Geometry & Topology Monographs 16 (2009) 215282 215

What precisely are E

1
ring spaces and E
1
ring spectra?
J P MAY
E
o
ring spectra were dened in 1972, but the term has since acquired several
alternative meanings. The same is true of several related terms. The new formulations
are not always known to be equivalent to the old ones and even when they are,
the notion of equivalence needs discussion: Quillen equivalent categories can be
quite seriously inequivalent. Part of the confusion stems from a gap in the modern
resurgence of interest in E
o
structures. E
o
ring spaces were also dened in 1972
and have never been redened. They were central to the early applications and they
tie in implicitly to modern applications. We summarize the relationships between
the old notions and various new ones, explaining what is and is not known. We take
the opportunity to rework and modernize many of the early results. New proofs and
perspectives are sprinkled throughout.
55P42, 55P43, 55P47, 55P48; 18C20, 18D50
Introduction
In the early 1970s, the theory of E
o
rings was intrinsically intertwined with a host of
constructions and calculations that centered around the relationship between E
o
ring
spectra and E
o
ring spaces (see Cohen, Lada and May [10] and May [27]). The two
notions were regarded as being on essentially the same footing, and it was understood
that the homotopy categories of ringlike E
o
ring spaces (
0
is a ring and not just a
semi-ring) and of connective E
o
ring spectra are equivalent.
In the mid 1990s, modern closed symmetric monoidal categories of spectra were
introduced, allowing one to dene a commutative ring spectrum to be a commutative
monoid in any such good category of spectra. The study of such rings is now central to
stable homotopy theory. Work of several people, especially Schwede and Shipley, shows
that, up to zigzags of Quillen equivalences, the resulting categories of commutative ring
spectra are all equivalent. In one of these good categories, commutative ring spectra
are equivalent to E
o
ring spectra. The terms E
o
ring spectra and commutative ring
spectra have therefore been used as synonyms in recent years. A variant notion of E
o
ring spectrum that can be dened in any such good category of spectra has also been
given the same name.
Published: 16 June 2009 DOI: 10.2140/gtm.2009.16.215
216 J P May
From the point of view of stable homotopy theory, this is perfectly acceptable, since
these notions are tied together by a web of Quillen equivalences. From the point of view
of homotopy theory as a whole, including both space and spectrum level structures, it is
not acceptable. Some of the Quillen equivalences in sight necessarily lose space level
information, and in particular lose the original connection between E
o
ring spectra
and E
o
ring spaces. Since some modern applications, especially those connected with
cohomological orientations and spectra of units, are best understood in terms of that
connection, it seems to me that it might be helpful to offer a thorough survey of the
structures in this general area of mathematics.
This will raise some questions. As we shall see, some new constructions are not at
present known to be equivalent, in any sense, to older constructions of objects with
the same name, and one certainly cannot deduce comparisons formally. It should also
correct some misconceptions. In some cases, an old name has been reappropriated for
a denitely inequivalent concept.
The paper divides conceptually into two parts. First, in Sections 110, we describe and
modernize additive and multiplicative innite loop space theory. Second, in Sections
1113, we explain how this early 1970s work ts into the modern framework of
symmetric monoidal categories of spectra. There will be two sequels (see May [32;
33]). In the rst, we recall how to construct E
o
ring spaces from bipermutative
categories. In the second, we review some of the early applications of E
o
ring spaces.
The following list of sections may help guide the reader.
1. The denition of E
o
ring spaces
2. Ispaces and the linear isometries operad
3. The canonical E
o
operad pair
4. The monadic interpretation of the denitions
5. The denition of E
o
ring prespectra and E
o
ring spectra
6. The monadic interpretation of E
o
ring spectra
7. The relationship between E
o
ring spaces and E
o
ring spectra
8. A categorical overview of the recognition principle
9. The additive and multiplicative innite loop space machine
10. Localizations of the special unit spectrum sl
1
R
11. E
o
ring spectra and commutative Salgebras
12. The comparison with commutative diagram ring spectra
13. Naive E
o
ring spectra
14. Appendix A. Monadicity of functors and comparisons of monads
15. Appendix B. Loop spaces of E
o
spaces and the recognition principle
Geometry & Topology Monographs, Volume 16 (2009)
What precisely are E
o
ring spaces and E
o
ring spectra? 217
We begin by dening E
o
ring spaces. As we shall see in Section 1, this is really quite
easy. The hard part is to produce examples, and that problem will be addressed in [32].
The denition requires a pair (C. G) of E
o
operads, with G acting in a suitable way
on C , and E
o
ring spaces might better be called (C. G)spaces. It is a truism taken
for granted since May [25] that all E
o
operads are suitably equivalent. However, for
E
o
ring theory, that is quite false. The precise geometry matters, and we must insist
that not all E
o
operads are alike. The operad C is thought of as additive, and the
operad G is thought of as multiplicative.
1
There is a standard canonical multiplicative operad L , namely the linear isometries
operad. We recall it and related structures that were the starting point of this area of
mathematics in Section 2. In our original theory, we often replaced L by an operad
OL , and we prefer to use a generic letter G for an operad thought of as appropriate
to the multiplicative role in the denition of E
o
ring spaces. The original denition
of E
o
ring spaces was obscured because the canonical additive operad C that was
needed for a clean description was only discovered later, by Steiner [43]. We recall
its denition and its relationship to L in Section 3. This gives us the canonical E
o
operad pair (C. L).
Actions by operads are equivalent to actions by an associated monad. As we explain in
Section 4, that remains true for actions by operad pairs. That is, just as E
o
spaces
can be described as algebras over a monad, so also E
o
ring spaces can be described
as algebras over a monad. In fact, the monadic version of the denition ts into a
beautiful categorical way of thinking about distributivity that was rst discovered by
Beck [5]. This helps make the denition feel denitively right.
As we also explain in Section 4, different monads can have the same categories of
algebras. This has been known for years, but it is a new observation that this fact can be
used to substantially simplify the mathematics. In the sequel [32], we will use this idea
to give an elementary construction of E
o
ring spaces from bipermutative categories
(and more general input data). We elaborate on this categorical observation and related
facts about maps of monads in Appendix A, which is written jointly with Michael
Shulman.
The early 1970s denition in May [27] of an E
o
ring spectrum was also obscure, this
time because the notion of twisted half-smash product that allows a clean description was
only introduced later, in [18]. The latter notion encapsulates operadically parametrized
internalizations of external smash products. As we recall in Section 5, E
o
ring spectra
are spectra in the sense of Lewis et al [18] and May [24], which we shall sometimes
1
As we recall in Sections 2 and 9, in many applications of additive innite loop space theory, we must
actually start with G , thinking of it as additive, and convert Gspaces to Cspaces before proceeding.
Geometry & Topology Monographs, Volume 16 (2009)
218 J P May
call LMS spectra for deniteness, with additional structure. Just as E
o
spaces can
be described in several ways as algebras over a monad, so also E
o
ring spectra can
be described in several ways as algebras over a monad. We explain this and relate the
space and spectrum level monads in Section 6.
There is a 0th space functor C
o
from spectra to spaces,
2
which is right adjoint to the
suspension spectrum functor
o
. A central feature of the denitions, both conceptually
and calculationally, is that the 0th space R
0
of an E
o
ring spectrum R is an E
o
ring space. Moreover, the space GL
1
R of unit components in R
0
and the component
SL
1
R of the identity are E
o
spaces, specically Lspaces.
3
We shall say more
about these spaces in Sections 7, 9 and 10.
There is also a functor from E
o
ring spaces to E
o
ring spectra. This is the point of
multiplicative innite loop space theory (see May [27; 29]). Together with the 0th space
functor, it gives the claimed equivalence between the homotopy categories of ringlike
E
o
ring spaces and of connective E
o
ring spectra. We recall this in Section 9.
The state of the art around 1975 was summarized in May [28], and it may help orient
Sections 110 of this paper to reproduce the diagram that survey focused on. (See
Figure 1.) Many of the applications alluded to above are also summarized in [28]. The
abbreviations at the top of the diagram refer to permutative categories and bipermutative
categories. We will recall and rework how the latter t into multiplicative innite loop
space theory in the sequel [32].
Passage through the black box is the subject of additive innite loop space theory
on the left and multiplicative innite loop space theory on the right. These provide
functors from E
o
spaces to spectra and from E
o
ring spaces to E
o
ring spectra.
We have written a single black box because the multiplicative functor is an enriched
specialization of the additive one. The black box gives a recognition principle: it tells
us how to recognize spectrum level objects on the space level.
We give a modernized description of these functors in Section 9. My early 1970s work
was then viewed as too categorical by older algebraic topologists.
4
In retrospect,
it was not nearly categorical enough for intuitive conceptual understanding. In the
2
Unfortunately for current readability, in [27] the notation
o
was used for the suspension prespec-
trum functor, the notation C
o
was used for the spectrication functor that has been denoted by L ever
since [18], and the notation Q
o
=C
o

o
was used for the current
o
.
3
GL
1
R and SL
1
R were called FR and SFR when they were introduced in [27]. These spaces
played a major role in that book, as we will explain in the second sequel [33]. As we also explain there, F
and GL
1
S are both tautologically the same and very different. The currently popular notations follow
Waldhausens later introduction [45] of the higher analogues GL
n
(R).
4
Sad to say, nearly all of the older people active then are now retired or dead.
Geometry & Topology Monographs, Volume 16 (2009)
What precisely are E
o
ring spaces and E
o
ring spectra? 219
PERM CATS
B

BIPERM CATS
B

E
o
SPACES

E
o
RING SPACES

SPACES
C

BLACK BOX
.

E
o
SPACES
C

SPECTRA
D
1

E
o
RING SPECTRA
D
1
_
Figure 1: Diagram from May [28]
expectation that I am addressing a more categorically sophisticated modern audience, I
explain in Section 8 how the theory is based on an analogy with the Beck monadicity
theorem. One key result, a commutation relation between taking loops and applying
the additive innite loop space machine, was obscure in my earlier work, and Ill give
a new proof in Appendix B.
The diagram above obscures an essential technical point. The two entries E
o
spaces
are different. The one on the upper left refers to spaces with actions by the additive
E
o
operad C , and spaces there mean based spaces with basepoint the unit for the
additive operadic product. The one on the right refers to spaces with actions by the
multiplicative E
o
operad G , and spaces there mean spaces with an operadic unit point
1 and a disjoint added basepoint 0. The functor C is the free C space functor, and it
takes Gspaces with 0 to E
o
ring spaces. This is a key to understanding the various
adjunctions hidden in the diagram. The functors labelled C and
o
are left adjoints.
The unit E
o
spaces GL
1
R and SL
1
R of an E
o
ring spectrum R can be fed
into the additive innite loop space machine to produce associated spectra gl
1
R and
sl
1
R. There is much current interest in understanding their structure. As we recall in
Section 10, one can exploit the interrelationship between the additive and multiplicative
structures to obtain a general theorem that describes the localizations of sl
1
R at sets of
primes in terms of purely multiplicative structure. The calculational force of the result
comes from applications to spectra arising from bipermutative categories, as we recall
Geometry & Topology Monographs, Volume 16 (2009)
220 J P May
and illustrate in the second sequel [33]. The reader may prefer to skip this section on
a rst reading, since it is not essential to the main line of development, but it gives a
good illustration of information about spectra of current interest that only makes sense
in terms of E
o
ring spaces.
Turning to the second part, we now jump ahead more than twenty years. In the 1990s,
several categories of spectra that are symmetric monoidal under their smash product
were introduced. This allows the denition of commutative ring spectra as commutative
monoids in a symmetric monoidal category of spectra. Anybody who has read this far
knows that the resulting theory of stable commutative topological rings has become
one of the central areas of study in modern algebraic topology. No matter how such
a modern category of spectra is constructed, the essential point is that there is some
kind of external smash product in sight, which is commutative and associative in an
external sense, and the problem that must be resolved is to gure out how to internalize
it without losing commutativity and associativity.
Starting from twisted half-smash products, this internalization was carried out by
Elmendorf, Kriz, Mandell and May (EKMM) in [13], where the symmetric monoidal
category of Smodules is constructed. We summarize some of the relevant theory in
Section 11. Because the construction there starts with twisted half-smash products,
the resulting commutative ring spectra are almost the same as E
o
ring spectra. The
almost is an important caveat. We didnt mention the unit condition in the previous
paragraph, and that plays an important and subtle role in [13] and in the comparisons
we shall make. As Lewis noted [17] and we will rephrase, one cannot have a symmetric
monoidal category of spectra that is as nicely related to spaces as one would ideally like.
The reason this is so stems from an old result of Moore, which says that a connected
commutative topological monoid is a product of EilenbergMac Lane spaces.
In diagram spectra, in particular symmetric spectra and orthogonal spectra (see Hovey,
Shipley and Smith [15] and Mandell et al [23]), the internalization is entirely different.
Application of the elementary categorical notion of left Kan extension replaces the
introduction of the twisted half-smash product, and there is no use of operads. However,
there is a series of papers by Mandell, May, Schwede and Shipley [22; 23; 38; 39; 41]
that lead to the striking conclusion that all reasonable categories of spectra that are
symmetric monoidal and have sensible Quillen model structures are Quillen equivalent.
Moreover, if one restricts to the commutative monoids, alias commutative ring spectra,
in these categories, we again obtain Quillen equivalent model categories.
Nevertheless, as we try to make clear in Section 12, these last Quillen equivalences
lose essential information. On the diagram spectrum side, one must throw away any
information about 0th spaces in order to obtain the Quillen equivalence with EKMM-
style commutative ring spectra. In effect, this means that diagram ring spectra do not
Geometry & Topology Monographs, Volume 16 (2009)
What precisely are E
o
ring spaces and E
o
ring spectra? 221
know about E
o
ring spaces and cannot be used to recover the original space level
results that were based on implications of that structure.
Philosophically, one conclusion is that fundamentally important homotopical informa-
tion can be accessible to one and inaccessible to the other of a pair of Quillen equivalent
model categories, contrary to current received wisdom. The homotopy categories of
connective commutative symmetric ring spectra and of ringlike E
o
ring spaces are
equivalent, but it seems impossible to know this without going through the homotopy
category of E
o
ring spectra, as originally dened.
We hasten to be clear. It does not follow that Smodules are better than symmetric or
orthogonal spectra. There is by now a huge literature manifesting just how convenient,
and in some contexts essential, diagram spectra are.
5
Rather, it does follow that to
have access to the full panoply of information and techniques this subject affords, one
simply must be eclectic. To use either approach alone is to approach modern stable
homotopy theory with blinders on.
A little parenthetically, there is also a quite different alternative notion of a naive E
o
ring spectrum (that is meant as a technical term, not a pejorative). For that, one starts
with internal iterated smash products and uses tensors with spaces to dene actions
by an E
o
operad. This makes sense in any good modern category of spectra, and the
geometric distinction between different choices of E
o
operad is irrelevant. Most such
categories of spectra do not know the difference between symmetric powers E
(j)
,
j
and homotopy symmetric powers (E
j
)

