commutative monoids,
and

N =F.
Remark 1.4 There is a trivial operad QP such that Q(0) is empty (violating our
usual assumption), Q(1) is a point, and Q(j ) is empty for j >1. The category

Q is
. Some of our denitions and constructions will be described in terms of categories
of operators, although they also apply to more general categories which contain but
not , or which map to F but not surjectively.
4
is Greek Upsilon and stands for unbased; we have discarded the injections from , keeping only
the surjections. The injections correspond to basepoint insertions in spaces {X
n
].
Geometry & Topology Monographs, Volume 16 (2009)
The construction of E
o
ring spaces from bipermutative categories 289
Denition 1.5 Let D be a category of operators. A Dspace Y in T is a continuous
functor D T , written n Y
n
. It is reduced if Y
0
is a point. It is special if the
following three conditions are satised.
(i) Y
0
is aspherical (equivalent to a point).
(ii) The maps : Y
n
Y
n
1
induced by the n projections
i
: n 1,
i
(j ) =
i,j
,
in are equivalences.
(iii) If : mn is an injection, then : X
m
X
n
is a

cobration.
It is very special if, further, the monoid
0
(Y
1
) is a group. A map : Y Z of D
spaces is a continuous natural transformation. It is an equivalence if each
n
: Y
n
Z
n
is an equivalence.
Except for the very special notion, the denition applies equally well if we only require
D and do not require the map to F to be a surjection.
Denition 1.6 Let DT | denote the category of Dspaces in T .
An Fspace structure on a space Y encodes products. The canonical map
n
:n1
that sends j to 1 for 1j n prescribes a map Y
0
Y
1
when n=0 and a canonical
nfold product Y
n
Y
1
when n >0. When Y is special, which is the case of interest,
this product induces a monoid structure on
0
(Y
1
), and similarly with F replaced
by a general category of operators. The cobration condition (iii) is minor, and a
whiskering construction given in [20, Appendix B] shows that it results in no loss of
generality: given a Y for which the condition fails, we can replace it by an equivalent
Y
t
for which the condition holds. In fact, the need for this condition and for the more
complicated analogues used in [16] will disappear from the picture in the next section.
For a based space X, there is a space RX that sends n to the cartesian power
X
n
and in particular sends 0 to a point; RX satises the cobration condition if the
basepoint of X is nondegenerate. The category encodes the operations that relate
the powers of a based space. The specialness conditions on Y state that its underlying
space behaves homotopically like RY
1
.
There is an evident functor L
t
from spaces to based spaces that sends Y
n
to Y
1
.
It was claimed in [20, 1.3] that L
t
is left adjoint to R, but that is false. There is a
unique map 0 1 in , and, since (RX)
0
is a point, naturality with respect to this
map shows that for any map of spaces Y RX, the map Y
1
(RX)
1
=X
must factor through the quotient Y
1
,Y
0
. The left adjoint L to R is rather the functor
Geometry & Topology Monographs, Volume 16 (2009)
290 J P May
that sends Y to Y
1
,Y
0
. In the applications, Y is often reduced, and we could restrict
attention to reduced Dspaces at the price of quotienting out by Y
0
whenever necessary.
Dening LY =Y
1
,Y
0
, we have the adjunction
(1-7) T |(LY. X) T (Y. RX).
This remains true for special spaces and nondegenerately based spaces.
There is a two-sided categorical bar construction
B(Y. D. X) =[B
+
(Y. D. X)[.
where D is a small topological category, X: D T is a covariant functor, and
Y : D T is a contravariant functor [10, Section 12]. If O is the set of objects of
D, then the space of qsimplices is
Y
O
D
O

O
D
O
X
or, more explicitly, the disjoint union over tuples of objects n
i
in O of
Y
n
q
D(n
q-1
. n
q
) D(n
0
. n
1
) X
n
0
.
The faces are given by the evaluation maps of Y , composition in D, and the evaluation
maps of X. The degeneracies are given by insertion of identity maps. This behaves
just like the analogous two-sided bar constructions of [17, Sections 89], and has the
same rationale. As there, we prefer to ignore model categorical considerations and
use various bar constructions to deal with change of homotopy categories in this paper.
The following result is [20, 1.8]. When specialized to c:

C F, it gives the upper
left pair of parallel arrows in Figure 1.
Theorem 1.8 Let v: D E be an equivalence of categories of operators. When
restricted to the full subcategories of special objects, the pullback of action functor
v
+
: E T | DT | induces an equivalence of homotopy categories.
Sketch proof Via v and the composition in E , each E (. n) is a contravariant functor
D T ; via the composition of E , each E (m. ) is a covariant functor E T .
For Y DT | , dene
(v
+
Y )
n
=B(E (. n). D. Y ).
This gives an extension of scalars functor v
+
: D E . Notice that v
+
Y is not reduced
even when Y is reduced. The following diagram displays a natural weak equivalence
between Y and v
+
v
+
Y .
Y B(D. D. Y )
B(,id id)

v
+
B(E . D. Y ) =v
+
v
+
Y.
Geometry & Topology Monographs, Volume 16 (2009)
The construction of E
o
ring spaces from bipermutative categories 291
Its left arrow has a natural homotopy inverse j. Similarly, for ZE T | , the following
composite displays a natural weak equivalence between v
+
v
+
Z and Z.
v
+
v
+
Z =B(E . D. v
+
Z)
B(id,,id)

B(E . E . Z)
t

Z.
The categorically minded reader will notice that these maps should be viewed as the
unit and counit of an adjunction fattened up by the bar construction.
While the functor v
+
takes us out of the subcategory of reduced objects, we could
recover reduced objects by quotienting out (v
+
Y )
0
. For our present emphasis, all we
really care about is the mere existence of the functor v
+
, since our goal is to create
input for the innite loop space machine that we described in [17, Section 9]. Thus the
distinction is of no great importance. However, it is thought provoking, and we show
how to eliminate it conceptually in the next section.
2 Monads associated to categories of operators
We are going to change our point of view now, since the change here in the one operad
case will illuminate the more substantial change in the two operad case. We recall the
following general and well-known result in the form that we gave it in [16, 5.7]. It
works in greater generality, but the form given there is still our focus here. Since this
by now should be standard category theory known by all algebraic topologists, we shall
not elaborate the details. We usually write j and j generically for the product and
unit of monads.
Construction 2.1 Let D be a topological category and let be a topologically
discrete subcategory with the same objects. Let U | denote the category of spaces
(functors U ) and let DU | denote the category of Dspaces (continuous
functors D U ). We construct a monad D in U | such that DU | is isomorphic
to the category of Dalgebras in U | . For an object n and a space Y , (DY )
n
is the categorical tensor product (or left Kan extension)
D(. n)

Y.
More explicitly, it is the coequalizer displayed in the diagram
`
: q-m
D(m. n) Y
q

`
m
D(m. n) Y
m

D(. n)

Y.
where the parallel arrows are given by action maps (q. m) Y
q
Y
m
and composi-
tion maps D(m. n) (q. m) D(q. n). Then DY is a Dspace (and in particular
a space) that extends the space Y . If Y is a Dspace, the inclusion of in
Geometry & Topology Monographs, Volume 16 (2009)
292 J P May
D induces a map DY D
D
Y Y that gives Y a structure of Dalgebra, and
conversely.
The point to be emphasized is that we can use varying subcategories of the same
category D, giving monads on different categories that have isomorphic categories of
algebras. In the previous section, we considered a category of operators D and focused
on = . Then Construction 2.1 gives the monad D on the category T | that
was used in [20]. In particular, when D =

C , it gives the monad denoted

C there.
We think of these as reduced monads. Their construction involves the injections in ,
which encode basepoint identications.
However, it greatly simplies the theory here if, when constructing a monad associated
to a category of operators D, we switch from to its subcategory of projections
and so eliminate the need for basepoint identications corresponding to injections. We
emphasize that we do not change D, so that we still insist that it contains . With this
switch, Construction 2.1 specializes to give an augmented monad D

on the category
U | . In particular, when D =

C , it gives a monad

C

. The following denitions

and results show that we are free to use D

instead of D for our present purposes;

compare Remark 3.11 below.
Denition 2.2 Let D be a category of operators. A Dspace Y in U is a continuous
functor D U , written n Y
n
. It is reduced if Y
0
is a point. It is special if the
following two conditions are satised.
(i) Y
0
is aspherical (equivalent to a point).
(ii) The maps : Y
n
Y
n
1
induced by the n projections
i
: n 1,
i
(j ) =
i,j
,
are equivalences.
It is very special if, further, the monoid
0
(Y
1
) is a group. A map : Y Z of D
spaces is a continuous natural transformation. It is an equivalence if each
n
: Y
n
Z
n
is an equivalence.
Denition 2.3 Let DU | denote the category of Dspaces in U .
For purposes of comparison, we temporarily adopt the following notations for the
categories of algebras over the two monads that are obtained from D by use of
Construction 2.1.
Denition 2.4 Let D be a category of operators.
Geometry & Topology Monographs, Volume 16 (2009)
The construction of E
o
ring spaces from bipermutative categories 293
(i) Let D

. U | denote the category of algebras over the monad on U | asso-

ciated to D.
(ii) Let D. T | denote the category of algebras over the monad on T | associ-
ated to D.
In fact, we have two other such categories of algebras over monads in sight. One is
D

. T | , which is isomorphic to D. T | and to the category of Dalgebras in

T . The other is D. U | , which is isomorphic to D

. U | and to the category of

Dalgebras in U .
The situation here is very much like that discussed in [17, Section 4]. If we have an
action of D on a space Y , then the maps 0 n of D, together with a
choice of basepoint in Y
0
, give the spaces Y
n
basepoints. The injections in also
give the unit properties of the products on a Dspace Y . Using and U rather than
and T means that we are not taking the basepoints of the Y
n
and the analogues of
insertion of basepoints induced by the injections in as preassigned.
The following result is analogous to [17, 4.4].
Proposition 2.5 Let D be a category of operators, such as

C for an operad C .
Consider the following four categories.
(i) The category DU | of Dspaces in U .
(ii) The category D

. U | of D

algebras in U | .
(iii) The category DT | of Dspaces in T .
(iv) The category D. T | of Dalgebras in T | .
The rst two are isomorphic and the last two are isomorphic. When restricted to reduced
objects (Y
0
= +), all four are isomorphic. In general, the forgetful functor sends
DT | isomorphically onto the subcategory of DU | that is obtained by preassigning
basepoints to 0th spaces Y
0
and therefore to all spaces Y
n
.
We have the analogue of Theorem 1.8, with the same proof.
Theorem 2.6 Let v: D E be an equivalence of categories of operators. When
restricted to the full subcategories of special objects, the pullback of action functor
v
+
: E U | DU | induces an equivalence of homotopy categories.
Geometry & Topology Monographs, Volume 16 (2009)
294 J P May
3 The comparison between Cspaces and
y
Cspaces
To begin with, let us abbreviate notations from the previous section. Let us write
V =U | for the category of spaces. This category plays a role analogous to U .
We then write DV | =D

. U | for a category of operators D. From here on out,

we shall always use augmented monads rather than reduced ones, and we therefore
drop the from the notations. This conicts with usage in the prequel [17] and in all
previous work in this area, but hopefully will not cause confusion here. We will never
work in a based context in the rest of this paper.
Now specialize to D =

C . Our change of perspective simplies the passage from

C
spaces to Cspaces of [20, Section 5]. For an unbased space X, dene (RX)
n
=X
n
,
with the evident projections. For an space Y , dene LY = Y
1
. Since we have
discarded the injection 0 1 in , there is no need to worry about the distinction
between reduced and unreduced spaces, and we have the adjunction
(3-1) V (LY. X) U (Y. RX).
Here the counit of the adjunction is the identity transformation LR Id, and the
unit : Y RLY is given by the maps : Y
n
Y
n
1
. The rst of the following
observations is repeated from [20, 5.25.4], and the second follows by inspection. The
reader may wish to compare the second with the analogous but more complicated result
[20, 5.5], which used and T instead of and U .
Notations 3.2 A morphism [ in F is effective if [
-1
(0) = 0; thus the effective
morphisms in are the injections, including the injections 0: 0 n for n 0. An
effective morphism [ is ordered if [(i ) < [(j ) implies i <j . Let E F denote
the subcategory of objects {n] and ordered effective morphisms [.
Lemma 3.3 Any morphism in F factors as a composite [ , where is a
projection and [ is effective, uniquely up to a permutation of the source of [. If
[: m n is effective, there is a permutation t
m
such that [ t is ordered.
If [ is ordered, then [ t is also ordered if and only if t ([)
m
, where
([) =
r
1

r
n
, r
j
=[[
-1
(j )[.
Lemma 3.4 For an space Y , (

CY )
0
=Y
0
and, for n 1,
(

CY )
n
=
a
)E (m,n)

Y
1jn
C([[
-1
(j )[)

())
Y
m
.
Geometry & Topology Monographs, Volume 16 (2009)
The construction of E
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The following analogues of [20, 5.65.8] are now easy. Since we have performed no
gluings along injections, at the price of retaining factors C(0) in the description of

CY , no cobration conditions are required.

Lemma 3.5 Assume that C is free, in the sense that each C(j ) is
j
free. If
: Y Y
t
is an equivalence of spaces, then so is

C .
Recall the monad C
U

on U from [17, 4.1]. In line with the conventions at the

beginning of this section, we abbreviate notation to C in this paper, so that
(3-6) CX =
a
m0
C(m)

m
X
m
.
Here and below, we must remember that the empty product of spaces is a point. For
m0,
m
is the unique effective morphism m1 (which is automatically ordered),
and the following result is clear.
Lemma 3.7 Let X U . Then L

CRX (

CRX)
1
= CX, and the natural map
:

CRX RL

CRX =RCX is an isomorphism.

Lemma 3.8 Assume that C is free. If Y is a special space, then so is

CY ,
hence

C restricts to a monad on the category of special space.
Proof Applying Lemma 3.5 to the horizontal arrows in the commutative diagram

CY
`
C

CRLY

RL

CY
RL
`
C

RL

CRLY
we see that its left vertical arrow is an equivalence.
We can now compare

Cspaces in V to Cspaces in U in the same way that we
compared the analogous categories of based spaces in [20, page 219]. We use the
two-sided monadic bar construction of [8], the properties of which are recalled in [17,
Section 8]. We recall relevant generalities relating monads to adjunctions in Appendix
A. We use properties of geometric realization proven in [8] and the following unbased
analogue of [8, 12.2], which has essentially the same proof.
Geometry & Topology Monographs, Volume 16 (2009)
296 J P May
Lemma 3.9 For simplicial objects Y in the category V , there is a natural isomorphism
v: [

CY [

C[Y [ such that the following diagrams commute.
[Y [
[)[

)

C
C
C
C
C
C
C
C
[

CY [

C[Y [
and
[

C

CY [
[[

`
C

CY [

C

C[Y [

CY
If (Y. ) is a simplicial

Calgebra, then ([Y [. [[ v
-1
) is a

Calgebra.
Theorem 3.10 If C is free, then the functor R induces an equivalence from the
homotopy category of Cspaces to the homotopy category of special

Cspaces.
Proof Lemma 3.7 puts us into one of the two contexts discussed in general categorical
terms in Proposition 14.3. Let X be a Cspace and Y be a

Cspace. By (iii) and (iv)
of Proposition 14.3, R embeds the category of Cspaces as the full subcategory of the
category of

Cspaces consisting of those

Cspaces with underlying space of the
form RX. By (i) and (ii) of Proposition 14.3, CL is a

Cfunctor and we can dene a
functor :

CU | CU | by
Y =B(CL.

C. Y ).
By Corollaries 14.4 and 14.5, together with general properties of the geometric realiza-
tion of simplicial spaces proven in [8], we have a diagram
Y
B(

C.

C. Y )

B(RCL.

C. Y ) RY
of

Cspaces in which the map c is a homotopy equivalence with natural homotopy
inverse j and the map =B(. id. id) is an equivalence when Y is special. Thus the
diagram displays a natural weak equivalence between Y and RY . When Y =RX,
the displayed diagram is obtained by applying R to the analogous diagram
X B(C. C. X)


t

B(CL.

C. RX) =RX
of Calgebras, in which c is a homotopy equivalence with natural inverse j.
Remark 3.11 In [20], the focus was on the generalization of the innite loop space
machine of [8] from Cspaces in T to

Cspaces in T . For that purpose, it was
essential to use the approximation theorem and therefore essential to use the monads
in T and T | that are constructed using basepoint type identications. It is that
theory that forced the use of the cobration condition Denition 1.5 (iii). However, we
Geometry & Topology Monographs, Volume 16 (2009)
The construction of E
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are here only concerned with the conversion of

Cspaces to Cspaces, and for that
purpose we are free to work with the simpler monads on U and V =U | whose
algebras are the Cspaces and

Cspaces in U . From the point of view of innite
loop space machines, we prefer to convert input data to Cspaces and then apply the
original machine of [8] rather than to generalize the machine to

Cspaces.
4 Pairs of operads and pairs of categories of operators
With this understanding of the additive theory, we now turn to the multiplicative theory.
We rst recall some basic denitions from [16, Section 1] since they are essential to
understanding the details. However, the reader should not let the notation obscure the
essential simplicity of the ideas. We are just parametrizing the structure of a ring space,
or more accurately rig space since their are no negatives, and then generalizing from
operations on products X
n
to operations on Y
n
, where, when Y is special, Y
n
looks
homotopically like Y
n
1
.
The category F is symmetric monoidal (indeed, bipermutative) under the wedge
and product. On objects, the operations are sum and product interpreted by ordering
elements in blocks and lexicographically. That is, the set m n is identied with
mn by identifying i with i for 1 i m and j with j m for 1 j n, and
the set m.n is identied with mn by identifying (i. j ), 1 i m and 1 j n
with ij , with the ordering ij < i
t
j
t
if i < i
t
or i =i
t
and j < j
t
. The wedge and
smash product of morphisms are forced by these identications. We x notations for
standard permutations.
Notations 4.1 Fix nonnegative integers k, j
r
for 1 r k, and i
r,q
for 1 r k
and 1 q j
r
.
(i) Let o
k
. Dene o(j
1
. . . . . j
k
) to be that permutation of j
1
j
k
elements
which corresponds under lexicographic identication to the permutation of smash
products
o: j
1
. .j
k
j
o
1
(1)
. .j
o
1
(k)
.
(ii) Let t
r

j
r
, 1 r k. Dene t
1
t
k
to be that permutation of j
1
j
k
elements which corresponds under lexicographic identication to the smash
product of permutations
t
1
. .t
k
: j
1
. .j
k
j
1
. .j
k
.
Geometry & Topology Monographs, Volume 16 (2009)
298 J P May
(iii) Let Q run over the set of sequences (q
1
. . . . . q
k
) such that 1 q
r
j
r
, ordered
lexicographically. Dene v =v({k. j
r
. i
r,q
]) to be that permutation of

Q
(
1rk
i
r,q
r
) =
1rk
(
1qj
r
i
r,q
)
elements which corresponds under block sum and lexicographic identications
on the left and right to the natural distributivity isomorphism
_
Q

^
1rk
i
r,q
r

^
1rk

_
1qj
r
i
r,q

.
Denition 4.2 Let C and G be operads with C(0) ={0] and G(0) ={1]. Write ;
for the structure maps of both operads and id for the unit elements in both C(1) and
G(1). An action of G on C consists of maps
z: G(k) C(j
1
) C(j
k
) C(j
1
j
k
)
for k 0 and j
r
0 which satisfy the following distributivity, unity, equivariance,
and nullity properties. Let
g G(k) and g
r
G(j
r
) for 1 r k
c C(j ) and c
r
C(j
r
) for 1 r k
c
r,q
C(i
r,q
) for 1 r k and 1 q j
r
.
Further, let
c
J
r
=(c
r,1
. . . . . c
r,j
r
) C(i
r,1
) C(i
r,j
r
)
and
c
Q
=(c
1,q
1
. . . . . c
k,q
k
) C(i
1,q
1
) C(i
k,q
k
).
(i) z(;(g: g
1
. . . . . g
k
): c
J
1
. . . . . c
J
k
) =z(g: z(g
1
: c
J
1
). . . . . z(g
k
: c
J
k
)).
(ii) ;(z(g: c
1
. . . . . c
k
):
Q
z(g: c
Q
)) v =z(g: ;(c
1
: c
J
1
). . . . . ;(c
k
: c
J
k
)).
(iii) z(id: c) =c.
(iv) z(g: id
k
) =id.
(v) z(go: c
1
. . . . . c
k
) =z(g: c
o
1
(1)
. . . . . c
o
1
(k)
) o(j
1
. . . . . j
k
).
(vi) z(g: c
1
t
1
. . . . . c
k
t
k
) =z(g: c
1
. . . . . c
k
) t
1
. . . t
k
.
(vii) z(1) =id C(1) when k =0.
(viii) z(g: c
1
. . . . . c
k
) =0 when any j
r
=0.
Geometry & Topology Monographs, Volume 16 (2009)
The construction of E
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Here (i), (iii), (v), and (vii) relate the z to the internal structure of G , while (ii), (iv),
(vi), and (viii) relate the z to the internal structure of C .
We have an analogous notion of an action of a category of operators K on a category
of operators D. Again, we x notations for some standard permutations.
Notations 4.3 Let : mn and [: n p be morphisms in F. For nonnegative
integers r
i
, 1i m, dene s
k
=
)(i)=k
r
i
. Dene o
k
([. ) to be that permutation
of s
k
letters which corresponds under lexicographic ordering to the bijection
^
)(i)=k
r
i