j
E
(j)
, and naive E
o
ring spectra in
such a good modern category of spectra are naturally equivalent to commutative ring
spectra in that category, as we explain in Section 13.
This summary raises some important compatibility questions. For example, there is
a construction, due to Schlichtkrull [36], of unit spectra associated to commutative
symmetric ring spectra. It is based on the use of certain diagrams of spaces that are
implicit in the structure of symmetric spectra. It is unclear that these unit spectra are
equivalent to those that we obtain from the 0th space of an equivalent E
o
ring
spectrum. Thus we now have two constructions, not known to be equivalent,
6
of objects
bearing the same name. Similarly, there is a construction of (naive) E
o
symmetric
ring spectra associated to op-lax bipermutative categories (which are not equivalent to
bipermutative categories as originally dened) that is due to Elmendorf and Mandell
[14]. It is again not known whether or not their construction (at least when specialized
to genuine bipermutative categories) gives symmetric ring spectra that are equivalent
5
Ive contributed to this in collaboration with Mandell, Schwede, Shipley, and Sigurdsson; see [22; 23;
34].
6
Since I wrote that, John Lind (at Chicago) has obtained an illuminating proof that they are.
Geometry & Topology Monographs, Volume 16 (2009)
222 J P May
to the E
o
ring spectra that are constructed from bipermutative categories via our
black box. Again, we have two constructions that are not known to be equivalent, both
thought of as giving the Ktheory commutative ring spectra associated to bipermutative
categories.
Answers to such questions are important if one wants to make consistent use of
the alternative constructions, especially since the earlier constructions are part of a
web of calculations that appear to be inaccessible with the newer constructions. The
constructions of [36] and [14] bear no relationship to E
o
ring spaces as they stand
and therefore cannot be used to retrieve the earlier applications or to achieve analogous
future applications. However, the new constructions have signicant advantages as well
as signicant disadvantages. Rigorous comparisons are needed. We must be consistent
as well as eclectic. There is work to be done!
For background, Thomason and I proved in [35], that any two innite loop space
machines (the additive black box in Diagram 1) are equivalent. The proof broke into
two quite different steps. In the rst, we compared input data. We showed that Segals
input data (special Ispaces) and the operadic input data of Boardman and Vogt and
myself (E
o
spaces) are each equivalent to a more general kind of input data, namely
an action of the category of operators

C associated to any chosen E
o
operad C . We
then showed that any two functors E from

Cspaces to spectra that satisfy a group
completion property on the 0th space level are equivalent. This property says that
there is a natural group completion map j: X E
0
X, and we will sketch how that
property appears in one innite loop space machine in Section 9.
No such uniqueness result is known for multiplicative innite loop space theory. As
we explain in [32], variant notions of bipermutative categories give possible choices of
input data that are denitely inequivalent. There are also equivalent but inequivalent
choices of output data, as I hope the discussion above makes clear. We might take
as target any good modern category of commutative ring spectra and then, thinking
purely stably, all choices are equivalent. However, the essential feature of [35] was the
compatibility statement on the 0th space level. There were no problematic choices
since the correct homotopical notion of spectrum is unambiguous, as is the correct ho-
motopical relationship between spectra and their 0th spaces (of brant approximations
model categorically). As we have indicated, understanding multiplicative innite loop
space theory on the 0th space level depends heavily on choosing the appropriate target
category.
7
With the black box that makes sense of Diagram 1, there are stronger comparisons of
input data and 0th spaces than the axiomatization prescribes. Modulo the inversion
7
In fact, this point was already emphasized in the introduction of [35].
Geometry & Topology Monographs, Volume 16 (2009)
What precisely are E
o
ring spaces and E
o
ring spectra? 223
of a natural homotopy equivalence, the map j is a map of E
o
spaces in the additive
theory and a map of E
o
ring spaces in the multiplicative theory. This property is
central to all of the applications of [10; 27]. For example, it is crucial to the analysis of
localizations of spectra of units in Section 10.
This paper contains relatively little that is technically new, although there are many
new perspectives and many improved arguments. It is intended to give an overview
of the global structure of this general area of mathematics, explaining the ideas in a
context uncluttered by technical details. It is a real pleasure to see how many terric
young mathematicians are interested in the theory of structured ring spectra, and my
primary purpose is to help explain to them what was known in the early stages of the
theory and how it relates to the current state of the art, in hopes that this might help
them see connections with things they are working on now. Such a retelling of an old
story seems especially needed since notations, denitions, and emphases have drifted
over the years and their are some current gaps in our understanding.
Another reason for writing this is that I plan to rework complete details of the analogous
equivariant story, a tale known decades ago but never written down. Without a more
up-to-date nonequivariant blueprint, that story would likely be quite unreadable. The
equivariant version will (or, less optimistically, may) give full details that will supersede
those in the 1970s sources.
Id like to thank the organizers of the Banff conference, Andy Baker and Birgit Richter,
who are entirely to blame for the existence of this paper and its sequels. They scheduled
me for an open-ended evening closing talk, asking me to talk about the early theory.
They then provided the audience with enough to drink to help alleviate the resulting
boredom. This work began with preparation for that talk. Id also like to thank John
Lind and an eagle-eyed anonymous referee for catching numerous misprints and thereby
sparing the reader much possible confusion.
1 The denition of E
1
ring spaces
We outline the denition of E
o
spaces and E
o
ring spaces. We will be careful about
basepoints throughout, since that is a key tricky point and the understanding here will
lead to a streamlined passage from alternative inputs, such as bipermutative categories,
to E
o
ring spaces in [32]. Aside from that, we focus on the intuition and refer the
reader to [25; 27; 29] for the combinatorial details. Let U denote the category of
(compactly generated) unbased spaces and T denote the category of based spaces. We
tacitly assume that based spaces X have nondegenerate basepoints, or are well-based,
which means that + X is a cobration.
Geometry & Topology Monographs, Volume 16 (2009)
224 J P May
We assume that the reader is familiar with the denition of an operad. The original
denition, and our focus here, is on operads in U , as in [25, pages 13], but the
denition applies equally well to dene operads in any symmetric monoidal category
(see May [30; 31]). As in [25], we insist that the operad O be reduced, in the sense that
O(0) is a point +. This is important to the handling of basepoints. Recall that there
is an element id O(1) that corresponds to the identity operation
8
and that the j th
space O(j ) has a right action of the symmetric group
j
. There are structure maps
;: O(k) O(j
1
) O(j
k
) O(j
1
j
k
)
that are suitably equivariant, unital, and associative. We say that O is an E
o
operad
if O(j ) is contractible and
j
acts freely.
The precise details of the denition are dictated by looking at the structure present on the
endomorphism operad End
X
of a based space X. This actually has two variants, End
T
X
and End
U
X
, depending on whether or not we restrict to based maps. The default will be
End
X
=End
T
X
. The j th space End
X
(j ) is the space of (based) maps X
j
X and
;(g:
1
. . . . .
k
) =g (
1

k
).
We interpret X
0
to be a point,
9
and End
X
(0) is the map given by the inclusion of the
basepoint. Of course, End
U
X
(0) =X, so the operad End
U
X
is not reduced.
An action 0 of O on X is a map of operads O End
X
. Adjointly, it is given by
suitably equivariant, unital, and associative action maps
0: O(j ) X
j
X.
We think of O(j ) as parametrizing a j fold product operation on X. The basepoint
of X must be 0(+), and there are two ways of thinking about it. We can start with
an unbased space X, and then 0(+) gives it a basepoint, xing a point in End
U
X
(0),
or we can think of the basepoint as preassigned, and then 0(+) =+ is required. With
j
1
= = j
k
= 0, the compatibility of 0 with the structure maps ; ensures that a
map of operads O End
U
X
necessarily lands in End
X
=End
T
X
.
Now consider a pair (C. G) of operads. Write C(0) ={0] and G(0) ={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
i
0 that satisfy certain equivariance, unit, and distributivity properties;
see [27, pages 142144], [29, pages 89], or the sequel [32, 4.2]. We will give an
8
The notation 1 is standard in the literature, but that would obscure things for us later.
9
This is reasonable since the product of the empty set of objects is the terminal object.
Geometry & Topology Monographs, Volume 16 (2009)
What precisely are E
o
ring spaces and E
o
ring spectra? 225
alternative perspective in Section 4 that dictates the details. To deal with basepoints,
we interpret the empty product of numbers to be 1 and, with k =0, we require that
z(1) =id C(1). We think of C as parametrizing addition and G as parametrizing
multiplication. For example, we have an operad N such that N (j ) =+ for all j . An
N space is precisely a commutative monoid. There is one and only one way N can
act on itself, and an (N . N ) space is precisely a commutative topological semi-ring
or rig space, a ring without negatives. We say that (C. G) is an E
o
operad pair if C
and G are both E
o
operads. We give a canonical example in Section 3.
Of course, a rig space X must have additive and multiplicative unit elements 0 and 1,
and they must be different for nontriviality. It is convenient to think of S
0
as {0. 1], so
that these points are encoded by a map e: S
0
X. In [27; 29], we thought of both
of these points as basepoints. Here we only think of 0 as a basepoint. This sounds
like a trivial distinction, but it leads to a signicant change of perspective when we
pass from operads to monads in Section 4. We let T
e
denote the category of spaces X
together with a map e: S
0
X. That is, it is the category of spaces under S
0
.
One would like to say that we have an endomorphism operad pair such that an action of
an operad pair is a map of operad pairs, but that is not quite how things work. Rather,
an action of (C. G) on X consists of an action 0 of C on (X. 0) and an action of G
on (X. 1) for which 0 is a strict zero, so that (g: y) =0 if any coordinate of y is 0,
and for which the parametrized version of the left distributivity law holds. In a rig space
X, for variables (x
r,1
. . . . . x
r,j
r
) X
j
r
, 1 r k, we set z
r
=x
r,1
x
r,j
r
and nd that
z
1
z
k
=
X
Q
x
1,q
1
. . . x
k,q
k
.
where the sum runs over the set of sequences Q=(q
1
. . . . . q
k
) such that 1 q
r
j
r
,
ordered lexicographically. The parametrized version required of a (C. G)space is
obtained by rst dening maps
(1-1) : G(k) C(j
1
) X
j
1
C(j
k
) X
j
k
C(j
1
j
k
) X
j
1
j
k
and then requiring the following diagram to commute.
(1-2)
G(k) C(j
1
) X
j
1
C(j
k
) X
j
k
id 0
k

G(k) X
k

C(j
1
j
k
) X
j
1
j
k
0

X
Geometry & Topology Monographs, Volume 16 (2009)
226 J P May
The promised map on the left is dened by
(1-3) (g: c
1
. y
1
. . . . . c
k
. y
k
) =(z(g: c
1
. . . . . c
k
):
Y
Q
(g: y
Q
))
where g G(k), c
r
C(j
r
), y
r
= (x
r,1
. . . . . x
r,j
r
), the product is taken over the
lexicographically ordered set of sequences Q, as above, and y
Q
=(x
1,q
1
. . . . . x
k,q
k
).
The following observation is trivial, but it will lead in the sequel [32] to signicant
technical simplications of [29].
Remark 1.4 All basepoint conditions, including the strict zero condition, are in fact
redundant. We have seen that the conditions C(0) =0 and G(0) =1 imply that the
additive and multiplicative operad actions specify the points 0 and 1 in X. If any
j
r
=0, then j
1
j
r
=0, we have no coordinates x
r,i
r
, and we must interpret in
(1-1) to be the unique map to the point C(0) X
0
. Then (1-2) asserts that the right
vertical arrow takes the value 0. With all but one j
r
=1 and the remaining j
r
=0,
this forces 0 to be a strict zero for .
2 Ispaces and the linear isometries operad
The canonical multiplicative operad is the linear isometries operad L , which was
introduced by Boardman and Vogt [6; 7]; see also [27, Section I.1]. It is an E
o
operad
that enjoys several very special geometric properties. In this brief section, we recall its
denition and that of related structures that give rise to Lspaces and L spectra.
Let I denote the topological category of nite dimensional real inner product spaces
and linear isometric isomorphisms and let I
c
denote the category of nite or countably
innite dimensional real inner product spaces and linear isometries.
10
For the latter,
we topologize inner product spaces as the colimits of their nite dimensional subspaces
and use the function space topologies on the I
c
(V. W). These are contractible spaces
when W is innite dimensional.
When V is nite dimensional and Y is a based space, we let S
V
denote the one-point
compactication of V and let C
V
Y =F(S
V
. Y ) denote the V fold loop space of Y .
In general F(X. Y ) denotes the space of based maps X Y . Let U =R
o
with
its standard inner product. Dene L(j ) = I
c
(U
j
. U), where U
j
is the sum of j
copies of U , with U
0
={0]. The element id L(1) is the identity isometry, the left
10
The notations I
+
and I were used for our I and I
c
in [27]. We are following [34].
Geometry & Topology Monographs, Volume 16 (2009)
What precisely are E
o
ring spaces and E
o
ring spectra? 227
action of
j
on U
j
induces the right action of
j
on L(j ), and the structure maps
; are dened by
;(g:
1
. . . . .
j
) =g (
1

j
).
Notice that L is a suboperad of the endomorphism operad of U .
For use in the next section and in the second sequel [33], we recall some related formal
notions from [27, Section I.1]. These reappeared and were given new names and
applications in May and Sigurdsson [34, Section 23.3], whose notations we follow. An
Ispace is a continuous functor F: I U . An IFCP (functor with cartesian
product) F is an Ispace that is a lax symmetric monoidal functor, where I is
symmetric monoidal under and U is symmetric monoidal under cartesian product.
This means that there are maps
o: F(V ) F(W) F(V W)
that give a natural transformation (F. F) F which is associative and
commutative up to coherent natural isomorphism. We require that F(0) =+ and that
o be the evident identication when V =0 or W =0. When F takes values in based
spaces, we require the maps FV F(V W) that send x to o(x. +) to be closed
inclusions. We say that (F. o) is monoid-valued if F takes values in the cartesian
monoidal category of monoids in U and o is given by maps of monoids. We then give
the monoids F(V ) their unit elements as basepoints and insist on the closed inclusion
property. All of the classical groups (and String) give examples. Since the classifying
space functor is product preserving, the spaces BF(V ) then give an IFCP BF.
We can dene analogous structures with I replaced throughout by I
c
. Clearly I
c