^
)(j)=k
^
(i)=j
r
i
that permutes the factors r
i
fromtheir order on the left (i increasing) to their order on the
right (j increasing and, for xed j , i increasing). Here s
k
=1 and o
k
([. ): 1 1
is the identity if there are no i such that [(i ) = k. Dene o([. ) to be the
isomorphism in
p
with coordinates the o
k
([. ). For morphisms : mn and
g: n p in a category of operators D, write o
k
(g. ) =o
k
(c(g). c( )), and write
o(g. ) for their product in D
p
.
Let D
0
be the trivial category, which has one object + and its identity morphism.
Denition 4.4 Let D and K be categories of operators. An action z of K on D
consists of functors z( ): D
m
D
n
for K (m. n) which satisfy the following
properties. Let c( ) =: mn.
(i) On objects, z( ) is specied by
z( )(r
1
. . . . . r
m
) =(s
1
. . . . . s
n
). where s
j
=.
(i)=j
r
i
.
(ii) On morphisms (y
1
. . . . . y
m
) of
m
D
m
, z( ) is specied by
z( )(y
1
. . . . . y
m
) =(o
1
. . . . . o
n
). where o
j
=.
(i)=j
y
i
.
(iii) On general morphisms (d
1
. . . . . d
m
) of D
m
, z( ) satises
c(z( )(d
1
. . . . . d
m
)) =(o
1
. . . . . o
n
). where o
j
=.
(i)=j
c(d
i
).
(iv) For morphisms : mn of K , z() is specied by
z()(d
1
. . . . . d
m
) =(d

1
(1)
. . . . . d

1
(n)
)
(v) For morphisms : mn and g: n p in K , the isomorphisms o(g. )
in
p
C
p
specify a natural isomorphism z(g ) z(g) z( ).
Geometry & Topology Monographs, Volume 16 (2009)
300 J P May
If
-1
(j ) is empty, then the j th coordinate of z( ) is 1 in (i) and the j th coordinate
is id C(1) in (ii)(iv). (Compare Denition 4.2 (vii)).
In what should by now be standard bicategorical language, the z(n), z( ), and o(g. )
specify a pseudofunctor z: K Cat . We do not assume familiarity with this, but it
shows that the denition is sensible formally. The denition itself species an action of
on any category of operators D and an action of any category of operators K on
both and F. However, our interest is in (

C.

G), where (C. G) is an operad pair
with G acting on C . To connect up denitions, we rst use Notations 4.3 to recall
how composition is dened in the category of operators

C associated to an operad C .
Notations 4.5 For an operad C , write (: c
1
. . . . . c
k
), or (: c) for short, for mor-
phisms in

C(m. n). Here : m n is a morphism in F and c
j
C([
-1
(j )[),
with c
j
=0 C(0) if
-1
(j ) is empty. For ([: d)

C(n. p), composition in

C is
specied by
([: d) (: c) =

[ :
1kp
;(d
k
:
)(j)=k
c
j
) o
k
([. )

.
Notations 4.6 Recall that we have canonical morphisms
n
: n 1 in F that send
j to 1 for 1 j n. Together with the morphisms of , they generate F under
the wedge sum. Notice that .
1rk

j
r
=
j
1
j
r
. We dene an embedding | of the
operad C in the category of operators

C by mapping c C(n) to the morphism
(
n
: c): n 1. Using wedges in F and cartesian products of spaces C(j ), we
dene maps

C(j
1
. 1)

C(j
k
. 1)

C(j
1
j
k
. k).
The operadic structure maps ; are recovered from these maps and composition

C(k. 1)

C(j
1
j
k
. k)

C(j
1
j
k
. 1).
The following result is [16, 1.9], and more details may be found there.
Proposition 4.7 An action z of an operad G on an operad C determines and is
determined by an action of

G on

C .
Sketch proof We have the embeddings | of C in

C and G in

G . An action z of G
on C is related to the corresponding action z of

G on

C by
(4-8) |z(g: c
1
. . c
k
) =z(|g: |c
1
. . . . . |c
k
).
Given z on the categories, this clearly determines z on the operads. Conversely, given
the combinatorics of how

G and

C are constructed from G and C , there is a unique
Geometry & Topology Monographs, Volume 16 (2009)
The construction of E
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ring spaces from bipermutative categories 301
way to extend (4-8) from the operads to the categories. Looking at Denition 4.4, we
see that if : mn is a morphism of

G with c( ) = and c
i
is a morphism of

C ,
1 i m, then the j th coordinate of z( : c
1
. . . . . c
m
) depends only on those c
i
with
(i ) = j , and has coordinates
j
G([
-1
(j )[) that allow use of the operadic
z to specify the categorical z. Details are in [16, 1.9]. Formulas (i), (iii), and (v) of
Denition 4.2 correspond to the requirement that the z( ) be functors. Formulas (ii),
(iv), and (vi) correspond to the naturality requirement of Denition 4.4(v). Formulas
(vii) and (viii) are needed for compatible treatment of 1 G(0) and 0 C(0).
5 Categories of ring operators and their actions
We can coalesce a pair of operator categories (D. K ) into a single wreath product
category K
R
D. The construction actually applies to any pseudofunctor z from any
category G to Cat , but we prefer to specialize in order to x notations.
Denition 5.1 Let z be an action of K on D, where K and D are categories of
operators. The objects of K
R
D are the ntuples of nite based sets (objects of F)
for n 0. We write objects as (n: S), where S = (s
1
. . . . . s
n
). There is a single
object, denoted (0: +), when n =0; we think of + as the empty sequence. The space
of morphisms (m: R) (n: S) in K
R
D is
a
F(m,n)
c
-1
()
Y
1jn
D

^
(i)=j
r
i
. s
j

. c: K F.
where the empty smash product is 1. Typical morphisms are written ( : d), where
K (m. n) and d =(d
1
. . . . . d
n
). If c( ) =, then d
j
D(.
(i)=j
r
i
. s
j
). For
a morphism (g: e): (n: S) (p: T), composition is specied by
(g: e) ( : d) =

g : e z(g)(d) o(g. )

.
More explicitly, with c(g) = [, the kth coordinate of e z(g)(d) o(g. ) is the
composite
V
)(i)=k
r
i
o
k
(),)

V
)(j)=k
V
(i)=j
r
i
z
k
(g)(
.j/Dk
d
j
)

V
)(j)=k
s
j
e
k

t
k
.
The object (0: +) is terminal, with unique morphism (m: R) (0: +) denoted (0: +);
the morphisms (0: +) (n: S) are of the form (0: d) = (id: d) (0: id
n
), where
0: 0 n, id: n n in F on the left, and id
n
C(1)
n
on the right.
We write the morphisms of
R
in the form (: y), where
y =(y
1
. . . . . y
n
): (r

1
(1)
. . . . . r

1
(n)
) (s
1
. . . . . s
n
).
Geometry & Topology Monographs, Volume 16 (2009)
302 J P May
Here either
-1
(j ) is a single element i or it is empty, in which case r

1
(n)
= 1.
We interpolate an analogous denition that is a followup to Remarks 1.3 and 1.4. It
will play an important role in our theory.
Denition 5.2 Let
R
denote the subcategory of
R
obtained by restricting
all morphisms to be in , thus using only projections. Similarly, dene
R
D and
K
R
exactly as in the previous denition, but starting from the actions of on D
and K on that are obtained by restricting the specications of Denition 4.4 from
to .
The following observation helps analyze the structure of K
R
D.
Lemma 5.3 There are inclusions of categories
D
R
D
R
D K
R
D K
R
K
R
K .
For maps (g: y): (n: S) (p: T) in K
R
and (: d): (m: R) (n: S) in

R
C ,
(g: y) (: d) =(1: y z(g)(d)) (g: o(g. )).
The subcategories
R
D and K
R
generate K
R
D under composition.
Proof All but the rst and last inclusions are obvious. The rst inclusion sends an
object n to (1: n) and a morphism d to (id: d). The last sends an object n to (n: 1
n
)
and a morphism : mn to ( : id
n
). As noted in [16, 1.6], the displayed formula
is obtained by composing the legs of the following commutative diagram, where, for
1 j n and 1 k p,
r
t
j
=r

1
(j)
. r
tt
k
=.
)(j)=k
r

1
(j)
. s
t
k
=.
)(j)=k
s
j
.
(p: R
tt
)
(id;z(g)(c))

J
J
J
J
J
J
J
J
J
(n: R
t
)
(g;id)

t
t
t
t
t
t
t
t
t
(id;c)

J
J
J
J
J
J
J
J
J
(p: S
t
)
(id;y)

I
I
I
I
I
I
I
I
I
(m: R)
(;c)

(;id)

t
t
t
t
t
t
t
t
t
(n: S)
(g,y)

(g;id)

t
t
t
t
t
t
t
t
t
(p: T)
Any morphism ( : d): (m: R) (n: S) factors as the composite
(m: R)
(( ;id
n
)

(n: R
t
)
(id;d)

(n: S)
where, with =c( ), r
t
j
=.
(i)=j
r
i
. This proves the last statement.
Geometry & Topology Monographs, Volume 16 (2009)
The construction of E
o
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With these constructions on hand, we dene a category of ring operators in analogy with
our denition of a category of operators. While our interest is in the case J =K
R
D,
the general concept is convenient conceptually. For an injection
(: y): (m: R) (n: S)
in
R
, dene (. y) to be the group of automorphisms (o: t): (n: S) (n: S)
such that (o: t)Im(: t) Im(: t).
Denition 5.4 A category of ring operators is a topological category J with objects
those of
R
such that the inclusion
R
F
R
F factors as the composite of
an inclusion
R
J and a surjection c: J F
R
F, both of which are the
identity on objects. We require the maps J((: Q). (m: R)) J((: Q). (n. S))
induced by an injection (. y): (m: R) (n: S) in
R
to be (: y)cobrations.
A map v: I J of categories of operators is a continuous functor v over F
R
F
and under
R
. It is an equivalence if each map
v: I((m: R). (n: S)) J((m: R). (n: S))
is an equivalence.
When J =

G
R

C for an operad pair (C. G), the cobration condition is automatically
satised since the maps in question are inclusions of components in disjoint unions. In
fact, with our new choice of details, the cobration condition is not actually needed for
the theory here. So far, we have been following [16], but we now diverge and things
begin to simplify. We dene Jspaces without cobration conditions and we ignore
basepoints, which take care of themselves.
Denition 5.5 Let J be a category of ring operators. A Jspace in U is a
continuous functor Z: J U , written (n: S) Z(n: S). It is reduced if Z(0: +)
and Z(1: 0) are single points. It is semispecial if the rst two of the following four
conditions hold, and it is special if all four conditions hold.
(i) Z(0: +) is aspherical.
(ii) The maps
tt
: Z(n: S)
Q
1jn
Z(1: s
j
) with coordinates induced by (
j
: id)
are equivalences.
(iii) Z(1: 0) is aspherical.
(iv) The maps
t
: Z(1: s) Z(1. 1)
s
with coordinates induced by (1:
j
) are
equivalences.
It is very special if, further, the rig
0
(Z(1: 1)) is a ring. Amap ZW of Jspaces
is an equivalence if each Z(n: S) W(n: S) is an equivalence.
Geometry & Topology Monographs, Volume 16 (2009)
304 J P May
Remark 5.6 When
J =

G
R

C
for an operad pair (C. G), the restriction of a Jspace Z to the subcategory

C of
J is a

Cspace Z

and the restriction of Z to the subcategory

G is a

Gspace Z

.
Denition 5.7 Let JU | denote the category of Jspaces in U .
Except for the very special notion, Denitions 5.5 and 5.7 apply equally well if we relax
our requirements on J to only require
R
, rather than
R
, to be contained in
J and do not require the map from J to F
R
F to be a surjection. This leads us to
our new choice of ground category for the multiplicative theory.
Denition 5.8 A (
R
)space is a functor
R
U , and we write W for the
category of (
R
)spaces. Changing notations from the additive theory, for a space
X we let RX denote the (
R
)space that sends (0: +) to a point and sends (n: S)
to X
s
1
s
n
for n 1.
Now comparisons of denitions give the following basic results, which are [16, 2.4
and 2.6], where more details may be found.
Proposition 5.9 Let J =F
R
F. Then the functor R: U W embeds the cate-
gory of commutative rig spaces X in the category of Jspaces as the full subcategory
of objects of the form RX.
Sketch proof For a Jspace RX, the maps induced by (id:
2
): (1: 2) (1: 1)
and (
2
: id): (2: 1
2
) (1: 1) give the addition and multiplication X X X.
The elements 0 X and 1 X are induced by the injections (0: +): (0: +) (1: 1)
and (0: id): (0: +) (1: 1) in
R
. There is a unique way to extend a given
(N . N )structure on X to an action of J on RX.
This result means that an (F
R
F)space structure on RX is determined by its re-
striction to a commutative rig space structure on X.
Proposition 5.10 Let J =

G
R

C for an operad pair (C. G). Then the functor
R: U W embeds the category of (C. G)spaces X in the category of Jspaces
as the full subcategory of objects of the form RX.
Geometry & Topology Monographs, Volume 16 (2009)
The construction of E
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ring spaces from bipermutative categories 305
Sketch proof The restriction of an action of J on RX to the operads C and G
embedded in the subcategories

C and

G give the additive and multiplicative operad
actions on X. There is a unique way to extend a (C. G)structure on X to an action
of J on RX.
Again, this means that a (

G
R

C)space structure on RX is determined by its restriction
to a (C. G)space structure on X.
By the same proof as those of Theorems 1.8 and 2.6, we have the following result. It
shows in particular that the homotopy category of special (F
R
F)spaces is equivalent
to the homotopy category of special (

G
R

C)spaces, where (C. G) is the canonical
E
o
operad pair of [17, Section 3].
Theorem 5.11 Let v: I J be an equivalence of categories of operators. When
restricted to the full subcategories of special objects, the pullback of action functor
v
+
: JU | IU | induces an equivalence of homotopy categories.
6 The denition of .
y
C;
y
G/spaces
Recall that we are writing V for the category of spaces and W for the category of
(
R
)spaces. We need a pair of adjunctions analogous to the adjunction relating U
and V (originally denoted (L. R)) that was used to compare monads in the additive
theory. Recall that we are now writing R for the evident functor U W . We can
factor R through V .
Denition 6.1 For a space X, write R
t
X ={X
n
] for the associated space. For
a space Y , let R
tt
Y be the (
R
)space that sends (0: +) to a point (the empty
product) and sends (n: S) to Y
s
1
Y
s
n
for n >0. Note that RX =R
tt
R
t
X. Let
L
t
Y be the space Y
1
(previously denoted LY ). For an (
R
)space Z, let L
tt
Z
be the space given by the spaces Z(1: s), s 0, and let LZ =L
t
L
tt
Z =Z(1: 1).
It is easy to see what these functors must do on morphisms. Some details are given in
[16, 4.1], but the adjunctions claimed in that result are in fact not adjunctions because
of basepoint and injection problems analogous to the mistake pointed out in Section 1.
The following result is an elementary unbased substitute. Its proof relies only on the
universal property of cartesian products.
Lemma 6.2 The diagram in Figure 2 displays two adjoint pairs of functors and their
composite.
Geometry & Topology Monographs, Volume 16 (2009)
306 J P May
U
R
0

V
R
00

L
0

W
L
00

Figure 2: Two adjoint pairs of functors and their composite.

Now let (C. G) be an operad pair and abbreviate J =

G
R

C . Proposition 5.10
suggests the following denition of the intermediate category mentioned in the intro-
duction.
5
Denition 6.3 Let J =

G
R

C for an operad pair (C. G). A (

C.

G)space is an
object Y V together with a Jspace structure on R
tt
Y . It is special if Y is special.
A map : Y Y
t
of (

C.

G)spaces is a map in V such that R
tt
is a map of
Jspaces. Thus, by denition, the functor R
tt
: V W embeds the category of
(

C.

G)spaces as the full subcategory of Jspaces of the form R
tt
Y .
7 The monad
x
J associated to the category J
To compare (

C.

G)spaces to Jspaces on the one hand and to (C. G)spaces on the
other, we must rst analyze the monad associated to a category of ring operators.
Denition 7.1 Let

J denote the monad on the category W such that the category
of Jspaces is isomorphic to the category of

Jalgebras in W . Dene functors

J: V V and J: U U by

J =L
tt

JR
tt
and J =L
t

JR
t
=L

JR.
The construction of

J is a special case of Construction 2.1. Ignoring the monadic
structure maps, we must nd an explicit description of the functor

J in order to relate it
to the adjunctions of Lemma 6.2. This is where the main simplication of [16] occurs.
We need some notations.
6
Recall the description of

C from Lemmas 3.3 and 3.4.
Remarks 7.2 Observe that an ordered effective morphism : mn in F decom-
poses uniquely as =
m
1

m
n
, where m
j
=[
-1
(j )[ and m
1
m
j
=m.
5
In [16], (

C.

G)spaces were called (

C. G)spaces to emphasize the partial use of actual products

implicit in their denition. I now feel that the earlier notation gives a misleading perspective.
6
The details to follow come from [16, Section 7], but the combinatorial mistakes related to injections
in
R
that begin in [16, 7.1(ii)] have been circumvented by avoiding basepoint identications.
Geometry & Topology Monographs, Volume 16 (2009)
The construction of E
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Such determine and are determined by partitions M = (m
1
. . . . . m
n
) of m. In
turn, for an object (m: R) of
R
, such a partition M determines a partition of
R = (r
1
. . . . . r
m
) into n blocks, R = (R
1
. . . . . R
n
), where R
j
is the j th block
subsequence of m
j
entries. When m = 0, we have a unique (ordered) effective
morphism 0: 0 n, a unique empty partition M of 0, and a unique empty sequence
R. There are no effective morphisms m0 when m>0.
Notations 7.3 Consider an object (m:R) of
R
, where m0 and R=(r
1
. . . . . r
m
)
with each r
i
0. In part (i), we use this notation but think of (m: R) as (m
j
: R
j
)
where 1 j n.
(i) Fix s 0. Say that a morphism y: .
1im
r
i
s in F is Reffective
if for every h, 1 h m, and every q, 1 q r
h
, there is a sequence
Q= (q
1
. . . . . q
m
) in which 1 q
i
r
i
for 1 i m such that q
h
= q and
y(Q) =0. Let E (R: s) denote the set of Reffective morphisms y, and dene
C(R: s) =
a
yE (R;s)
Y
1ts
C([y
-1
(t )[).
Further, dene (m: R) to be the group of automorphisms of (m: R) in
R
.
(ii) Fix S =(s
1
. . . . . s
n
), where s
j
0. For a partition M =(m
1
. . . . . m
n
) of m
with derived partition R=(R
1
. . . . . R
n
) of R, dene
J(M: R. S) =
Y
1jn
G(m
j
) C(R
j
: s
j
).
Further, dene (M: R) =
Q
1jn
(m
j
: R
j
) (m: R).
Remark 7.4 We clarify some special cases. When s =0 in (i) and when n =0 in (ii),
empty products of spaces are interpreted to be a single point. If m=0 in (i), the smash
product over the empty sequence R is interpreted as 1 and we allow y to be 0: 1 0
or any injection 1 s in F. If m > 0 and any one r
i
= 0, then Reffectiveness
forces all r
i
=0 and we allow y =0: 0 s.
Remark 7.5 For later reference, we record when an Reffective map y in (i) can be
in F in the cases s =0 and s =1. When s =0, we can only have m=0 and
y =0: 1 0 or m>0, all r
i
=0, and y =id =0: 0 0. When s =1, we can
only have m=0 and y =id: 1 1 or m>0, all r
i
=1, and y =id: 1 1.
Geometry & Topology Monographs, Volume 16 (2009)
308 J P May
Proposition 7.6 Let Z W . Then (

JZ)(0: +) =Z(0: +) and, for n > 0 and S =

(s
1
. . . . . s
n
),
(

JZ)(n: S) =
a
(M;R)
J(M: R. S)
(M,R)
Z(m: R).
where the union runs over all partitions M = (m
1
. . . . . m
n
) of all m 0 and all
sequences R=(r
1
. . . . . r
m
).
Proof We prove this by extracting correct details from [16, Section 7]. To begin with,
observe that if y
t
: .
1im
r
t
i
s is a map in F that is not R
t
effective, then it is
a composite y .
1im
o
i
where o
i
: r
t
i
r
i
is a projection and y is Reffective.
Indeed, suppose that y
t
(Q) =0 for all sequences Q with hth term q, where 1 q r
h
.
Then y
t
= (y
t
.
1im
o
i
) .
1im
v
i
, where v
i
= o
i
= id: r
t
i
r
t
i
for i = h,
v
h
: r
t
h
r
t
h
1 is the projection that sends q to 0 and is otherwise ordered, and
o
h
: r
t
h
1 r
t
h
is the ordered injection that misses q. The required factorization is
obtained by repeating this construction inductively.
By Construction 2.1 and Denitions 1.2 and 5.1, (