FCPs restrict to IFCPs. Conversely, our closed inclusion requirement allows us to

pass to colimits over inclusions V V
t
of subspaces in any given countably innite
dimensional inner product space to obtain I
c
FCPs from IFCPs. Formally, we
have an equivalence between the category of IFCPs and the category of I
c
FCPs.
Details are given in [27, Sections I.1 and VII.2] and [34, Section 23.6], and we will
illustrate the argument by example in the next section. When we evaluate an I
c
FCP
F on U , we obtain an L space F(U), often abbreviated to F when there is no
danger of confusion. The structure maps
0: L(j ) F(U)
j
F(U)
are obtained by rst using o to dene F(U)
j
F(U
j
) and then using the evaluation
maps
I
c
(U
j
. U) F(U
j
) F(U)
Geometry & Topology Monographs, Volume 16 (2009)
228 J P May
of the functor F. This simple source of E
o
spaces is fundamental to the geometric
applications, as we recall in Section 10 and the second sequel [33]. We can feed these
examples into the additive innite loop space machine to obtain spectra.
There is a closely related notion of an IFSP (functor with smash product).
11
For this,
we again start with an Ispace T: I T , but we now regard T as symmetric
monoidal under the smash product rather than the cartesian product. The sphere functor
S is specied by S(V ) = S
V
and is strong symmetric monoidal: S(0) = S
0
and
S
V
.S
W
S
V W
. An IFSP is a lax symmetric monoidal functor T together
with a unit map S T . This structure is given by maps
o: T(V ) .T(W) T(V W)
and j: S
V
T(V ). When W = 0, we require o (id . j) to be the obvious
identication
T(V ) .S
0
T(V 0).
The Thom spaces TO(V ) of the universal O(V )bundles give the Thom IFSP TO,
and the other classical groups give analogous Thom IFSPs. A full understanding of
the relationship between IFCPs and IFSPs requires the notion of parametrized
IFSPs, as dened and illustrated by examples in [34, 23.2], but we shall say no
more about that here.
We shall dene E
o
ring prespectra, or L prespectra, in Section 5. The denition
codies structure that is implicit in the notion of an IFSP, so these give examples.
That is, we have a functor from IFSPs to L prespectra. The simple observation
that the classical Thom prespectra arise in nature from IFSPs is the starting point of
E
o
ring theory and thus of this whole area of mathematics. We shall also dene E
o
ring spectra, or L spectra, in Section 5, and we shall describe a spectrication functor
from L prespectra to L spectra. Up to language and clarication of details, these
constructions date from 1972 and are given in [27]. It was noticed over twenty-ve years
later that IFSPs are exactly equivalent to (commutative) orthogonal ring spectra.
This gives an alternative way of processing the simple input data of IFSPs, as we
shall explain. However, we next return to E
o
ring spaces and explain the canonical
operad pair that acts on the 0th spaces of L spectra, such as Thom spectra.
11
These were called I
+
prefunctors when they were rst dened in [27, Section IV.2]; the more
sensible name FSP was introduced later by Bkstedt [8]. For simplicity, we restrict attention to commu-
tative IFSPs in this paper. In analogy with IFCPs, the denition in [27, Section IV.2] required a
technically convenient inclusion condition, but it is best not to insist on that.
Geometry & Topology Monographs, Volume 16 (2009)
What precisely are E
o
ring spaces and E
o
ring spectra? 229
3 The canonical E
1
operad pair
The canonical additive E
o
operad is much less obvious than L . We rst recall the
little cubes operads C
n
, which were also introduced by Boardman and Vogt [6; 7], and
the little discs operads D
V
. We then explain why neither is adequate for our purposes.
For an open subspace X of a nite dimensional inner product space V , dene the
embeddings operad Emb
X
as follows. Let E
X
denote the space of (topological)
embeddings X X. Let Emb
X
(j ) E
j
X
be the space of j tuples of embeddings
with disjoint images. Regard such a j tuple as an embedding
j
X X, where
j
X
denotes the disjoint union of j copies of X (where
0
X is the empty space). The
element id Emb
X
(1) is the identity embedding, the group
j
acts on Emb
X
(j ) by
permuting embeddings, and the structure maps
(3-1) ;: Emb
X
(k) Emb
X
(j
1
) Emb
X
(j
k
) Emb
X
(j
1
j
k
)
are dened as follows. Let g =(g
1
. . . . . g
k
) Emb
X
(k) and
r
=(
r,1
. . . . .
r,j
r
)
Emb
X
(j
r
), 1 r k. Then the r th block of j
r
embeddings in ;(g:
1
. . . . .
j
) is
given by the composites g
r

r,s
, 1 s j
r
.
Taking X =(0. 1)
n
R
n
, we obtain a suboperad C
n
of Emb
X
by restricting to the
little ncubes, namely those embeddings : X X such that =
1

n
,
where
i
(t ) =a
i
t b
i
for real numbers a
i
>0 and b
i
0.
For a general V , taking X to be the open unit disc D(V ) V , we obtain a sub-
operad D
V
of Emb
V
by restricting to the little V discs, namely those embeddings
: D(V ) D(V ) such that (:) =a: b for some real number a >0 and some
element b D(V ).
It is easily checked that these denitions do give well-dened suboperads. Let F(X. j )
denote the conguration space of j tuples of distinct elements of X, with its permu-
tation action by
j
. These spaces do not t together to form an operad, and C
n
and
D
V
specify fattened up equivalents that do form operads. By restricting little ncubes
or little V discs to their values at the center point of (0. 1)
n
or D(V ), we obtain

j
equivariant deformation retractions
C
n
(j ) F((0. 1)
n
. j ) F(R
n
. j ) and D
V
(j ) F(D(V ). j ) F(V. j ).
This gives control over homotopy types.
For a little ncube , the product id gives a little (n1)cube. Applying this to
all little ncubes gives a suspension map of operads C
n
C
n1
. We can pass to
colimits over n to construct the innite little cubes operad C
o
, and it is an E
o
operad.
Geometry & Topology Monographs, Volume 16 (2009)
230 J P May
However, little ncubes are clearly too square to allow an action by the orthogonal
group O(n), and we cannot dene an action of L on C
o
.
For a little V disc and an element g O(V ), we obtain another little V disc
gg
-1
. Thus the group O(V ) acts on the operad D
V
. However, for V W , so that
W =V (W V ) where W V is the orthogonal complement of V in W , little
V discs are clearly too round for the product id to be a little W disc. We can
send a little V disc : a: b to the little W disc n an b, but that is not
compatible with the decomposition S
W
S
V
.S
W-V
used to identify C
W
Y with
C
W-V
C
V
Y . Therefore we cannot suspend.
In [27], the solution was to introduce the little convex bodies partial operads. They
werent actually operads because the structure maps ; were only dened on subspaces
of the expected domain spaces. The use of partial operads introduced quite unpleasant
complications. Steiner [43] found a remarkable construction of operads K
V
which
combine all of the good properties of the C
n
and the D
V
. His operads, which we
call the Steiner operads, are dened in terms of paths of embeddings rather than just
embeddings.
Dene R
V
E
V
= Emb
V
(1) to be the subspace of distance reducing embeddings
: V V . This means that [ (:) (n)[ [: n[ for all :. n V . Dene a
Steiner path to be a map h: I R
V
such that h(1) =id and let P
V
be the space of
Steiner paths. Dene : P
V
R
V
by evaluation at 0, (h) =h(0). Dene K
V
(j )
to be the space of j tuples (h
1
. . . . . h
j
) of Steiner paths such that the (h
r
) have
disjoint images. The element id K
V
(1) is the constant path at the identity embedding,
the group
j
acts on K
V
(j ) by permutations, and the structure maps ; are dened
pointwise in the same way as those of Emb
V
. That is, for g =(g
1
. . . . . g
k
) K
V
(k)
and
r
=(
r,1
. . . . .
r,j
r
) K (j
r
), ;(g:
1
. . . . .
j
) is given in by the embeddings
g
r
(t )
r,s
(t ), in order. This gives well dened operads, and application of to
Steiner paths gives a map of operads : K
V
Emb
V
.
By pullback along , any space with an action by Emb
V
inherits an action by K
V
.
As in [25, Section 5] or [27, Section VII.2], Emb
V
acts naturally on C
V
Y . A j tuple
of embeddings V V with disjoint images determines a map from S
V
to the wedge
of j copies of S
V
by collapsing points of S
V
not in any of the images to the point
at innity and using the inverses of the given embeddings to blow up their images to
full size. A j tuple of based maps S
V
Y then gives a map from the wedge of the
S
V
to Y . Thus the resulting action 0
V
of K
V
on C
V
Y is given by composites
K
V
(j )(C
V
Y )
j
tid

Emb
V
(j )(C
V
Y )
j
C
V
(
j
S
V
)(C
V
Y )
j
C
V
Y.
Geometry & Topology Monographs, Volume 16 (2009)
What precisely are E
o
ring spaces and E
o
ring spectra? 231
Evaluation of embeddings at 0 V gives maps : Emb
V
(j ) F(V. j ). Steiner
determines the homotopy types of the K
V
(j ) by proving that the composite maps
: K
V
(j ) F(V. j ) are
j
equivariant deformation retractions.
The operads K
V
have extra geometric structure that make them ideally suited for our
purposes. Rewriting F(V ) =F
V
, we see that E, R, and P above are monoid-valued
IFCPs. The monoid products are given (or induced pointwise) by composition of
embeddings, and the maps o are given by cartesian products of embeddings. For the
functoriality, if : V V is an embedding and g: V W is a linear isometric
isomorphism, then we obtain an embedding gg
-1
: W W which is distance
reducing if is. We have an inclusion R E of monoid-valued IFCPs, and
evaluation at 0 gives a map : P RE of monoid-valued IFCPs. The operad
structure maps of Emb
V
and K
V
are induced by the monoid products, as is clear from
the specication of ; after (3-1).
The essential point is that, in analogy with (3-1), we have maps
(3-2)
z: I(V
1
V
k
. W) Emb
V
1
(j
1
) Emb
V
k
(j
k
) Emb
W
(j
1
j
k
)
dened as follows. Let g: V
1
V
k
W be a linear isometric isomorphism
and let
r
= (
r,1
. . . . .
r,j
r
) Emb
V
r
(j
r
), 1 r k. Again consider the set of
sequences Q=(q
1
. . . . . q
k
), 1 q
r
j
r
, ordered lexicographically. Identifying direct
sums with direct products, the Qth embedding of z(g:
1
. . . . .
k
) is the composite
g
Q
g
-1
, where
Q
=
1,q
1

k,q
k
. Restricting to distance reducing embeddings

r,s
and applying the result pointwise to Steiner paths, there result maps
(3-3) z: I(V
1
V
k
. W) K
V
1
(j
1
) K
V
k
(j
k
) K
W
(j
1
j
k
).
Passing to colimits over inclusions V V
t
of subspaces in any given countably innite
dimensional inner product space, such as U , we obtain structure exactly like that
just described, but now dened on all of I
c
, rather than just on I . (Compare [27,
Sections I.1 and VII.2] and [34, Section 23.6]). For example, suppose that the V
r
and W in (3-3) are innite dimensional. Since the spaces Emb
V
r
(1) are obtained
by passage to colimits over the nite dimensional subspaces of the V
r
, for each of
the embeddings
r,s
, there is a nite dimensional subspace A
r,s
such that
r,s
is
the identity on the orthogonal complement V
r
A
r,s
. Therefore, all of the
Q
are
the identity on V
1
V
k
B for a sufciently large B. On nite dimensional
subspaces gV W , we dene z as before, using the maps g
Q
g
-1
. On the orthogonal
complement W gV for V large enough to contain B, we can and must dene the
Qth embedding to be the identity map for each Q.
Geometry & Topology Monographs, Volume 16 (2009)
232 J P May
Finally, we dene the canonical additive E
o
operad, denoted C , to be the Steiner
operad K
U
. Taking V
1
= =V
k
=U , we have the required maps
z: L(k) C(j
1
) C(j
k
) C(j
1
j
k
).
They make the (unspecied) distributivity diagrams commute, and we next explain the
signicance of those diagrams.
4 The monadic interpretation of the denitions
We assume that the reader has seen the denition of a monad. Fixing a ground category
V , a quick denition is that a monad (C. j. j) on V is a monoid in the category of
functors V V . Thus C: V V is a functor, j: CC C and j: Id C are
natural transformations, and j is associative with j as a two-sided unit. A Calgebra
is an object X V with a unital and associative action map : CX X. We let
CV | denote the category of Calgebras in V .
Similarly, when O is an operad in V , we write OV | for the category of Oalgebras
in V . We note an important philosophical distinction. Monads are very general,
and their algebras in principle depend only on V , without reference to any further
structure that category might have. In contrast, Oalgebras depend on a choice of
symmetric monoidal structure on V , and that might vary. We sometimes write OV . |
to emphasize this dependence.
For an operad O of unbased spaces with O(0) =+, as before, there are two monads
in sight, both of which have the same algebras as the operad O.
12
Viewing operads as
acting on unbased spaces, we obtain a monad O
U

on U with
(4-1) O
U

X =
a
j0
O(j )

j
X
j
.
Here j(x) =(1. x) O(1)X, and j is obtained by taking coproducts of maps on or-
bits induced by the structure maps ; . If X has an action 0 by O, then : O
U