JZ)(n: S) is a quotient of
a
(m;R)
a
(;y)
Y
1jn
( G([
-1
(j )[)
Y
1ts
j
C([y
-1
j
(t )[) ) Z(m: R).
where (: y) runs over the morphisms (m: R) (n: S) in F
R
F, which means that
F(m. n) and y =(y
1
. . . . . y
n
), where y
j
F(.
(i)=j
r
i
. s
j
). The quotient is
obtained using identications that are induced by the morphisms of
R
, namely the
projections, which we think of as composites of proper projections and permutations.
In the description just given, we may restrict attention to those (: y) such that =

m
1

m
n
for some partition M of m and y =(y
1
. . . . . y
n
), where y
j
is R
j

effective. Indeed, if (
t
: y
t
) is not of this form, it factors as (: y)([: o) where (: y)
is of this form and [ and the coordinates of o are projections. To construct [ and o,
we use the observation above and record which elements other than 0 of the sets m and
the r
j
are sent to 0 by
t
and the y
t
j
. Then [
-1
(j )[ =[(
t
)
-1
(j )[ for 1 j n,
[y
-1
(t )[ =[(y
t
)
-1
(t )[ for 1 t s
j
, and any morphism (g
t
: c
t
): (m
t
: R
t
) (n: S)
in J such that c(g
t
: c
t
) =(
t
: y
t
) factors as (g: c)([: o) for some morphism (g: c)
such that c(g: c) =(: y). Up to permutations, (g: c) =(g
t
: c
t
) as elements of
Y
1jn
G([
-1
(j )[)
Y
1ts
j
C([y
-1
(t )[).
This reduction takes account of the identications dened using proper projections but
ignoring permutations; the identications dened using permutations are taken account
of by passage to orbits over the (M: R).
Geometry & Topology Monographs, Volume 16 (2009)
The construction of E
o
ring spaces from bipermutative categories 309
Specializing (n: S) to (1: s) and then specializing (1: s) to (1: 1) we obtain the follow-
ing descriptions of the functors

J =L
tt

JR
tt
and J =L
t

JR
t
.
Corollary 7.7 Let Y V and X U . Then
(

JY )
s
=
a
(m;R)
( G(m) C(R: s) )
(m;R)
Y
r
1
Y
r
m
and JX is obtained by setting s =1 and replacing Y
r
by X
r
.
The passage to orbits in Proposition 7.6 is well-behaved by the following observation.
It is [16, 7.4], and the proof is a straightforward inspection.
Lemma 7.8 Assume that C and G are free. Then the action of (m: R) on
G(m) C(R: s) is free. Therefore the action of (M: R) on J(M: R. S) is free.
This implies the following analogue of Lemma 3.5.
Proposition 7.9 Assume that C and G are free. If : Z Z
t
is an equivalence
of (
R
)spaces, then so is

J . Therefore, if : Y Y
t
is an equivalence of
spaces, then so is

J , and if : X X is an equivalence of spaces, then so is
J : JX J Y .
8 The comparison of .
y
C;
y
G/spaces and Jspaces
We can now compare (

C.

G)spaces and Jspaces by mimicking the comparison of
Cspaces with

Cspaces given in Lemmas 3.7 and 3.8 and Theorem 3.10. We need
three preliminary results.
Proposition 8.1 Let Y V . Then the natural map

tt
:

JR
tt
Y R
tt
L
tt

JR
tt
Y R
tt

JY
is an isomorphism. Therefore

J inherits a structure of monad from

J and the functor
R
tt
embeds the category of

Jalgebras as the full subcategory of the category of

Jalgebras consisting of those

Jalgebras of the form R
tt
Y .
Proof By our description of

J , we see that (

JR
tt
Y )(0: +) is a point and, for n >0,
(

JR
tt
Y )(n: S) =
a
(M;R)
Y
1jn
( G(m
j
) C(R
j
: s
j
) )
(m;R)
Y
r
1
Y
r
m
.
Geometry & Topology Monographs, Volume 16 (2009)
310 J P May
On the other hand,
(R
tt

JY )(n: S) =
Y
1jn
a
(m;R)
( G(m) C(R: s
j
) )
(m;R)
Y
r
1
Y
r
m
.
The map
tt
gives the identication that is obtained by commuting disjoint unions past
cartesian products and assembling block partitions. By Proposition 14.3, the second
statement is a formal consequence of the rst.
We restate the second statement since it is pivotal to our later comparison of (

C.

G)
spaces and (C. G)spaces.
Corollary 8.2 The categories of (

C.

G)spaces and

Jalgebras are isomorphic.
There are other comparisons of functors that one might hope to make and that fail.
We record some of them. These failures dictate the conceptual outline of the theory.
They clarify why we must introduce the notion of a semispecial (
R
)space and
why we must use the intermediate category of (

C.

G)spaces rather than compare
(C. G)spaces and Jspaces directly.
Remark 8.3 Note that (

JZ)(1: s) depends on all Z(m: R) and not just the Z(1: s).
Therefore L
tt

JZ is not isomorphic to

JL
tt
Z, in contrast to Lemma 3.7. Similarly,
(

JY )
1
depends on all Y
s
and not just Y
1
. Therefore L
t

JY is not isomorphic to JL
t
Y .
Again, for a space X,

JR
t
X is not isomorphic to R
t
JX. In fact, (

JR
t
X)
n
is not
even equivalent to (JX)
n
. Thus

JZ need not be special when Z is special and

JY
need not be special when Y is special.
Proposition 8.4 Assume that C and G are free. If Z is a semispecial (
R
)
space then so is

JZ, hence

J restricts to a monad on the category of semispecial
(
R
)spaces.
Proof Applying Proposition 7.9 to the horizontal arrows in the diagram
(

JZ)(n: S)

J
00

00

JR
tt
L
tt
Z)(n: S)

00

(R
tt
L
tt

JZ)(n: S)
R
00
L
00

J
00

(R
tt
L
tt

JR
tt
L
tt
Z)(n: S)
we see that its left vertical arrow is an equivalence.
Geometry & Topology Monographs, Volume 16 (2009)
The construction of E
o
ring spaces from bipermutative categories 311
As promised, we can now compare (

C.

G)spaces in V to

Jspaces in W by simply
repeating the proof of Theorem 3.10. We again use the two-sided monadic bar construc-
tion of [8] together with the monadic generalities in Appendix A, general properties of
geometric realization, and the following analogue of Lemma 3.9, whose proof is just
like that of [8, 12.2].
Lemma 8.5 For simplicial objects Z in the category W , there is a natural isomor-
phism v: [

JZ[

J[Z[ such that the following diagrams commute.
[Z[
[)[

)

D
D
D
D
D
D
D
D
[

JZ[

J[Z[
and
[

J

JZ[
[[

JZ[

J

J[Z[

JZ
If (Z. ) is a simplicial

Jalgebra, then ([Z[. [[ v
-1
) is a

Jalgebra.
Theorem 8.6 If C and G are free, then the functor R
tt
: V W induces an
equivalence from the homotopy category of special (

C.

G)spaces to the homotopy
category of special Jspaces.
Proof We repeat the proof of Theorem 3.10. Again, Proposition 8.1 puts us into
one of the two contexts discussed in general terms in Proposition 14.3. Let Y be a
(

C.

G)space and Z be a Jspace. By Proposition 14.3,

JL
tt
is a

Jfunctor, and we
can dene a functor
tt
:

JW |

JV | by sending a

Jalgebra Z to the

Jalgebra

tt
Z =B(

JL
tt
.

J. Z).
By Corollaries 14.4 and 14.5, together with general properties of the geometric realiza-
tion of simplicial spaces proven in [8], we have a diagram
Z
B(

J.

J. Z)

00

B(R
tt

JL
tt
.

J. Z) R
tt

tt
Z
of

Jspaces in which the map c is a homotopy equivalence with natural homotopy
inverse j and the map
tt
= B(
tt
. id. id) is an equivalence when Z is semispecial.
Thus the diagram displays a natural weak equivalence between Z and R
tt

tt
Z. When
Z=R
tt
Y , the displayed diagram is obtained by applying R
tt
to the analogous diagram
Y
B(

J.

J. X)


t

B(

JL
tt
.

J. R
tt
Y ) =
tt
R
tt
Y
of

Jalgebras, in which c is a homotopy equivalence with natural inverse j.
Geometry & Topology Monographs, Volume 16 (2009)
312 J P May
9 Some comparisons of monads
To clarify ideas and to set up the comparison of (C. G)spaces and (

C.

G)spaces, we
dene and compare several other monads and functors related to those already specied.
We again x J =

G
R

C with associated monad

J on W . Recall that

J =L
tt

JR
tt
and J = L
t

JR
t
= L

JR. Taking C or G to be the operad Q of Remark 1.4, the

following denition is a special case of Denition 7.1.
Denition 9.1 Let

C denote the monad on W whose algebras are the
R

Cspaces
and let

G denote the monad on W whose algebras are the

G
R
spaces.
Similarly, the following result is a special case of Proposition 8.1.
Proposition 9.2 Let Y V . The natural maps

tt
:

CR
tt
Y R
tt
L
tt

CR
tt
Y and
tt
:

GR
tt
Y R
tt
L
tt

GR
tt
Y
are isomorphisms. Therefore the monad structures on

C and

G induce monad struc-
tures on L
tt

CR
tt
and L
tt

GR
tt
such that the functor R
tt
embeds the category of
L
tt

CR
tt
algebras Y isomorphically onto the full subcategory of

Calgebras of the
form R
tt
Y and embeds the category of L
tt

GR
tt
algebras Y isomorphically onto the
full subcategory of

Galgebras of the form R
tt
Y .
Proposition 9.3 The monad L
tt

CR
tt
can be identied with the monad

C .
Proof Inspection of the case G =Q of Corollary 7.7 makes clear that the underlying
functors can be identied. The structure maps of the monads agree under the identica-
tions since they are induced by the structure maps of the operad C .
The analogue for G is not true, and we introduce an abbreviated notation.
Denition 9.4 Dene

G to be the monad L
tt

GR
tt
and let

GV | denote the category
of

Galgebras in V .
We now appeal to Becks results on distributivity and monads, which are summarized
in Theorem 15.2 below. We used monads in V and W to compare (

C.

G)spaces
to Jspaces and we will use monads in GU | and

GV | to compare (C. G)spaces
to (

C.