X X
is given by the action maps 0: O(j )

j
X
j
X, and conversely. The subscript
on the monad is intended to indicate that it is augmented, rather than reduced.
The superscript U is intended to indicate that the operad from which the monad is
constructed is an operad of unbased spaces.
Viewing operads as acting on spaces with preassigned basepoints, we construct a
reduced monad O =O
U
on T by quotienting O
U

X by basepoint identications.
12
Further categorical perspective on the material of this section is given in Appendix A.
Geometry & Topology Monographs, Volume 16 (2009)
What precisely are E
o
ring spaces and E
o
ring spectra? 233
There are degeneracy operations o
i
: O(j ) O(j 1) given by
o
i
(c) =;(c: (id)
i-1
. +. (id)
j-i
)
for 1 i j , and there are maps s
i
: X
j-1
X
j
obtained by inserting the basepoint
in the i th position. We set
(4-2) OX O
U
X =O
U

X,(~).
where (c. s
i
y) ~ (o
i
c. y) for c O(j ) and y X
j-1
. Observations in Section 1
explain why these two operads have the same algebras. The reduced monad O =O
U
is more general than the augmented monad O
U

since the latter can be obtained by

applying the former to spaces of the form X

. That is, for unbased spaces X,

(4-3) O
U
(X

) O
U

X
as Ospaces. To keep track of adjunctions later, we note that the functor ()

is left
adjoint to the functor i : T U that is obtained by forgetting basepoints.
The reduced monad O =O
U
on T is of primary topological interest, but the idea
that there is a choice will simplify the multiplicative theory. Here we diverge from the
original sources [27; 29].
13
Summarizing, we have the following result.
Proposition 4.4 The following categories are isomorphic.
(i) The category OU . | =OT . | of Ospaces.
(ii) The category O
U

U | of O
U

algebras in U .
(iii) The category OT | O
U
T | of Oalgebras in T .
We have another pair of monads associated to an operad O. Recall again that operads
and operad actions make sense in any symmetric monoidal category V . Above, in (i),
we are thinking of T as cartesian monoidal, and we are entitled to use the alternative
notations OU | and OT | since Oalgebras in U can equally well be regarded as O
algebras in T . The algebras in Proposition 4.4 have parametrized products X
j
X
that are dened on cartesian powers X
j
.
However, we can change ground categories to the symmetric monoidal category T
under its smash product, with unit S
0
. We write X
(j)
for the j fold smash power
of a space (or, later, spectrum) X, with X
(0)
=S
0
. Remembering that X

.Y

(X Y )

, we can adjoin disjoint basepoints to the spaces O(j ) to obtain an operad

13
I am indebted to helpful conversations with Bob Thomason and Tyler Lawson, some twenty-ve
years apart, for the changed perspective.
Geometry & Topology Monographs, Volume 16 (2009)
234 J P May
O

with spaces O

(j ) =O(j )

in T ; in particular, O

(0) =S
0
. The actions of
O

parametrize j fold products X

(j)
X, and we have the category O

T | of
O

spaces in T .
Now recall that T
e
denotes the category of spaces X under S
0
, with given map
e: S
0
X. In [27; 29], we dened an Ospace with zero, or O
0
space, to be an
Ospace (X. ) in T
e
such that 0 acts as a strict zero, so that ( : x
1
. . . . . x
j
) =0
if any x
i
= 0. That is exactly the same structure as an O

space in T . The only

difference is that now we think of : S
0
= O

(0) X as building in the map

e: S
0
X, which is no longer preassigned. We are entitled to use the alternative
notation O

T
e
| for O

T | .
We construct an (augmented) monad O

=O
T

on T such that an O

space in T
is the same as an O

algebra in T by setting
(4-5) O

X O
T

X =
_
j0
O(j )

j
X
(j)
.
This and (4-1) are special cases of a general denition that applies to operads in any
cocomplete symmetric monoidal category, as is discussed in detail in [30; 31].
As a digression, thinking homotopically and letting E
j
be any contractible free

j
space, one denes the extended j fold smash power D
j
X of a based space X
by
(4-6) D
j
X =(E
j
)

j
X
(j)
.
These spaces have many applications. Homotopically, when O is an E
o
operad,
O

X is a model for the wedge over j of the spaces D

j
X.
Here we have not viewed the element 1 of a space under S
0
as a basepoint. However,
we can use basepoint identications to take account of the unit properties of 1 in an
action by O

. We then obtain a reduced monad O

T
on T
e
with the same algebras
as the monad O
T

on T . It is again more general than the monad O

. For a based
space X, S
0
X is the space under S
0
obtained from the based space X by adjoining
a point 1 (not regarded as a basepoint). This gives the left adjoint to the inclusion
i : T
e
T that is obtained by forgetting the point 1, and
(4-7) O
T
(S
0
X) O
T

(X)
as O

spaces. We summarize again.

Proposition 4.8 The following categories are isomorphic.
(i) The category O

T . .| =O

T
e
. .| of O

spaces.
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What precisely are E
o
ring spaces and E
o
ring spectra? 235
(ii) The category O

T | O
T

T | of O

algebras in T .
(iii) The category O
T
T
e
| of O
T
algebras in T
e
.
In [27; 29], we viewed the multiplicative structure of E
o
ring spaces as dened on the
base category T
e
, and we used the monad G
T
on T
e
instead of the monad G

on
T . That unnecessarily complicated the details in [29], where different kinds of input
data to multiplicative innite loop space theory were compared, and we now prefer to
use G

. That is convenient both here and in the sequel [32].

With this as preamble, consider an operad pair (C. G), such as the canonical one
(C. L) from the previous section. We have several monads in sight whose algebras
are the (C. G)spaces X. We single out the one most convenient for the comparison
of E
o
ring spaces and E
o
ring spectra by focusing on the additive monad C on
T , where the basepoint is denoted 0 and is the unit for the operadic product, and the
multiplicative monad G

, which is also dened on T . A different choice will be

more convenient in the sequel [32].
The diagrams that we omitted in our outline denition of an action of G on C are
exactly those that are needed to make the following result true.
Proposition 4.9 The monad C on T restricts to a monad on the category G

T | of
G

spaces in T . A (C. G)space is the same structure as a Calgebra in G

T | .
Sketch proof The details of the denition of an action of G on C are designed to
ensure that, for a G

space (X. ), the maps of (1-3) induce maps

: G(k)

.(CX)
(k)
CX
that give CX a structure of G

space in T such that

j: CCX CX and j: X CX
are maps of G

spaces in T . Then the diagram (1-2) asserts that a (C. G)space is

the same as a Calgebra in G

T | . Details are in [27, Section VI.1].

We have the two monads (C. j

. j

) and (G

. j

. j

), both dened on T , such

that C restricts to a monad on G

T | . This puts things in a general categorical context

that was studied by Beck [5]. A summary of his results is given in [29, 5.6] and in
[32, Appendix B]. He gives several equivalent characterizations of a pair (C. G

) of
monads related in this fashion. One is that the composite functor CG

is itself a monad
with certain properties. Another is that there is a natural interchange transformation
G

C CG

such that appropriate diagrams commute. Category theorists know

Geometry & Topology Monographs, Volume 16 (2009)
236 J P May
well that this is denitively the right context in which to study generalized distributivity
phenomena. While I dened E
o
ring spaces before I knew of Becks work, his context
makes it clear that this is a very natural and conceptual denition.
5 The denition of E
1
ring prespectra and E
1
ring spectra
We rst recall the categories of (LMS) prespectra and spectra from Lewis et al [18]. As
before, we let U =R
o
with its standard inner product. Dene an indexing space to be
a nite dimensional subspace of U with the induced inner product. A (coordinate free)
prespectrum T consists of based spaces TV and based maps o: TV C
W-V
T W
for V W , where W V is the orthogonal complement of V and S
W-V
is its
one point compactication; o must be the identity when V = W , and the obvious
transitivity condition must hold when V W Z. A spectrum is a prespectrum such
that each map o is a homeomorphism; we then usually write E rather than T .
We let P and S denote the categories of prespectra and spectra. Then S is a full
subcategory of P, with inclusion : S P, and there is a left adjoint spectrication
functor L: PS . When T is an inclusion prespectrum, meaning that each o is
an inclusion, (LT)(V ) =colim
V W
C
W-V
T W .
We may restrict attention to any conal set of indexing spaces V in U ; we require 0
to be in V and we require the union of the V in V to be all of U . Up to isomorphism,
the category S is independent of the choice of V . The default is V =A. We can
dene prespectra and spectra in the same way in any countably innite dimensional real
inner product space U , and we write P(U) when we wish to remember the universe.
The default is U =R
o
.
For X T , we dene QX to be colimC
V

V
X, and we let j: X QX be the
natural inclusion. We dene (
o
X)(V ) =Q
V
X. Since S
W
S
V
.S
W-V
, we
have identications Q
V
X C
W-V
Q
W
X, so that
o
X is a spectrum. For a
spectrum E, we dene C
o
E = E(0); we often write it as E
0
. The functors
o
and C
o
are adjoint, QX is C
o

o
X, and j is the unit of the adjunction. We let
c:
o
C
o
E E be the counit of the adjunction.
As holds for any adjoint pair of functors, we have a monad (Q. j. j) on T , where
j: QQ Q is C
o
c
o
, and the functor C
o
takes values in Qalgebras via
the action map C
o
c: C
o

o
C
o
E C
o
E. These observations are categorical
trivialities, but they are central to the theory and will be exploited heavily. We will see
later that this adjunction restricts to an adjunction in L

T | . That fact will lead to the

proof that the 0th space of an E
o
ring spectrum is an E
o
ring space.
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What precisely are E
o
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o
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As already said, the starting point of this area of mathematics was the observation that
the classical Thom prespectra, such as T U , appear in nature as IFSPs and therefore
as E
o
ring prespectra in the sense we are about to describe. To distinguish, we will
write MU for the corresponding spectrum. Quite recently, Strickland [44, Appendix]
has observed similarly that the periodic Thom spectrum, PMU =MUx. x
-1
| , also
arises in nature as the spectrum associated to an E
o
ring prespectrum. To dene this
concept, we must consider smash products and change of universe.
We have an external smash product T ZT
t
that takes a pair of prespectra indexed on
A in U to a prespectrum indexed on the indexing spaces V V
t
in U U . It is
specied by
(T ZT
t
)(V. V
t
) =TV .T
t
V
t
with evident structure maps induced by those of T and T
t
. This product is commutative
and associative in an evident sense; for the commutativity, we must take into account
the interchange isomorphism t: U U U U . If we think of spaces as prespectra
indexed on 0, then the space S
0
is a unit object. Formally, taking the disjoint union
over j 0 of the categories P(U
j
), we obtain a perfectly good symmetric monoidal
category. This was understood in the 1960s. A well-structured treatment of spectra
from this external point of view was given by Elmendorf [12].
For a linear isometry : U U
t
, we have an obvious change of universe functor