G)spaces in the next section. Becks results will allow us to complete that
comparison conceptually.
Geometry & Topology Monographs, Volume 16 (2009)
The construction of E
o
ring spaces from bipermutative categories 313
For any pair of monads, C and G say, on the same category V , there is a notion of an
action of G on C , spelled out in Denition 15.1. When G acts on C , C restricts to a
monad on GV | . As is made precise in Theorem 15.2, it is equivalent that CG is a
monad on V such that CGalgebras in V are the same as Calgebras in GV | . The
shift in perspective that this allows is crucial to our intermediate use of (

C.

G)spaces.
As Theorem 15.2 also makes precise, a third equivalent condition is that G acts
on C if and only if there is a natural map ,: GC CG that makes appropriate
diagrams commute. We agree to call such a map , a distributivity map since it encodes
distributivity data. We have three results that arise from this perspective and tie things
together. The rst two will be given here and the third in the next section. It may
be helpful to the reader if we rst list the relevant endofunctors on our three ground
categories.

C.

G.

C

G.

J on W (9-5)

C.

G.

C

G.

J on V (9-6)
C. G. CG. J. on U . (9-7)
All of these functors except J are monads, as we shall see, and inclusions of operad
pairs induce a number of obvious maps between them. Our promised three results, one
for each of W , V , and U , show how these monads and maps are related.
Theorem 9.8 There is a distributivity map ,:

G

C

C

G which makes the following
diagram commute.

C
p

C

G

J

J

J

J

J

The composite

G

C

J in the diagram is an isomorphism of monads on W .
Sketch proof Modulo our variant monads, this is a version of [16, 6.12], where more
details can be found. The diagram and the constructions of the monads dictate the
denition of ,, and a precise formula for the map is dictated by the commutation
relation in Lemma 5.3. Diagram chases show that , satises the properties of a
distributivity map specied in Theorem 15.2(iii). It follows that

G acts on

C , so
that

C is a monad on

GW | and

C

G is a monad on W with the same algebras. The
displayed diagram itself implies that its composite is a map of monads. It is surjective
because G
R
and
R
C generate J under composition, and inspection shows that
it is injective. More conceptually, a

Jspace Z is a

C

G by pullback and, conversely,
Geometry & Topology Monographs, Volume 16 (2009)
314 J P May
suitably compatible actions of G
R
and
R
C on Z determine an action of J on
Z. This implies that a

C

Galgebra is the same thing as a

Jalgebra, so that the two
monads have the same algebras. In turn, by the monadicity of the forgetful functor
from Jspaces to W (as in [17, Appendix A]), that implies that the map of monads

C

G

J is an isomorphism.
Using Theorem 15.2 together with Propositions 9.2, and 14.3, we nd that the following
result, which is a version of [16, 6.13], is a formal consequence of the previous one.
Theorem 9.9 The composite displayed in the following diagram is a distributivity
map ,:

G

C

C

G.

C =L
tt

GR
tt
L
tt

CR
tt
p

(L
00

G
00

CR
00
)
1

L
tt

CR
tt
L
00
pR
00

C

G =L
tt

CR
tt
L
tt

GR
tt
L
tt

C

GR
tt
L
00

C
00

GR
00

The natural composite

C

G

J

J

J
is an isomorphism of monads

C

G

J on V .
The following key result is now immediate from Corollary 8.2 and Theorem 15.2.
Corollary 9.10 The categories of (

C.

G)spaces, of

C

Galgebras in V , and of

C
algebras in

GV | are isomorphic.
10 The comparison of .C; G/spaces and .
y
C;
y
G/spaces
Again, let (C. G) be an operad pair, so that G acts on C . We have the monads C
and G on U of the prequel [17, 4.1]. As a reminder, recall that we have changed
notations from there, so that CX here is as specied in (3-6), and similarly for GX.
We have isomorphisms of categories CU | CU | and GU | GU | . In [17], we
gave a monadic description of E
o
ring spaces using monads that take account of
the basepoint 0 and its role as a zero for the multiplication. Here we are ignoring
basepoints and, using the language of Appendix B, we have the following alternative
version of [17, 4.8]. Again, the difference is just a question of whether or not basepoints
are thought of as preassigned.
Geometry & Topology Monographs, Volume 16 (2009)
The construction of E
o
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Proposition 10.1 The monad G on U acts on the monad C , so that C induces
a monad, also denoted C , on the category GU | of Galgebras. The category of
(C. G)spaces is isomorphic to the category of Calgebras in GU | .
Proof Taking [17, 1.4] into account, the proof is the same as that of [17, 4.8], whose
missing details (from [11, Section VI.1]) can be read off directly from Denition 4.2.
Since G acts on C , Theorem 15.2 gives a corresponding distributivity map. The
following result, which combines versions of [16, 6.11 and 6.13], describes it.
Theorem 10.2 The distributivity map ,: GC CG is the composite of the maps
,
1
and ,
2
dened by the commutativity of the upper and lower rectangles in the
following diagram.
GC =L
t

GR
t
L
t

CR
t
p
1

(L
0

G
0
`
CR
0
)
1

L
t

CR
t
L
0
pR
0

J =L
t

JR
t
p
2

L
t

C

GR
t

L
0
`
C
0

GR
0

CG
L
t

CR
t
L
t

GR
t
Thinking of (C. G)spaces as multiplicatively enriched Cspaces, we have in effect
changed ground categories from U to GU | . Since

G acts on

C , as explained in
Section 4, one might well expect the monad

G to act on the monad

C , but that is
false.
7
However, as we saw in the previous section, the monad

G does act on

C . We
could extract an explicit description of

G by specializing the explicit description of
L
tt

JR
tt
given in Corollary 7.7 and using the generalization of Remark 7.5 to s >1.
We omit the details since we have no need for them. However, we observe that Remark
7.5 implies the following result.
Lemma 10.3 For Y V , the space (

GY )
0
can be identied with G(Y
0
), and the
space L
t
Y =(

GY )
1
can be identied with G(Y
1
).
This implies the following analogue of Lemma 3.7. Recall that (L. R) there is the
same as (L
t
. R
t
) here.
7
Vigleik Angeltveit showed me convincingly exactly how this fails, and he pointed out some faulty
details in a purported description of the monad

G given in an earlier draft of this paper.
Geometry & Topology Monographs, Volume 16 (2009)
316 J P May
Lemma 10.4 Let X U . Then (

GR
t
X)
0
= G(+), L
t

GR
t
X (

GR
t
X)
1
= GX,
and the natural map L
t

G: L
t

GY L
t

GR
t
L
t
Y =GL
t
Y is an isomorphism.
Of the previous three results, only the last statement of Lemma 10.4 is on the main line
of development. Returning to the desired comparison of (C. G)spaces and (

C.

G)
spaces, the following result puts us into the framework of Appendix A.
Lemma 10.5 The adjunction (L
t
. R
t
) induces an adjunction
GU |(L
t
Y. X)

GV |(Y. R
t
X).
Proof It is obvious that L
t
takes

Galgebras Y to Galgebras since L
t

GY =GL
t
Y
and we can restrict the action maps accordingly. We claim that R
t
takes Galgebras
X to

Galgebras R
t
X. To see this conceptually, we can modify slightly the denition
of an operad pair by allowing C(0) to be empty. Then, quite trivially, any operad
G acts on our operad Q such that

Q =. Clearly, we can identify (Q. G)spaces
with Galgebras in U , and these can then be identied with

G
R

Qspaces of the
form RX =R
tt
R
t
X, as in Proposition 5.10. As in Denition 6.3, we dene (

Q.

G)
spaces Y to be (

G
R

Q)spaces of the form R
tt
Y , and it is then obvious that R
t
X
is a (

Q.

G)space. Finally, as in Corollary 9.10, we see that (

Q.

G)spaces can be
identied with

G algebras in V .
Since (

C.

G)spaces are the same as

Calgebras in the category of

Galgebras, by
Corollary 9.10, Theorem 3.10 admits the following multiplicative elaboration. In effect,
we just change ground categories from U and V to GU | and

GV | . Otherwise the
proof is exactly the same.
Theorem 10.6 If C is free, then the functor R
t
: U V induces an equivalence
from the homotopy category of (C. G)spaces to the homotopy category of special
(

C.

G)spaces.
This gives the bottom right pair of parallel arrows in Figure 1.
11 Permutative categories in innite loop space theory
We assume familiarity with the notion of a symmetric monoidal category. That is
just a (topological) category A with a product and a unit object which satisfy the
associativity, commutativity, and unit laws up to coherent natural isomorphism. If the
Geometry & Topology Monographs, Volume 16 (2009)
The construction of E
o
ring spaces from bipermutative categories 317
associativity and unit laws hold strictly, then A is said to be permutative.
8
There is no
loss of generality in restricting to permutative categories since any (small) symmetric
monoidal category is equivalent to a permutative category [5; 9]. One cannot also
make the commutativity law hold strictly, and it is the lack of strict commutativity that
leads to the higher homotopies implicit in innite loop space theory. Thus permutative
categories are the strictest kind of symmetric monoidal category that one can dene
without loss of generality.
Precisely, a permutative category A has an associative product with strict two-sided
unit object u and a natural commutativity involution c: AB BA such that
c =id: A=uAAu =A and the following diagram commutes.
ABC
c

idc

O
O
O
O
O
O
O
O
O
O
O
CAB
ACB
cid

o
o
o
o
o
o
o
o
o
o
o
More generally, rather than having a set of objects, A might be an internal category
in U , so that it has a space of objects and continuous source, target, identity, and
composition maps.
A functor F: A B between symmetric monoidal categories is lax symmetric
monoidal if there is a map : u
B
F(u
A
) and a natural transformation
:
B
F F F
A
of functors A A B satisfying appropriate coherence conditions. An op-lax
functor is dened similarly, but with maps going in the other direction. We say that F
is strong (instead of lax or op-lax) if and are isomorphisms and that F is strict
if and are identities. The strict notion is only interesting when A and B are
permutative.
The relationship between permutative categories and spectra was axiomatized in [14;
15]. An innite loop space machine dened on the category PC of permutative
categories is a functor E from PC to any good category of spectra (say Cprespectra
for simplicity) together with a natural group completion |: BA E
0
A , where
BA is the classifying space of A . Up to natural equivalence, there is a unique such
machine (E. |) [14, Theorem 3]. We have omitted the specication of the morphisms
of PC . Strict morphisms were used in [14]. However, there is a functor from the
category of permutative categories and lax morphisms to the category of permutative
8
I believe that this pleasant and appropriate name is due to Don Anderson [1].
Geometry & Topology Monographs, Volume 16 (2009)
318 J P May
categories and strict morphisms that can be used to show that the uniqueness theorem
remains valid when the morphisms in PC are taken to be lax; see [15, 4.3].
There are several constructions of such a machine (E. |). There is an E
o
operad

in Cat whose j th category

j
is the translation category of the symmetric group
j
.
It was dened in [9, Section 4] and, in more detail (and with a minor correction) in
[11, Section 4]. As observed in these sources, there are functors