+
: P(U
t
) P(U) specied by (
+
T
t
)(V ) =T
t
( V ), with evident structure
maps. It restricts to a change of universe functor
+
: S(U
t
) S(U). These
functors have left adjoints
+
. When is an isomorphism,
+
= (
-1
)
+
. For a
general , it is not hard to work out an explicit denition; see [18, Section II.1]. The
left adjoint on the spectrum level is L
+
. This is the way one constructs a spectrum
level left adjoint from any prespectrum level left adjoint. The external smash product
of spectra is dened similarly, EZE
t
=L(EZE
t
).
The reader should have in mind the Thom spaces TO(V ) or, using complex inner
product spaces, T U(V ) associated to well chosen universal V plane bundles. In
contrast to the original denitions of [27; 18], we restrict attention to the linear isometries
operad L . There seems to be no useful purpose in allowing more general operads in
this exposition.
We agree to write T
jj
for external j fold smash powers. An E
o
prespectrum, or
L prespectrum, T has an action of L that is specied by maps of prespectra

j
( ): T
jj

+
T
or, equivalently,
+
T
jj
T , for all L(j ) that are suitably continuous and
are compatible with the operad structure on L . The compatibility conditions are
Geometry & Topology Monographs, Volume 16 (2009)
238 J P May
that
j
( t) =
j
( ) t
+
for t
j
(where t is thought of as a linear isomorphism
U
j
U
j
),
1
(id) =id, and

j
1
j
k
(;(g:
1
. . . . .
k
)) =
k
(g) (
j
1
(
1
) Z Z
j
k
(
k
))
for g L(k) and
r
L(j
r
).
For the continuity condition, let V = V
1
V
j
and let A(V. W) L(j ) be
the subspace of linear isometries such that (V ) W , where the V
r
and W are
indexing spaces. We have a map A(V. W) V A(V. W) W of bundles over
A(V. W) that sends (. :) to (. (:)). Its image is a subbundle, and we let T(V. W)
be the Thom complex of its complementary bundle. Dene a function
=(V. W): T(V. W) .TV
1
. .TV
j
T W
by
((. n). y
1
. . . . . y
j
) =o(
j
( )(y
1
. .y
j
) .n)
for A(V. W), n W (V ), and y
r
TV
r
; on the right, o is the structure
map T( V ) .S
W-( V
T W . The functions (V. W) must all be continuous.
This is a very simple notion. As already noted in Section 2, it is easy to see that
IFSPs and thus Thom prespectra give examples. However, the continuity condition
requires a more conceptual description. We want to think of the maps
j
( ) as j fold
products parametrized by L(j ), and we want to collect them together into a single
global map. The intuition is that we should be able to construct a twisted half-smash
product prespectrum L(j ) T
jj
indexed on U by setting
(L(j ) T
jj
)(W) =T(V. W) .TV
1
. .TV
j
.
This doesnt quite make sense because of the various possible choices for the V
r
,
but it does work in exactly this way if we choose appropriate conal sequences of
indexing spaces in U
j
and U . The resulting construction L() on the spectrum
level is independent of choices. Another intuition is that we are suitably topologizing
the disjoint union over L(j ) of the prespectra
+
T
jj
indexed on U .
These intuitions are made precise in [18, Chapter VI] and, more conceptually but perhaps
less intuitively, [13, Appendix]. The construction of twisted half-smash products of
spectra is the starting point of the EKMM approach to the stable homotopy category
[13], but for now we are focusing on earlier work. With this construction in place, we
have an equivalent denition of an L prespectrum in terms of action maps

j
: L(j ) T
jj
T
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What precisely are E
o
ring spaces and E
o
ring spectra? 239
such that equivariance, unit, and associativity diagrams commute. The diagrams are
exactly like those in the original denition of an action of an operad on a space.
In more detail, and focusing on the spectrum level, the construction of twisted half-
smash products actually gives functors
(5-1) AE
1
Z ZE
j
for any spectra E
r
indexed on U and any map AL(j ). There are many relations
among these functors as j and A vary. In particular there are canonical maps
(5-2)
L(k) (L(j
1
) E
j
1
j
Z ZL(j
k
) E
j
k
j
)

(L(k) L(j
1
) L(j
k
)) E
jj
;id

L(j ) E
jj
where j =j
1
j
k
. Using such maps we can make sense out of the denition of
an E
o
ring spectrum in terms of an action by the operad L .
Denition 5.3 An E
o
ring spectrum, or L spectrum, is a spectrum R with an
action of L given by an equivariant, unital, and associative system of maps
(5-4) L(j ) R
jj
R.
Lemma 5.5 If T is an L prespectrum, then LT is an L spectrum.
This holds since the spectrication functor L: PS satises
L(L(j ) T
jj
) L(j ) (LT)
jj
.
6 The monadic interpretation of E
1
ring spectra
At this point, we face an unfortunate clash of notations, and for the moment the reader
is asked to forget all about prespectra and the spectrication functor L: P S .
We focus solely on spectra in this section.
In analogy with Section 4, we explain that there are two monads in sight, both of whose
algebras coincide with E
o
ring spectra.
14
One is in [18], but the one more relevant to
our current explanations is new.
14
Again, further categorical perspective is supplied in Appendix A.
Geometry & Topology Monographs, Volume 16 (2009)
240 J P May
In thinking about external smash products, we took spectra indexed on zero to be spaces.
Since L(0) is the inclusion 0 U , the change of universe functor L(0) ()
can and must be interpreted as the functor
o
: T S . Similarly, the zero fold
external smash power E
0j
should be interpreted as the space S
0
. Since
o
S
0
is the
sphere spectrum S, the 0th structure map in (5-4) is a map e: S R. We have the
same dichotomy as in Section 4. We can either think of the map e as preassigned, in
which case we think of our ground category as the category S
e
of spectra under S, or
we can think of e =
0
as part of the structure of an E
o
ring spectrum, in which case
we think of our ground category as S .
In analogy with the space level monad L

we dene a monad L

on the category of
spectra by letting
L

E =
_
j0
L(j )

j
E
jj
.
The 0th term is S, and j: S L

E is the inclusion. The product j is induced by

passage to orbits and wedges from the canonical maps (5-2).
We also have a reduced monad L dened on the category S
e
. The monad L on T
e
is obtained from the monad L

on T by basepoint identications. The construction

can be formalized in terms of coequalizer diagrams. We obtain the analogous monad
L on S
e
by base sphere identications that are formalized by precisely analogous
coequalizer diagrams [18, Section VII.3]. In analogy with (4-7), the spectrum level
monads L and L

are related by a natural isomorphism

(6-1) L(S E) L

E.
This isomorphism, like (4-7), is monadic. This means that the isomorphisms are
compatible with the structure maps of the two monads, as is made precise in Denition
14.1. The algebras of L are the same as those of L

, and we have the following

analogue of Propositions 4.4 and 4.8.
Proposition 6.2 The following categories are isomorphic.
(i) The category LS| =LS
e
| of L spectra.
(ii) The category L

S| of L

algebras in S .
(iii) The category LS
e
| of Lalgebras in S
e
.
A central feature of twisted half-smash products is that there is a natural untwisting
isomorphism
(6-3) A(
o
X
1
Z Z
o
X
j
)
o
(A

.X
1
. X
j
).
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What precisely are E
o
ring spaces and E
o
ring spectra? 241
for any space A over L(j ). Using this, we obtain a monadic natural isomorphism
(6-4) L

o
X
o
L

X
relating the space and spectrum level monads L

.
It has become usual to compress notation by dening
(6-5)
o

X =
o
(X

)
for a space X, ignoring its given basepoint if it has one. When X has a (nondegenerate)
basepoint, the equivalence (S
0
X) .S
1
X implies an equivalence
15
(6-6)
o

X .S
o
X
under S. With the notation of (6-5), the relationship between the space and spectrum
level monads L is given by a monadic natural isomorphism
(6-7) L
o

X
o

LX.
(See [18, VII.3.5]). Remarkably, as we shall show in Appendix A, the commutation
relations (6-1), (6-4), and (6-7) are formal, independent of calculational understanding
of the monads in question.
We digress to recall a calculationally important splitting theorem that is implicit in
these formalities. Using nothing but (6-1) and (6-3)(6-7), we nd
S
o
LX .
o

LX
L
o

X
.L(S
o
X)
L

o
X
=
_
j0
L(j )

j
(
o
X)
jj
S
_
j1

o
(L(j )

j
X
(j)
).
Quotienting out S and recalling (4-6), we obtain a natural equivalence
(6-8)
o
LX .
_
j1

o
D
j
X.
15
In contrast with the isomorphisms appearing in this discussion, this equivalence plays no role in our
formal theory; we only recall it for use in a digression that we are about to insert.
Geometry & Topology Monographs, Volume 16 (2009)
242 J P May
We may replace LX by CX for any other E
o
operad C , such as the Steiner operad,
and then CX .QX when X is connected. Thus, if X is connected,
(6-9)
o
QX .
_
j1

o
D
j
X.
This beautiful argument was discovered by Ralph Cohen [11]. As explained in [18,
Section VII.5], we can exploit the projection O L L to obtain a splitting
theorem for OX analogous to (6-8), where O is any operad, not necessarily E
o
.
7 The relationship between E
1
ring spaces and E
1
ring
spectra
We show that the 0th space of an E
o
ring spectrum is an E
o
ring space. This
observation is at the conceptual heart of what we want to convey. It was central to the
1970s applications, but it seems to have dropped off the radar screen, and this has led
to some confusion in the modern literature. One reason may be that the only proof
ever published is in the original source [27], which preceded the denition of twisted
half-smash products and the full understanding of the category of spectra. Since this
is the part of [27] that is perhaps most obscured by later changes of notations and
denitions, we give a cleaner updated treatment that gives more explicit information.
Recall that L

T | L

T | and LS| L

S| denote the categories of L

spaces, or Lspaces with zero, and of L spectra, thought of as identied with the
categories of L

algebras in T and of L

algebras in S . To distinguish, we write

X for based spaces, Z for L

spaces, E for spectra, and R for L spectra.

Proposition 7.1 The (topological) adjunction
S(
o
X. E) T (X. C
o
E)
induces a (topological) adjunction
L

S|(
o
Z. R) L

T |(Z. C
o
R).
Therefore the monad Q on T restricts to a monad Q on L

T | and, when restricted

to L spectra, the functor C
o
takes values in algebras over L

T | .
Proof This is a formal consequence of the fact that the isomorphism (6-4) is monadic,
as is explained in general categorical terms in Proposition 14.4. If (Z. ) is an L

algebra, then
o
Z is an L

algebra with structure map

L

o
Z

o
L

o
Z.
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What precisely are E
o
ring spaces and E
o
ring spectra? 243
The isomorphism (6-4) and the adjunction give a natural composite :
L

C
o
E
)

C
o

o
L

C
o
E


C
o
L

o
C
o
E
D
1
L
C
t

C
o
L

E.
If (R. ) is an L

algebra, then C
o
R is an L

algebra with structure map

L

C
o
R

C
o
L

R
D
1

C
o
R.
Diagram chases show that the unit j and counit c of the adjunction are maps of L

algebras when Z and R are L

algebras. The last statement is another instance of

an already cited categorical triviality about the monad associated to an adjunction, and
we shall return to the relevant categorical observations in the next section.
In the notation of algebras over operads, this has the following consequence.
Corollary 7.2 The adjunction of Proposition 7.1 induces an adjunction
LS|(
o

Y. R) LU |(Y. C
o
R)
between the category of Lspaces and the category of L spectra.
Proof Recall that the functor i : T U given by forgetting the basepoint induces
an isomorphism LT | LU | since maps of Lspaces must preserve the basepoints
given by the operad action. Also, since adjoining a disjoint basepoint 0 to an Lspace
Y gives an Lspace with zero, or L

space in T , while forgetting the basepoint 0

of an L

space in T gives an Lspace in U , the evident adjunction

T (U

. X) U (U. iX)
for based spaces X and unbased spaces U induces an adjunction
L

T |(Y

. Z) LU |(Y. Z)
for Lspaces Y and L

spaces Z. Taking Z = C
o
R, the result follows by
composing with the adjunction of Proposition 7.1.
Now let us bring the Steiner operads into play. For an indexing space V U , K
V
acts
naturally on V fold loop spaces C
V
Y . These actions are compatible for V W , and
by passage to colimits we obtain a natural action 0 of the Steiner operad C =K
U
on
C
o
E for all spectra E. For spaces X, we dene : CX QX to be the composite
(7-3) CX
C)

CC
o

o
X
0

C
o

o
X =QX.
Geometry & Topology Monographs, Volume 16 (2009)
244 J P May
Another purely formal argument shows that is a map of monads in T [25, 5.2]. This
observation is central to the entire theory.
We have seen in Propositions 4.9 and 7.1 that C and Q also dene monads on L

T | ,
and we have the following crucial compatibility.
Proposition 7.4 The map : C Q of monads on T restricts to a map of monads
on L

T | .
Sketch proof We must show that : CX QX is a map of L

spaces when X
is an L

space. Since it is clear by naturality that Cj: CX CQX is a map of

L

spaces, it sufces to show that C0: CQX QX is a map of L

spaces. We
may use embeddings operads rather than Steiner operads since the action of K
V
on
C
V
Y is obtained by pullback of the action of Emb
V
. The argument given for the little
convex bodies operad in [27, VII.2.4, page 179] applies verbatim here. It is a passage
to colimits argument similar to that sketched at the end of Section 3.
Recall that (C. L)spaces are the same as Calgebras in L

T | , and these are our

E
o
ring spaces. Similarly, our E
o
ring spectra are the same as L spectra, and the
0th space functor takes L spectra to Qalgebras in L

T | . By pullback along ,
this gives the following promised conclusion.
Corollary 7.5 The 0th spaces of E
o
ring spectra are naturally E
o
ring spaces.
In particular, the 0th space of an E
o
ring spectrum is both a C space and an L space
with 0. The interplay between the DyerLashof homology operations constructed from
the two operad actions is essential to the caculational applications, and for that we
must use all of the components. However, to apply innite loop space theory using the
multiplicative operad L , we must at least delete the component of 0, and it makes
sense to also delete the other nonunit components.
Denition 7.6 The 0th space R
0
of an (up to homotopy) ring spectrum R is an (up
to homotopy) ring space, and
0
(C
o
R) is a ring. Dene GL
1
R to be the subspace
of R
0
that consists of the components of the unit elements. Dene SL
1
R to be the
component of the identity element.
16
For a space X, X

. GL
1
R| is the group of
units in the (unreduced) cohomology ring R
0
(X).
Corollary 7.7 If R is an E
o
ring spectrum, then the unit spaces GL
1
R and SL
1
R
are Lspaces.
Again, we emphasize how simply and naturally these denitions t together.
16
To repeat, these spaces were introduced in [27], where they were denoted FR and SFR.
Geometry & Topology Monographs, Volume 16 (2009)
What precisely are E
o
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o
ring spectra? 245
8 A categorical overview of the recognition principle
The passage from space level to spectrum level information through the black box of 1
admits a simple conceptual outline. We explain it here.
17
We consider two categories, V and W , with an adjoint pair of functors (. ) between
them.
18
We write
j: Id and c: Id
for the unit and counit of the adjunction. The reader should be thinking of (
n
. C
n
),
where V =T and W is the category of nfold loop sequences {C
i
Y [0 i n] and
maps {C
i
[0 i n]. This is a copy of T , but we want to remember how it encodes
nfold loop spaces. It is analogous and more relevant to the present theory to think of
(
o
. C
o
), where V =T and W =S .
As we have already noted several times, we have the monad
(. j. j)
on V , where j =c. We also have a right action of the monad on the functor
and a left action of on . These are given by the natural maps
c: and c: .
Actions of monads on functors satisfy unit and associativity diagrams just like those
that dene the action of a monad on an object. If we think of an object X V as a
functor from the trivial one object category + to V , then an action of a monad on the
object X is the same as a left action of the monad on the functor X.
Now suppose we also have some monad C on V and a map of monads : C .
By pullback, we inherit a right action of C on and a left action of C on , which
we denote by , and z. Thus
, =c : C and z =c : C.
For a Calgebra X in V , we seek an object EX in W such that X is weakly
equivalent to EX as a Calgebra. This is the general situation addressed by our
black box, and we rst remind ourselves of how we would approach the problem if
were looking for a categorical analogue. We will reprove a special case of Becks
monadicity theorem (see Mac Lane [21, Section VI.7]), but in a way that suggests how
to proceed homotopically.
17
I am indebted to Saunders Mac Lane, Gaunce Lewis, and Matt Ando for ideas I learned from them
some thirty-ve years apart.
18
is the capital Greek letter Xi; and are meant to look a little like and C.
Geometry & Topology Monographs, Volume 16 (2009)
246 J P May
Assume that W is cocomplete. For any right C functor : V W with right
action , and left C functor : U V with left action z, where U is any other
category, we have the monadic tensor product
C
: U W that is dened on
objects U U as the coequalizer displayed in the diagram
(8-1) (C)(U)
p

z
()(U)

(
C
)(U).
We are interested primarily in the case when U = + and = X for a Calgebra
(X. ) in V , and we then write
C
X. Specializing to the case =, this is the
coequalizer in W displayed in the diagram
(8-2) CX
p

C
X.
For comparison, we have the canonical split coequalizer
(8-3) CCX

C
CX

X
in V , which is split by j
CX
: CX CCX and j
X
: X CX.
Let CV | denote the category of Calgebras in V . Then
C
() is a functor
CV | W , and our original adjunction restricts to an adjunction
(8-4) W (
C
X. Y ) CV |(X. Y ).
Indeed, consider a map : X Y of Calgebras, so that =z C . Taking
its adjoint

: X Y , we see by a little diagram chase that it equalizes the pair of
maps and , and therefore induces the required adjoint map
C
X Y .
This construction applies in particular to the monad , and for the moment we take
C = . The Beck monadicity theorem says that the adjunction (8-4) is then an
adjoint equivalence under appropriate hypotheses, which we now explain.
Consider those parallel pairs of arrows (. g) in the diagram
X
(

g
Y
h
_ _ _
Z
in W such that there exists a split coequalizer diagram
X
(

g
Y
j

V
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o
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o
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in V . Assume that preserves and reects coequalizers of such pairs (. g) of
parallel arrows. Preservation means that if h is a coequalizer of and g, then
h: Y Z is a coequalizer of and g. It follows that there is a unique
isomorphism i : V Z such that i j = h. Reection means that there is a
coequalizer h of and g and an isomorphism i : V Z such that i j =h.
By preservation, if (X. ) is a Calgebra, then j: X (
C
X) must be an
isomorphism because application of to the arrows , =c and of (8-2) gives
the pair of maps that have the split coequalizer (8-3). By reection, if Y is an object
of W , then c:
C
Y Y must be an isomorphism since if we apply to the
coequalizer (8-2) with X =Y we obtain the split coequalizer (8-3) for the Calgebra
Y . This proves that the adjunction (8-4) is an adjoint equivalence.
Returning to our original map of monads : C , but thinking homotopically,
one might hope that (
C
X) is equivalent to X under reasonable hypotheses.
However, since coequalizers usually behave badly homotopically, we need a homotopi-
cal approximation. Here thinking model categorically seems more of a hindrance than
a help, and we instead use the two-sided monadic bar construction of [25]. It can be
dened in the generality of (8-1) as a functor
B(. C. ): U W .
but we restrict attention to the case of a Calgebra X, where it is
B(. C. X).
We have a simplicial object B
+
(. C. X) in W whose object of qsimplices is
B
q
(. C. X) =C
q
X.
The right action C induces the 0th face map, the map C
i-1
j, 1 i < q,
induces the i th face map, and the action CX X induces the qth face map. The
maps C
i
j, 0 i q induce the degeneracy maps. We need to realize simplicial
objects Y
+
in V and W to objects of V and W , and we need to do so compatibly.
For that, we need a covariant standard simplex functor ^
+
: ^ V , where ^ is
the standard category of nite sets n and monotonic maps.
19
We compose with to
obtain a standard simplex functor ^
+
: ^W . We dene
[X
+
[
V
=X
+

Z
^
+
19
Equivalently, n is the ordered set 0 <1 < <n, and maps preserve order.
Geometry & Topology Monographs, Volume 16 (2009)
248 J P May
for simplicial objects X
+
in V , and similarly for W . In the cases of interest, realization
is a left adjoint. We dene
B(. C. X) =[B
+
(. C. X)[.
Commuting the left adjoint C past realization, we nd that
(8-5) [CX
+
[
V
C[X
+
[
V
and conclude that the realization of a simplicial Calgebra is a Calgebra. The
iterated action map
q1
: C
q1
X X gives a map c
+
from B
+
(C. C. X) to the
constant simplicial object at X. Passing to realizations, c
+
gives a natural map of C
algebras c: B(C. C. X) X. Forgetting the C action, c
+
is a simplicial homotopy
equivalence in the category of simplicial objects in W , in the combinatorial sense that
is dened for simplicial objects in any category. In reasonable situations, for example
categories tensored over spaces, passage to realization converts this to a homotopy
equivalence in W . Commuting coequalizers past realization, we nd
(8-6) B(. C. X)
C
B(C. C. X).
This has the same avor as applying cobrant approximation to X and then apply-
ing
C
(), but the two-sided bar construction has considerable advantages. For
example, in our topological situations, it is a continuous functor, whereas cobrant
approximations generally are not. Also, starting from a general object X, one could not
expect something as strong as an underlying homotopy equivalence from a cobrant
approximation X
t
X. More fundamentally, the functoriality in all three variables is
central to the arguments. Presumably, for good model categories V , if one restricts X
to be cobrant in V , then B(C. C. X) is cobrant in V T | , at least up to homotopy
equivalence, and thus can be viewed as an especially nice substitute for cobrant
approximation, but Ive never gone back to work out such model categorical details.
20
Now the black box works as follows. We take V and W to be categories with a
reasonable homotopy theory, such as model categories. Our candidate for EX is
B(. C. X). There are three steps that are needed to obtain an equivalence between a
suitable Calgebra X and EX. The fundamental one is to prove an approximation
theorem to the effect that C is a homotopical approximation to . This step has
nothing to do with monads, depending only on the homotopical properties of the
comparison map .
20
This exercise has recently been carried out in [1]. It is to be emphasized that nothing in the outline we
are giving simplies in the slightest. Rather, the details sketched here are reinterpreted model-theoretically.
As anticipated, the essential point is to observe that B(C. C. X) is of the homotopy type of a cobrant
object when X is cobrant.
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Step 8.7 For suitable objects X V , : CX X is a weak equivalence.
The second step is a general homotopical property of realization that also has nothing
to do with the monadic framework. It implies that the good homotopical behavior of
is preserved by various maps between bar constructions.
Step 8.8 For suitable simplicial objects Y
+
and Y
t
+
in W , if
+
: Y
+
Y
t
+
is a map
such that each
q
is a weak equivalence, then [
+
[ is a weak equivalence in W , and
similarly for V .
In the space level cases of interest, the weaker property of being a group completion
will generalize being a weak equivalence in Steps 8.7 and 8.8, but we defer discussion
of that until the next section.
The third step is an analogue of commuting past coequalizers in the categorical
argument we are mimicking. It has two parts, one homotopical and the other monadic.
Step 8.9 For suitable simplicial objects Y
+
in W , the canonical natural map
: [Y
+
[
V
[Y
+
[
W
is both a weak equivalence and a map of Calgebras.
Here Y
+
is obtained by applying levelwise to simplicial objects. To construct the
canonical map , we rst obtain a natural isomorphism
(8-10) [X
+
[
V
[X
+
[
W
by commuting left adjoints, where X
+
is a simplicial object in V . Applying this with
X
+
replaced by Y
+
we obtain
[c[
W
: [Y
+
[
V
[Y
+
[
W
[Y
+
[
W
.
Its adjoint is the required natural map : [Y
+
[
V
[Y
+
[
W
.
Assuming that these steps have been taken, the black box works as follows to relate the
homotopy categories of CV | and W . For a Calgebra X in V , we have the diagram
of maps of Calgebras
X B(C. C. X)
t

B(,id,id)

B(. C. X)

B(. C. X) =EX
in which all maps are weak equivalences (or group completions). On the left, we
have a map, but not a C map, j: X B(C. C. X) which is an inverse homotopy
Geometry & Topology Monographs, Volume 16 (2009)
250 J P May
equivalence to c. We also write j for the resulting composite X EX. This is
the analogue of the map j: X (
C
X) in our categorical sketch.
For an object Y in W , observe that
B
q
(. . Y ) =()
q1
Y
and the maps ()
q1
Y obtained by iterating c give a map of simplicial objects
from B
+
(. . Y ) to the constant simplicial object at Y . On passage to realization,
we obtain a composite natural map
EY =B(. C. Y )
B(id,,id)

B(. . Y )
t

Y.
which we also write as c. This is the analogue of c:
C
Y Y in our categorical
sketch. We have the following commutative diagram in which all maps except the top
two are weak equivalences, hence they are too.
EY =B(. C. Y )
B(id,,id)

B(. . Y )
t

Y
B(. C. Y )
B(id,,id)

B(. . Y )

p
p
p
p
p
p
p
p
p
p
p
p
We do not expect to reect weak equivalences in general, so we do not expect
EY .Y in general, but we do expect this on suitably restricted Y . We conclude
that the adjunction (8-4) induces an adjoint equivalence of suitably restricted homotopy
categories.
9 The additive and multiplicative innite loop space machine
The original use of this approach in [25] took (. ) to be (
n
. C
n
) and C to be the
monad associated to the little ncubes operad C
n
. It took to be the map of monads
given by the composites 0 C
n
j: C
n
X C
n

n
X. For connected C
n
spaces X,
details of all steps may be found in [25].
For the nonconnected case, we say that an Hmonoid
21
is grouplike if
0
(X) is a
group under the induced multiplication. We say that a map : X Y between
homotopy commutative Hmonoids is a group completion if Y is grouplike and
two things hold. First,
0
(Y ) is the group completion of
0
(X) in the sense that
any map of monoids from
0
(X) to a group G factors uniquely through a group
21
This is just a convenient abbreviated way of writing homotopy associative Hspace.
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What precisely are E
o
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o
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homomorphism
0
(Y ) G. Second, for any commutative ring of coefcients, or
equivalently any eld of coefcients, the homomorphism
+
: H
+
(X) H
+
(Y ) of
graded commutative rings is a localization (in the classical algebraic sense) at the
submonoid
0
(X) of H
0
(X). That is, H
+
(Y ) is H
+
(X)
0
(X)
-1
| .
For n 2,
n
is a group completion for all C
n
spaces X by calculations of Fred
Cohen [10] or by an argument due to Segal [40]. Thus
n
is a weak equivalence for
all grouplike X. This gives Step 8.7, and Steps 8.8 and 8.9 are dealt with in May [26].
As Gaunce Lewis pointed out to me many years ago and Ando et al [1] rediscovered, in
the stable case n =o we can compare spaces and spectra directly. We take V =T
and W = S , we take (. ) to be (
o
. C
o
), and we take C to be the monad
associated to the Steiner operad (for U =R
o
); in the additive theory, we could equally
well use the innite little cubes operad C
o
. As recalled in (7-3), we have a map of
monads : C Q. The calculations needed to prove that is a group completion
preceded those in the deeper case of nite n [10], and we have Step 8.7. Step 8.8
is given for spaces in [25, Chapter 11] and [26, Appendix] and for spectra in [13,
Chapter X]; see [13, X.1.3 and X.2.4].
We need to say a little about Step 8.9. For simplicial spaces X
+
such that each X
q
is
connected (which holds when we apply this step) and which satisfy the usual (Reedy)
cobrancy condition (which follows in our examples from the assumed nondegeneracy
of basepoints), the map : [CX
+
[ C[X
+
[ is a weak equivalence by [25, 12.3]. That
is the hardest thing in [25]. Moreover, the nfold iterate
n
: [C
n
X
+
[ C
n
[X
+
[ is
a map of C
n
spaces by [25, 12.4]. The latter argument works equally well with C
n
replaced by the Steiner operad K
R
n , so we have Step 8.9 for each C
n
. For sufciently
nice simplicial spectra E
+
, such as those relevant here, the canonical map
(9-1) : [C
o
E
+
[
T
C
o
[E
+
[
S
can be identied with the colimit of the iterated canonical maps
(9-2)
n
: [C
n
(E
n
)
+
[
T
C
n
[(E
n
)
+
[
T
.
and Step 8.9 for C
o
follows by passage to colimits from Step 8.9 for the C
n
, applied
to simplicial (n1)connected spaces. Here, for simplicity of comparison with [25],
we have indexed spectra sequentially, that is on the conal sequence R
n
in U . The nth
spaces (E
n
)
+
of the simplicial spectrum E
+
give a simplicial space and the C
n
(E
n
)
+
are compatibly isomorphic to (E
0
)
+
. Thus, on the left side, the colimit is [C
o
E
+
[
T
.
When E
+
= LT
+
for a simplicial inclusion prespectrum T
+
, the right side can be
computed as L[T
+
[
P
, where the prespectrum level realization is dened levelwise.
One checks that [T
+
[
P
is again an inclusion prespectrum, and the identication of the
colimit on the right with C
o
[E
+
[
S
follows.
Geometry & Topology Monographs, Volume 16 (2009)
252 J P May
Granting these details, we have the following additive innite loop space machine.
Recall that a spectrum is connective if its negative homotopy groups are zero and that
a map of connective spectra is a weak equivalence if and only if C
o
is a weak
equivalence.
Theorem 9.3 For a Cspace, dene EX =B(
o
. C. X). Then EX is connective
and there is a natural diagram of maps of Cspaces
X B(C. C. X)
t