j
A
j
A that
specify an action of

on A . Passing to classifying spaces, we have an action of the
E
o
operad D =B

of spaces on the space BA . As recalled in [17, 9.6], D is the

topological version of the BarrattEccles operad [2]. The additive innite loop space
machine of [8], as described in [17, Section 9], gives the required machine (E. |).
Alternatively, there are at least two ways, one combinatorial and the other conceptual,
to construct a special Fcategory from a permutative category. Application of the
classifying space functor then gives a special Fspace, to which Segals innite loop
space machine [21] can be applied. The combinatorial construction is due to Segal
[21]. Full details are supplied in May [14, Construction 10]. It is essential to the
uniqueness theorem there that the construction actually gives a functor from PC to
the category FPC of special functors F PC . A defect of the construction is
that it is functorial only on strict rather than lax morphisms of permutative categories.
The conceptual construction is an application of Streets rst construction from [22]
and is spelled out in May [15, Sections 34]. It does not give a functor to FPC , but
it is functorial on lax morphisms. We say a bit more about it, or rather its bipermutative
analogue, in the next section.
While there is an essentially unique way to construct spectra frompermutative categories,
there is another consistency statement that is of considerable importance in some of
the topological applications. In [17, Section 2], we recalled the notion of a monoid-
valued IFCP (functor with cartesian product) from [11, Section I.1] and the more
modern source [19, Chapter 23]. As explained in those sources, such a functor G
can be extended from the category I of nite dimensional inner product spaces to
the category I
c
of countably innite dimensional inner product spaces by passage to
colimits. Then G(R
o
) is an Lspace, where L is the linear isometries E
o
operad,
and so is BG BG(R
o
). These can be fed into the additive innite loop space
machine of [17, Section 9]. On the other hand, the G(R
n
) are the morphism spaces
of a permutative category with object set {n[n 0] and no morphisms mn for
m=n. It is proven in [13] that the spectrum obtained from the Lspace BG is the
connected cover of the spectrum obtained from the permutative category l
n0
G(R
n
),
whose 0th space is equivalent to BG Z.
Geometry & Topology Monographs, Volume 16 (2009)
The construction of E
o
ring spaces from bipermutative categories 319
12 What precisely are bipermutative categories?
We would like to assume familiarity with the notion of a symmetric bimonoidal category,
but the categorical literature on this important topic is strangely meager. Intuitively,
we have a category A with two symmetric monoidal products, and , with unit
objects denoted 0 and 1. The distributivity laws must hold, at least up to coherent
natural transformation. As usual, the notion of coherence has to be made precise
in order to have a sensible denition, and a coherence theorem is necessary for the
notion to be made rigorous. The only systematic study of coherence and the only
coherence theorems that I know of in this context are those of Laplaza [6; 7]. The
essential starting point is to formulate distributivity precisely. Laplaza requires a left
9
distributivity monomorphism
(12-1) : A(B C) (AB) (AC).
If we dene F
A
() =A() and think of (A. ) as a symmetric monoidal category,
then coherence says in part that F
A
is an op-lax symmetric monoidal functor under
and the evident unit isomorphism. Therefore, we might say that LaPlaza requires
a semi op-lax distributivity law. A fully op-lax distributivity law would delete the
monomorphism requirement. A lax distributivity law would have the arrow point the
other way. In the interesting examples, is a natural isomorphism, and I prefer to
require that in the denition, as I did in [11, page 153]. Perhaps we should then call these
strong symmetric bimonoidal categories. In any case, the left and right distributivity
laws, and
t
say, must determine each other by the following commutative diagram,
in which c

is the commutativity isomorphism for .

(12-2) A(B C)

(AB) (AC)
c

(B C) A

(B A) (C A)
As originally specied in [11, Section VI.3], bipermutative categories give the strictest
kind of strong symmetric bimonoidal category that one can dene without loss of
generality. They are permutative under both and , 0 is a strict two-sided zero
object for the functor , and the right distributivity law holds strictly, so that
(AB) C =(AC) (B C).
This equality must be a permutative functor with respect to , so that c

id =c

.
The left distributivity law is specied by (12-2), with
t
=id, and cannot be expected
9
In algebra, the left distributivity law states that a(b c) =ab ac , so that left multiplication by a is
linear. Curiously, [4] has left and right reversed, viewing (12-1) as right distributivity.
Geometry & Topology Monographs, Volume 16 (2009)
320 J P May
to hold strictly. Only one additional coherence diagram is required to commute, namely
(AB) (C D)

((AB) C) ((AB) D)
(AC) (B C) (AD) (B D)
Id c

Id

(A(C D)) (B (C D))

(AC) (AD) (B C) (B D).

Since bipermutative categories are a specialization of Laplazas symmetric bimon-
oidal categories, his work resolves their coherence problem. The assymmetry in
the distributive laws is intrinsic, and the strictness of the right rather than the left
distributivity law meshes with our use of lexicographic orderings in specifying the
notion of an action of an operad pair. It is proven in [11, Section VI.3] that any (small)
strong symmetric bimonoidal category is equivalent to a bipermutative category, so
that there is no loss of generality in restricting attention to bipermutative categories.
Scholium 12.3 Regrettably, the term bipermutative category was redened in [4] to
mean a weaker and denitely inequivalent notion, which we call a lax bipermutative
category. It has two permutative structures, but it only has lax distributivity maps.
That is, it has a map like that of (12-1) and therefore its companion map of (12-2),
but with the arrows pointing in the opposite direction. It is stated on [4, page 178]
that Laplazas symmetric bimonoidal categories are more general even than our
bipermutative categories, and since they can be rectied to equivalent bipermutative
categories in Mays sense, so can ours. This statement is wrong on two counts. Lax
bipermutative categories are not special cases of Laplazas semi op-lax symmetric
bimonoidal categories, and neither the latter nor the former can be rectied unless the
distributivity maps are isomorphisms. We note that no precise denition or coherence
theorem has been formulated for lax symmetric bimonoidal categories, and it is unclear
that such objects can be rectied to the lax bipermutative categories of [4].
From the point of view of our applications, these differences do not much matter.
The interesting examples are strong symmetric bimonoidal and can be rectied to
bipermutative categories as originally dened. The latter give rise to (F
R
F)spaces,
as I recall in the next section. In fact, as I will explain, any sensible notion of lax or
op-lax bipermutative category works for that. By the earlier sections of this paper,
(F
R
F)spaces give rise to E
o
ring spaces. By the theory recalled in the prequel
[17], E
o
ring spaces give rise to E
o
ring spectra.
Geometry & Topology Monographs, Volume 16 (2009)
The construction of E
o
ring spaces from bipermutative categories 321
From the point of view of mathematical philosophy and comparisons of constructions,
these differences do matter. The theory of [4] constructs symmetric ring spectra from
lax bipermutative categories and, as it stands, cannot recover the applications of [11]
(and other more recent applications), that depend on the use of E
o
ring spaces. We
need a comparison theorem to the effect that if we start with a bipermutative category
and process it to an E
o
ring spectrum and thus to a commutative Salgebra by going
through the theory here and in [17], then the result is equivalent to what we get by
using [4] to construct a symmetric ring spectrum and converting that to a commutative
Salgebra. This should be true, but it is not at all obvious.
10
Before continuing, we highlight the mistake in [11] which led to the need for the theory
that we are describing here.
Scholium 12.4 In [11, VI.2.3, VI.2.6, and VI.4.4], it is claimed that (M. M) and
(D. D) are operad pairs and that (D. D) acts on the classifying spaces of bipermutative
categories. These assertions are incorrect, as is explained in detail in [16, Appendix
A], and for this reason there seems to be no elementary shortcut showing that the
classifying spaces of bipermutative categories are E
o
ring spaces. The use of D alone
in the theory of permutative categories is unaffected by the mistake.
13 The construction of .F
R
F/categories from bipermuta-
tive categories
There are notions of lax, strong, and strict morphisms between symmetric bimonoidal
categories, in analogy with the corresponding notions for symmetric monoidal cate-
gories. Again, the strict notion is only interesting in the bipermutative case. We recall
a problem that was left open in [16, page 16].
Conjecture 13.1 There is a functor on the bipermutative category level that replaces
lax morphisms by strict morphisms, in a sense analogous to the corresponding result
[15, 4.3] for permutative categories.
In analogy with the permutative category situation, there are two functors, one combi-
natorial and one conceptual, that construct (F
R
F)categories from bipermutative
categories. The combinatorial construction is due to Woolfson [23] and entails use of
a more complicated category that contains F
R
F. It is spelled out in detail in [16,
Appendix D]. It is only functorial on strict morphisms. In the absence of a proof of
Conjecture 13.1, this makes it less useful than its permutative category analogue.
10
I now have a sketch proof that looks convincing.
Geometry & Topology Monographs, Volume 16 (2009)
322 J P May
The conceptual construction is given in [16, Section 4] and is again an application of
Streets rst construction from [22]. A detailed restatement of the properties of the
construction is given in [15, 3.4]. In brief, for any (small) category G , it gives a functor
from the category of either lax or op-lax functors G Cat and lax or op-lax natural
transformations to the category of genuine functors and genuine natural transformations
G Cat , together with a comparison of the input and output up to natural homotopy.
11
To apply this general categorical construction to our situation, we need only construct
an op-lax (or lax) functor A: F
R
F Cat from a bipermutative category A . That
is very easy to do. We recall the details from [16, Section 3] to emphasize the role
of the distributivity law and explain why a lax or op-lax law would work just as well
as a strict or strong law.
12
We start by specifying A(n: S) = A
s
1
A
s
n
on
objects, where A(0: +) is the trival category + and A
0
is the trivial category 0. For a
morphism (: y): (m: R) (n: S) in F
R
F, we specify the functor
A(: y): A(m: R) A(n: S)
by the formula
A(: y)(
m
i=1

r
i
u=1
a
i,u
) =
n
j=1

s
j
=1
M
y
j
(U)=
O
(i)=j
a
i,u
i
on both objects and morphisms. Here U runs over the lexicographically ordered set
(l.o.s) of sequences with i th term u
i
satisfying 1 u
i
r
i
for i
-1
(j ); this set
can be identied with .
(i)=j
r
i
{0]. That is the same formula that we would have
used if we had given a complete proof of Proposition 5.9, describing rig spaces as
F
R
Fspaces explicitly. In that context, we would have strict commutativity and
distributivity and the formula would give a functor F
R
F U . In the present
context, we have coherence isomorphisms that give lax functoriality. Note rst that
A takes identity morphisms to identity functors. However, for a second morphism
([: o): (n: S) (p: T) in F
R
F, we have
A([ : )(
m
i=1