B(,id,id)

B(Q. C. X)

C
o
EX
in which c is a homotopy equivalence with natural homotopy inverse j, is a weak
equivalence, and B(. id. id) is a group completion. The composite j: X C
o
EX
is therefore a group completion and thus a weak equivalence if X is grouplike. For a
spectrum Y , there is a composite natural map of spectra
c: EC
o
Y
B(id,,id)

B(
o
. Q. C
o
Y )
t

Y.
and the induced maps of Cspaces
C
o
c: C
o
EC
o
Y
D
1
B(id,,id)

C
o
B(
o
. Q. C
o
Y )
D
1
t

C
o
Y
are weak equivalences. Therefore E and C
o
induce an adjoint equivalence between the
homotopy category of grouplike E
o
spaces and the homotopy category of connective
spectra.
The previous theorem refers only to the Steiner operad C , for canonicity, but we can
apply it equally well to any other E
o
operad O. We can form the product operad
P =C O, and the j th levels of its projections
1
: P C and
2
: P O
are
j
equivariant homotopy equivalences. While the monad P associated to P is
not a product, the induced projections of monads P C and P O are natural
weak equivalences. This allows us to replace C by P in the previous theorem. If X is
an Ospace, then it is a Pspace by pullback along
2
, and
o
is a right Pfunctor
by pullback along
1
.
There is another way to think about this trick that I now nd preferable. Instead of
repeating Theorem 9.3 with C replaced by P, one can rst change input data and
then apply Theorem 9.3 as it stands. Here we again use the two-sided bar construction.
For Ospaces X regarded by pullback as Pspaces, we have a pair of natural weak
equivalences of Pspaces
(9-4) X B(P. P. X)
t

B(t
1
,id,id)

B(C. P. X).
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What precisely are E
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o
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where B(C. P. X) is a C space regarded as a Pspace by pullback along
1
. The
same maps show that if X is a Pspace, then it is weakly equivalent as a Pspace to
B(C. P. X). Thus the categories of Ospaces and Pspaces can be used interchange-
ably. Reversing the roles of C and O, the categories of Cspaces and Pspaces can
also be used interchangeably. We conclude that Ospaces for any E
o
operad O can
be used as input to the additive innite loop space machine. We have the following
conclusion.
Corollary 9.5 For any E
o
operad O, the additive innite loop space machine E
and the 0th space functor C
o
induce an adjoint equivalence between the homotopy
category of grouplike Ospaces and the homotopy category of connective spectra.
In particular, many of the interesting examples are Lspaces. We can apply the
additive innite loop space machine to them, ignoring the special role of L in the
multiplicative theory. As we recall in the second sequel [33], examples include various
stable classifying spaces and homogeneous spaces of geometric interest. By Corollary
7.7, they also include the unit spaces GL
1
R and SL
1
R of an E
o
ring spectrum
R. The importance of these spaces in geometric topology is explained in [33]. The
following denitions and results highlight their importance in stable homotopy theory
and play a signicant role in [1].
22
We start with a reinterpretation of the adjunction of
Corollary 7.2 for grouplike Lspaces Y .
Lemma 9.6 If Y is a grouplike L space, then
LS|(
o

Y. R) LU |(Y. GL
1
R)
Proof A map of Lspaces Y C
o
R must take values in GL
1
R since the group

0
Y must map to the group of units of the ring
0
C
o
R.
The notations of the following denition have recently become standard, although the
denition itself dates back to [27].
Denition 9.7 Let R be an L spectrum. Using the operad L in the additive innite
loop space machine, dene gl
1
R and sl
1
R to be the spectra obtained from the L
spaces GL
1
R and SL
1
R, so that C
o
gl
1
R.GL
1
R and C
o
sl
1
R.SL
1
R.
Corollary 9.8 On homotopy categories, the functor gl
1
from E
o
ring spectra to
spectra is right adjoint to the functor
o

C
o
from spectra to E
o
ring spectra.
22
That paper reads in part like a sequel to this one. However, aside from a very brief remark that
merely acknowledges their existence, E
o
ring spaces are deliberately avoided there.
Geometry & Topology Monographs, Volume 16 (2009)
254 J P May
Proof Here we implicitly replace the C space C
o
R by a weakly equivalent L
space, as above. Using this replacement, we can view the functor
o

C
o
as taking
values in L spectra. Now the conclusion is obtained by composing the equivalence
of Corollary 9.5 with the adjunction of Lemma 9.6 and using that, in the homotopy
category, maps from a spectrum E to a connective spectrum F, such as gl
1
R, are the
same as maps from the connective cover cE of E into F, while C
o
cE .C
o
E.
Note that we replaced L by C to dene GL
1
R as a C space valued functor before
applying E, and we replaced C by L to dene C
o
as an L space valued functor.
Another important example of an E
o
operad should also be mentioned.
Remark 9.9 There is a categorical operad, denoted D, that is obtained by applying the
classifying space functor to the translation categories of the groups
j
. This operad acts
on the classifying spaces of permutative categories, as we recall from [26, Section 4] in
[32]. Another construction of the same E
o
operad is obtained by applying a certain
product-preserving functor from spaces to contractible spaces to the operad M that
denes monoids; see [25, page 161] and [26, 4.8]. The second construction shows
that M is a suboperad, so that a Dspace has a canonical product that makes it a
topological monoid. The simplicial version of D is called the BarrattEccles operad in
view of Barratt and Eccles use of it in [2; 3; 4].
Remark 9.10 In the applications, one often uses the consistency statement that, for an
E
o
space X, there is a natural map of spectra ECX EX that is an equivalence
if X is connected. This is proven in [25, Section 14], with improvements in [26,
Section 3] and [27, VI.3.4]. The result is considerably less obvious than it seems, and
the cited proofs are rather impenetrable, even to me. I found a considerably simpler
conceptual proof while writing this paper. Since this is irrelevant to our multiplicative
story, Ill avoid interrupting the ow here by deferring the new proof to Appendix B.
Now we add in the multiplicative structure, and we nd that it is startlingly easy to
do so. Let us say that a rig space X is ringlike if it is grouplike under its additive
Hmonoid structure. A map of rig spaces : X Y is a ring completion if Y is
ringlike and is a group completion of the additive structure. Replacing T and S
by L

T | and L

S| and using that : C Q is a map of monads on L

T | ,
the formal structure of the previous section still applies verbatim, and the homotopy
properties depend only on the additive structure. The only point that needs mentioning
is that, for the monadic part of Step 8.9, we now identify the map of (9-1) with the
colimit over V of the maps
(9-11)
V
: [C
V
(EV )
+
[
T
C
V
[(EV )
+
[
T
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What precisely are E
o
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o
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and see that is a map of L

spaces because of the naturality of the colimit system

(9-11) with respect to linear isometries. Therefore the additive innite loop space
machine specializes to a multiplicative innite loop space machine.
Theorem 9.12 For a (C. L)space X, dene EX =B(
o
. C. X). Then EX is a
connective L spectrum and all maps in the diagram
X B(C. C. X)
t

B(,id,id)

B(Q. C. X)

C
o
EX
of the additive innite loop space machine are maps of (C. L)spaces. Therefore the
composite j: X C
o
EX is a ring completion. For an L spectrum R, the maps
c: EC
o
R
B(id,,id)

B(
o
. Q. C
o
R)
t

R
are maps of L spectra and the maps
C
o
c: C
o
EC
o
R
D
1
B(id,,id)

C
o
B(
o
. Q. C
o
R)
D
1
t

C
o
R
are maps of (C. L)spaces. Therefore E and C
o
induce an adjoint equivalence
between the homotopy category of ringlike E
o
ring spaces and the homotopy category
of connective E
o
ring spectra.
Again, we emphasize how simply and naturally these structures t together. However,
here we face an embarrassment. We would like to apply this machine to construct new
E
o
ring spectra, and the problem is that the only operad pairs we have in sight are
(C. L) and (N . N ). We could apply the product of operads trick to operad pairs if
we only had examples to which to apply it. We return to this point in the sequel [32],
where we show how to convert such naturally occurring data as bipermutative categories
to E
o
ring spaces, but the theory in this paper is independent of that problem.
10 Localizations of the special unit spectrum sl
1
R
The BarrattPriddyQuillen theorem tells us how to construct the sphere spectrum from
symmetric groups. This result is built into the additive innite loop space machine.
A multiplicative elaboration is also built into the innite loop space machine, as
we explain here. For a C space or (C. L)space X, we abbreviate notation by
writing IX =C
o
EX, using notations like I
1
X to indicate components. We write
j: X IX for the group completion map of Theorem 9.3.
23
Since it is the composite
23
The letter I is chosen as a reminder of the group completion property.
Geometry & Topology Monographs, Volume 16 (2009)
256 J P May
of E
o
maps or, multiplicatively, E
o
ring maps and the homotopy inverse of such a
map, we may think of it as an E
o
or E
o
ring map.
Theorem 10.1 For a based space Y , and the left map j are group completions and
I and the right map j are equivalences in the commutative diagram
CY

QY
)

ICY
I

IQY
If Y is an L

space, then this is a diagram of E

o
ring spaces.
Replacing Y by Y

, we see by inspection that C(Y

) is the disjoint union over j 0

of the spaces C(j )

j
Y
j
, and of course C(j ) is a model for E
j
. When Y =BG
for a topological group G, these are classifying spaces B(
j
R
G). When Y =+ and
thus Y

=S
0
, they are classifying spaces B
j
, and we see that the 0th space of the
sphere spectrum is the group completion of the Hmonoid l
j0
B
j
. This is one
version of the BarrattQuillen theorem.
For an E
o
space X with a map S
0
X, there is a natural map of monoids from
the additive monoid Z
0
of nonnegative integers to the monoid
0
(X). It is obtained
by passage to
0
from the composite CS
0
CX X. We assume that it is a
monomorphism, as holds in the interesting cases. Write X
m
for the mth component.
Translation by an element in X
n
(using the Hspace structure induced by the operad
action) induces a map n: X
m
X
mn
. We have the homotopy commutative ladder
X
0
1

X
1

)

X
n-1
)

X
n
)

I
0
X
1

I
1
X



I
n-1
X
1

I
n
X


Write

X for the telescope of the top row. The maps on the bottom row are homotopy
equivalences, so the ladder induces a map j:

X I
0
X. Since j is a group com-
pletion, j induces an isomorphism on homology. Taking X =CS
0
, it follows that
j: B
o
Q
0
S
0
is a homology isomorphism and therefore that Q
0
S
0
is the plus
construction on B
o
. This is another version of the BarrattQuillen theorem.
We describe a multiplicative analogue of this argument and result, due to Tornehave
in the case of QS
0
and generalized in [27, VII.5.3], where full details may be found.
Recall that we write gl
1
R and sl
1
R for the spectra EGL
1
R and ESL
1
R that the
Geometry & Topology Monographs, Volume 16 (2009)
What precisely are E
o
ring spaces and E
o
ring spectra? 257
black box associates to the E
o
spaces SL
1
R GL
1
R, where R is an E
o
ring
spectrum. The map sl
1
R gl
1
R is a connected cover. It is usually not an easy
matter to identify sl
1
R explicitly. The cited result gives a general step in this direction.
The point to emphasize is that the result intrinsically concerns the relationship between
the additive and multiplicative E
o
space structures on an E
o
ring space. Even if
ones focus is solely on understanding the spectrum sl
1
R associated to the E
o
ring
spectra R, one cannot see a result like this without introducing E
o
ring spaces.
As in [27, Section VII.5], we start with an E
o
ring space X and we assume that
the canonical map of rigs from the rig Z
0
of nonnegative integers to
0
(X) is a
monomorphism. When X = C
o
R, EX is equivalent as an E
o
ring spectrum to
the connective cover of R and GL
1
E(X) is equivalent as an E
o
space to GL
1
R.
The general case is especially interesting when X is the classifying space BA of a
bipermutative category A (as dened in [27, Section VI.3]; see the sequel [32]).
Let M be a multiplicative submonoid of Z
0
that does not contain zero. For example,
M might be {p
i
] for a prime p, or it might be the set of positive integers prime to
p. Let Z
M
=ZM
-1
| denote the localization of Z at M; thus Z
M
=Zp
-1
| in our
rst example and Z
M
=Z
(p)
in the second. Let X
M
denote the disjoint union of the
components X
m
with m M. Often, especially when X = BA , we have a good
understanding of X
M
.
Clearly X
M
is a sub-L space of X. Converting it to a C space and applying the
additive innite loop space machine or, equivalently, applying the additive innite loop
space machine constructed starting with C L , we obtain a connective spectrum
E(X
M
) = E(X
M
. ). The alternative notation highlights that the spectrum comes
from the multiplicative operad action on X. This gives us the innite loop space
I
1
(X
M
. ) =C
o
1
E(X
M
. ), which depends only on X
M
.
We shall relate this to SL
1
E(X) =C
o
1
E(X. 0). The alternative notation highlights
that E(X. 0) is constructed from the additive operad action on X and has multiplicative
structure inherited from the multiplicative operad action.
A key example to have in mind is X = BGL(R), where GL(R) is the general
linear bipermutative category of a commutative ring R; its objects are the n 0 and
its morphisms are the general linear groups GL(n. R). In that case EX = KR is
the algebraic Ktheory E
o
ring spectrum of R. The construction still makes sense
when R is a topological ring. We can take R=R or R=C, and we can restrict to
orthogonal or unitary matrices without changing the homotopy type. Then KR=kO
and KC =kU are the real and complex connective topological Ktheory spectra with
special linear spaces BO

and BU

.
Geometry & Topology Monographs, Volume 16 (2009)
258 J P May
To establish the desired relation between SL
1
E(X) and I
1
(X
M
. ), we need a mild
homological hypothesis on X, namely that X is convergent at M. It always holds
when X is ringlike, when X =CY for an L

space Y , and when X =BA for the

usual bipermutative categories A ; see [27, VII.5.2]. We specify it during the sketch
proof of the following result.
Theorem 10.2 If X is convergent at M, then, as an E
o
space, the localization of
C
o
1
E(X. 0) at M is equivalent to the basepoint component C
o
1
E(X
M
. ).
Thus, although E(X. 0) is constructed using the C space structure on X, the local-
izations of sl
1
E(X) depend only on X as an L space. When X =C
o
R, E(X. 0)
is equivalent to the connective cover of R and sl
1
EX is equivalent to sl
1
R.
Corollary 10.3 For an E
o
ring spectrum R, the localization (sl
1
R)
M
is equivalent
to the connected cover of E((C
o
R)
M
. ).
Sketch proof of Theorem 10.2 We repeat the key diagram from [27, page 196]:
I(X
1
. )
Ii