r
i
u=1
a
i,u
) =
p
k=1

t
k
u=1
M

k
(Y )=u
O
())(i)=k
a
i,y
i
.
11
It is generally understood in bicategory theory that lax functors F should have comparison natural
transformations F([) F() F([ ); op-lax functors should have the arrows reversed. Street
[22] uses lax functors and calls them that; in view of the freedom to replace G by G
op
, his construction
applies equally well to op-lax functors. Unfortunately, in [15; 16], I used op-lax functors but called them
lax functors. Ill call them op-lax functors here. The natural homotopies of [15; 16] are special cases of
what are called modications in the bicategorical literature.
12
I learned this from Michael Shulman.
Geometry & Topology Monographs, Volume 16 (2009)
The construction of E
o
ring spaces from bipermutative categories 323
where
k
=o
k
(.
)(j)=k
y
j
) o
k
([. ) and Y runs through the l.o.s of sequences
with 1 y
i
r
i
for i ([)
-1
(k), regarded as elements of .
)(i)=k
r
i
, whereas
A([: o)A(: y)(
m
i=1

r
i
u=1
a
i,u
)
=
p
k=1

t
k
u=1
M
o
k
(V )=u
O
)(j)=k

M
y
j
(U)=
j
O
(i)=j
a
i,u
i

.
where U run through the l.o.s. of sequences with 1 u
i
r
i
for i
-1
(j ) and V
runs through the l.o.s. of sequences with 1 :
j
s
j
for j [
-1
(k), regarded as
elements of .
(i)=j
r
i
and .
)(j)=k
s
j
respectively. The commutativity isomorphisms
c

and c

, together with the strict right distributivity law, induce a natural isomorphism
o(([: o). (: y)): A([ : ) A([: o)A(: y).
The coherence in the denition of a bicategory gives the coherence with respect to
the associativity and unity of composition that are implicit in the assertion that this
denition does give an op-lax functor.
Clearly, we can reverse the arrow, and then we have a lax rather than op-lax functor.
With the proper specication of coherence data, dictated by the requirement that the
denition give a lax or op-lax functor, we see that there is no need for a strict or even a
strong distributivity law. We conclude that, with proper denitions, we can obtain a lax
or op-lax functor from a lax or op-lax bipermutative category. Streets construction
applies to rectify either to a (special) functor F
R
F Cat . Thus we rst construct
a lax or op-lax functor that uses actual cartesian products on objects, and we then use
Streets construction to convert it to a genuine functor, but one that no longer uses actual
cartesian products of objects. Streets construction is ideally suited to convert the kind
of structured categories that we encounter in nature to the kind of structured categories
that we know how to convert to E
o
ring spaces after passage to their classifying
spaces.
14 Appendix A. Generalities on monads
To make this paper reasonably self-contained, we repeat some results from [20, p. 219]
and [16, Section 5]; the elementary categorical proofs may be found there.
Let L: W V and R: V W be an adjoint pair of functors with counit LR=
Id and unit : Id RL. We have a pair of propositions and corollaries relating
monad structures on functors C: V V and D: W W . They differ due to the
assymmetry of our assumptions on L and R. The next result is [16, 5.1].
Geometry & Topology Monographs, Volume 16 (2009)
324 J P May
Proposition 14.1 Let (C. j. j) be a monad on V , let (F. ,) be a (right) Cfunctor
in some category V
t
, and let (X. ) be a Calgebra in V . Dene D =RCL.
(i) D is a monad on W with unit and product the composites
Id

RL
R)L

RCL=D
DD =RCLRCL=RCCL
RL

RCL.
(ii) FL is a Dfunctor in V
t
with right action
,L: FLD =FLRCL=FCLFL.
(iii) RX is a Dalgebra in W with action
R: DRX =RCLRX =RCX RX.
In the present generality, we state results about bar constructions simplicially. After
geometric realization in our space level situations, they give corresponding results
about the actual bar constructions of interest.
Corollary 14.2 The simplicial two-sided bar construction satises
B
+
(F. C. X) =B
+
(FL. D. RX)
for a Calgebra X and Cfunctor F, where D =RCL.
The following result combines the two results [16, 5.2 and 5.3].
Proposition 14.3 Let (D. v. ) be a monad on W . Dene C =LDR and let

: D
RCL denote the common composite in the following diagram:
D
D

L
L
L
L
L
L
DRL
DRL

RLD
RLD

RLDRL
Assume that one of the following two natural maps is an isomorphism:
DR=

R: DRRC or LD =L

: LD CL.
Geometry & Topology Monographs, Volume 16 (2009)
The construction of E
o
ring spaces from bipermutative categories 325
(i) C is a monad on V with unit and product the composites
Id =LR
LR

LDR
and
CC =LDRLDR
(LDDR)
1

LDDR
LR

LDR=C.
and

: D RCL is a map of monads on W .
(ii) If (F. ,) is a Cfunctor, then (FL. ,L FL

) is a Dfunctor. In particular,
RCL is a Dfunctor and

: D RCL is a map of Dfunctors.
(iii) If (X. ) is a Calgebra, then (RX. R

R) is a Dalgebra. In particular, for

Y W , RCLY is a Dalgebra and

: DY RCLY is a map of Dalgebras.
(iv) If (RX. [) is a Dalgebra, then (X. L[) is a Calgebra, and R embeds CV |
into DW | as the full subcategory of Dalgebras of the form RX.
(v) When LD: LD CL is an isomorphism, if (Y. [) is a Dalgebra in W ,
then (LY. L[ (LD)
-1
) is a Calgebra in V and : Y RLY is a map
of Dalgebras.
Corollary 14.4 Let D be a monad on W and let C =LDR.
(i) If

R: DRRC is an isomorphism, then
B
+
(GR. C. X) B
+
(G. D. RX)
for a Calgebra X and Dfunctor G and therefore
B
+
(F. C. X) B
+
(FL. D. RX)
for a Cfunctor F.
(ii) If LD: LD CL is an isomorphism, then
B
+
(F. C. LY ) B
+
(FL. D. Y )
for a Dalgebra Y and a Cfunctor F.
Recall from [8, 9.8] that we always have a map
c
+
: B
+
(C. C. X) X
+
of simplicial Calgebras that is a simplicial homotopy equivalence, where X
+
is the
constant simplicial object at X.
Geometry & Topology Monographs, Volume 16 (2009)
326 J P May
Corollary 14.5 Under the hypotheses of Proposition 14.3,

+
=B
+
(

. id. id): B
+
(D. D. Y ) B
+
(RCL. D. Y ) =RB
+
(CL. D. Y )
is a map of simplicial Dalgebras. If

R: DRRC is an isomorphism and Y =RX
for a Calgebra X, then the diagram
Y
+
B
+
(D. D. Y )

RB
+
(CL. D. Y )
of simplicial Dalgebras is obtained by applying R to the evident diagram
X
+
B
+
(C. C. X)


t

B
+
(CL. D. RX)
of simplicial Calgebras.
15 Appendix B. Monads and distributivity
Consider two monads, (C. j

. j

) and (G. j

. j

), on the same category V . As

the notation indicates, we think of C as additive and G as multiplicative. We want to
understand a monadic distributivity law for an action of G on C . This was obtained in
an elegant paper of Beck [3], as I only learned after reproducing many of its results
in the course of working out multiplicative innite loop space theory [16, Section 5].
Since this theory is central to understanding, we repeat it here in abbreviated form,
referring the reader to [3] for detailed verications.
Let CV | and GV | denote the categories of Calgebras and Galgebras in V .
Denition 15.1 An action of G on C is a structure of monad on GV | induced by
the monad C on V . In detail, for an action of G on X, there is a prescribed functorial
induced action of G on CX (and thus on CCX by iteration) such that j

: X CX
and j

: CCX CX are maps of Galgebras.

Recall that the following diagram commutes for composable pairs of functors (B. A)
and (D. C) and for natural transformations : AC and : B D.
BC
C

F
F
F
F
F
F
F
F
BA
B

y
y
y
y
y
y
y
y
A

E
E
E
E
E
E
E
E
DC
DA
D

x
x
x
x
x
x
x
x
Geometry & Topology Monographs, Volume 16 (2009)
The construction of E
o
ring spaces from bipermutative categories 327
In the categorical literature the common composite is generally written or . It
is just the horizontal composition of the 2category Cat , but we shall be explicit.
Theorem 15.2 The following data relating the monads C and G are equivalent.
(i) An action of G on C .
(ii) A natural transformation j: CGCG CG with the following properties.
(a) (CG. j. j) is a monad on V , where j =j

G j

: Id CG.
(b) Cj

: C CG and j

G: G CG are maps of monads.

(c) The following composite is the identity natural transformation.
CG
C)

CCG
C)

CG

CGCG

CG
(iii) A natural transformation ,: GC CG such that the following two diagrams
commute.
G
G)

{
{
{
{
{
{
{
{
)

C
C
C
C
C
C
C
C
GC
p

CG
C
)

C
C
C
C
C
C
C
C
C C)

{
{
{
{
{
{
{
{
and
GCC
G

pC

CGC
Cp

CCG

GC
p

CG
GGC
Gp

GCG
pG

CGG
C

When given such data, the category CGV || of Calgebras in GV | is isomorphic to

the category CGV | of CGalgebras in V .
Sketch proof Details are in [3]. We relate (i) to (iii) and (ii) to (iii). Given the data of
(i), we obtain the data of (iii) by dening , to be the composite
GC
GC)

GCG

CG.
Geometry & Topology Monographs, Volume 16 (2009)
328 J P May
where, for X V , is the action of G on CGX induced from the canonical action
of G on GX. Given the map , as in (iii) and given a Galgebra (X. ), the following
composite species a natural action of G on CX that satises (i).
GCX
p

CGX
C

CX
Given j satisfying (ii), the following composite is a map , satisfying (iii).
GC
GC)

GCG
)

GCG

CGCG

CG
Given , satisfying (iii), the following composite is a map j satisfying (ii).
CGCG
CpG

CCGG
CC

CCG

CG
Given these equivalent data, a Calgebra (X. . 0) in GV | determines a CGalgebra
(X. [) in V by letting [ be the composite
CGX
C

CX
0

X.
and a CGalgebra (X. [) determines a Calgebra (X. . 0) in GV | by letting and
0 be the pullbacks of [ along the maps of monads j

G and Cj

of (iib).
Ackowledgments
It is a pleasure to thank Vigleik Angeltveik, John Lind, and an anonymous referee for
catching errors and suggesting improvements.
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Department of Mathematics, The University of Chicago
Chicago, Illinois 60637, USA
may@math.uchicago.edu
http://www.math.uchicago.edu/
~
may
Received: 14 September 2008
Geometry & Topology Monographs, Volume 16 (2009)