I)

I(X
M
. )
I)

X
1
i

k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
X
M
)

j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
I(I
1
(X. 0). )
Ii

I(I
M
(X. 0). )
I
1
(X. 0)
i

k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
I
M
(X. 0)
)

j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
Figure 2: Diagram of Lspaces and homotopy inverses of equivalences that
are maps of Lspaces
Remember that IX =C
o
EX; I
1
X and I
M
X denote the component of 1 and the
disjoint union of the components I
m
X for mM. We have distinguished applications
of the innite loop space machine with respect to actions 0 of C and of L , and
we write j

and j

for the corresponding group completions. The letter i always

denotes an inclusion. Figure 2 is a diagram of Lspaces and homotopy inverses of
Geometry & Topology Monographs, Volume 16 (2009)
What precisely are E
o
ring spaces and E
o
ring spectra? 259
equivalences that are maps of Lspaces. Note that the spaces on the left face are
connected, so that their images on the right face lie in the respective components of 1.
There are two steps.
(i) If X is ringlike, then the composite Ii j

: X
1
I
1
(X
M
. ) on the top face
is a localization of X
1
at M.
(ii) If X is convergent, then the vertical map Ij

labelled . at the top right is a

weak equivalence.
Applying (i) to the bottom face, as we may, we see that the (zigzag) composite from
I
1
(X. 0) at the bottom left to I
1
(X
M
. ) at the top right is a localization at M.
To prove (i) and (ii), write the elements of M in increasing order, {1. m
1
. m
2
. . . .], let
n
i
=m
1
m
i
, and consider the following homotopy commutative ladder.
X
1
m
1

X
n
1

X
n
i1
m
i

X
n
i
)

I
1
(X
M
. )
m
1

I
n
1
(X
M
. )



I
n
i1
(X
M
. )
m
i

I
n
i
(X
M
. )


Here the translations by m
i
mean multiplication (using the Hspace structure induced
by the action ) by an element of X
m
i
.
Let

X
M
be the telescope of the top row. Take homology with coefcients in a commu-
tative ring. The m
i
in the bottom row are equivalences since I(X
M
. ) is grouplike,
so the diagram gives a map

X I
1
(X
M
. ). The homological denition of a group
completion, applied to j

, implies that this map is a homology isomorphism. Note that

the leftmost arrow j

factors through I(X

1
. ) =I
1
(X
1
. ). Exploiting formulas that
relate the additive and multiplicative Pontryagin products on H
+
(X), we can check
that

H
+
(

X: F
p
) =0 if p divides an element m M. The point is that m is the sum
of m copies of 1, and there is a distributivity formula for x m in terms of the additive
Hspace structure +. This implies that the space I
1
(X
M
. ) is Mlocal.
For (i), write +(n): X
n
X
0
for the equivalence given by using + to send x to
x +y
-n
for a point y
-n
X
-n
. Formulas in the denition of a (C. L)space imply
that the following ladder is homotopy commutative.
X
1
m
1

+(-1)

X
n
1

+(-n
1
)

X
n
i1
m
i

+(-n
i1
)

X
n
i
+(-n
i
)

X
0
m
1

X
0



X
0
m
i

X
0


Geometry & Topology Monographs, Volume 16 (2009)
260 J P May
A standard construction of localizations of Hspaces gives that the telescope of the
bottom row is a localization X
0
(X
0
)M
-1
| , hence so is the telescope X
1

X
M
of the top row, hence so is its composite with

X
M
I
1
(X
M
. ). This proves (i).
For (ii), we consider the additive analogue of our rst ladder:
X
1
m
1

X
n
1

X
n
i1
m
i

X
n
i
)

I
1
(X. 0)
m
1

I
n
1
(X. 0)



I
n
i1
(X. 0)
m
i

I
n
i
(X. 0)


We say that X is convergent at M if, for each prime p which does not divide any
element of M, there is an eventually increasing sequence n
i
(p) such that
(j

)
+
: H
j
(X
i
: F
p
) H
j
(I
i
(X. 0): F
p
)
is an isomorphism for all j n
i
(p). With this condition, the induced map of telescopes
is a mod p homology isomorphism for such primes p. This implies the same statement
for the map
I
1
j

: I
1
(X
M
. ) I
1
(I
M
(X. 0). ).
Since this is a map between Mlocal spaces, it is an equivalence. This proves (ii) on
components of 1, and it follows on other components.
For a general example, consider CY for an L

space Y .
Corollary 10.4 There is a natural commutative diagram of E
o
spaces
I
1
(CY. 0)

I
1

I
1
(QY. 0)

Q
1
Y
)

I
1
(C
M
Y. )
I
1

I
1
(Q
M
Y. ) I
1
(Q
M
Y. )
in which the horizontal arrows are weak equivalences and the vertical arrows are
localizations at M.
Now specialize to the case Y = S
0
. Then Q
1
S
0
, the unit component of the 0th
space of the sphere spectrum, is the space SL
1
S =SF of degree 1 stable homotopy
equivalences of spheres. We see that its localization at M is the innite loop space
constructed from the L space C
M
S
0
. The latter space is the disjoint union of the
EilenbergMac Lane spaces C(m),
m
= K(
m
. 1), given an E
o
space structure
that realizes the products
m

mn
determined by lexicographically ordering
Geometry & Topology Monographs, Volume 16 (2009)
What precisely are E
o
ring spaces and E
o
ring spectra? 261
the products of sets of m and n elements for m. n M. Thus the localizations of
SF can be recovered from symmetric groups in a way that captures their innite loop
structures.
11 E
1
ring spectra and commutative Salgebras
Jumping ahead over twenty years, we here review the basic denitions of Elmendorf
et al [13], leaving all details to that source. However, to establish context, let us rst
recall the following result of Lewis [17].
Theorem 11.1 Let S be any category that is enriched in based topological spaces
and satises the following three properties.
(i) S is closed symmetric monoidal under continuous smash product and function
spectra functors . and F that satisfy the topological adjunction
S(E.E
t
. E
tt
) S(E. F(E
t
. E
tt
)).
(ii) There are continuous functors
o
and C
o
between spaces and spectra that
satisfy the topological adjunction
S(
o
X. E) T (X. C
o
E).
(iii) The unit for the smash product in S is S
o
S
0
.
Then, for any commutative monoid R in S , such as S itself, the component SL
1
(R)
of the identity element in C
o
R is a product of EilenbergMac Lane spaces.
Proof The enrichment of the adjunctions means that the displayed isomorphisms
are homeomorphisms. By [17, 3.4], the hypotheses imply that S is tensored over
T . In turn, by [17, 3.2], this implies that the functor C
o
is lax symmetric monoidal
with respect to the unit j: S
0
C
o

o
S
0
=C
o
S of the adjunction and a natural
transformation
: C
o
D.C
o
E C
o
(D.E).
Now let D =E =R with product j and unit j: S R. The adjoint of j is a map
S
0
C
o
R, and we let 1 C
o
R be the image of 1 S
0
. The composite
C
o
EC
o
R

C
o
R.C
o
R

C
o
(R.R)
D
1

C
o
R
gives C
o
R a structure of commutative topological monoid with unit 1. Restricting to
the component SL
1
R of 1, we have a connected commutative topological monoid,
and Moores theorem (eg [25, 3.6]) gives the conclusion.
Geometry & Topology Monographs, Volume 16 (2009)
262 J P May
As Lewis goes on to say, if C
o

o
X is homeomorphic under X to QX, as we have
seen holds for the category S of (LMS) spectra, and if (i)(iii) hold, then we can
conclude in particular that SF =SL
1
S is a product of EilenbergMac Lane spaces,
which is false. We interpolate a model theoretic variant of this contradiction.
Remark 11.2 The sphere spectrum S is a commutative ring spectrum in any symmet-
ric monoidal category of spectra S with unit object S. Suppose that S is cobrant
in some model structure on S whose homotopy category is equivalent to the stable
homotopy category and whose brant objects are Cprespectra. More precisely, we
require an underlying prespectrum functor U: S P such that UE is an C
prespectrum if E is brant, and we also require the resulting 0th space functor U
0
to
be lax symmetric monoidal. Then we cannot construct a model category of commutative
ring spectra by letting the weak equivalences and brations be the maps that are weak
equivalences and brations in S . If we could, a brant approximation of S as a
commutative ring spectrum would be an Cspectrum whose 0th space is equivalent to
QS
0
. Its 1component would be a connected commutative monoid equivalent to SF.
All good modern categories of spectra satisfy (i) and (iii) (or their simplicially enriched
analogues) and therefore cannot satisfy (ii). However, as our summary so far should
make clear, one must not let go of (ii) lightly. One needs something like it to avoid
severing the relationship between spectrum and space level homotopy theory. Our
summaries of modern denitions will focus on the relationship between spectra and
spaces. Since we are now switching towards a focus on stable homotopy theory, we
start to keep track of model structures. Returning to our xed category S of (LMS)
spectra, we shall describe a sequence of Quillen equivalences, in which the right adjoints
labelled are both inclusions of subcategories.
76 54
01 23
P
L

76 54
01 23
S
L

?> =<
89 :;
LS|
S.
L
(-)

?> =<
89 :;
M
S
F
L
(S,-)

The category P of prespectra has a level model structure whose weak equivalences
and brations are dened levelwise, and it has a stable model structure whose weak
Geometry & Topology Monographs, Volume 16 (2009)
What precisely are E
o
ring spaces and E
o
ring spectra? 263
equivalences are the maps that induce isomorphisms of (stabilized) homotopy groups
and whose cobrations are the level cobrations; its brant objects are the Cprespectra.
The category S of spectra is a model category whose level model structure and
stable model structures coincide. That is, the weak equivalences and brations are
dened levelwise, and these are already the correct stable weak equivalences because
the colimits that dene the homotopy groups of a spectrum run over a system of
isomorphisms. The spectrication functor L and inclusion give a Quillen equivalence
between P and S .
Of course, S satises (ii) but not (i) and (iii). We take the main step towards the latter
properties by introducing the category LS| of Lspectra.
The space L(1) is a monoid under composition, and we have the notion of an action
of L(1) on a spectrum E. It is given by a map
: LE =L(1) E E
that is unital and associative in the evident sense. Since L(1) is contractible, the unit
condition implies that must be a weak equivalence. Moreover, LE is an Lspectrum
for any spectrum E, and the action map : LE E is a map of Lspectra for any
Lspectrum E. The inclusion : LS| S forgets the action maps. It is right
adjoint to the free Lspectrum functor L: S LS| . Dene the weak equivalences
and brations of Lspectra to be the maps such that is a weak equivalence or
bration. Then LS| is a model category and (L. ) is a Quillen equivalence between
S and LS| . Indeed, the unit j: E LE and counit : LE E of the
adjunction are weak equivalences, and every object in both categories is brant, a very
convenient property.
Using the untwisting isomorphism (6-3) and the projection L(1)

S
0
, we see that
the spectra
o
X are naturally Lspectra. However S =
o
S
0
, which is cobrant
in S , is not cobrant in LS| ; rather, LS is a cobrant approximation.
We have a commutative and associative
24
smash product E.
L
E
t
in LS| ; we write
t for the commutativity isomorphism E.
L
E
t
E
t
.
L
E. The smash product is
dened as a coequalizer L(2)
L(1)L(1)
E.E
t
, but we refer the reader to [13] for
details. There is a natural unit map z: S .
L
E E. It is a weak equivalence for all
Lspectra E, but it is not in general an isomorphism. That is, S is only a weak unit.
Moreover, there is a natural isomorphism of Lspectra

o
(X .Y )
o
X .
L

o
Y.
24
This crucial property is a consequence of a remarkable motivating observation, due to Mike Hopkins,
about special properties of the structure maps of the linear isometries operad.
Geometry & Topology Monographs, Volume 16 (2009)
264 J P May
Note, however, that the (
o
. C
o
) adjunction must now change. We may reasonably
continue to write C
o
for the 0th space functor C
o
, but its left adjoint is now the
composite L
o
.
While Lewis contradictory desiderata do not hold, we are not too far off since we still
have a sensible 0th space functor. We are also very close to a description of E
o
ring
spectra as commutative monoids in a symmetric monoidal category.
Denition 11.3 A commutative monoid in LS| is an Lspectrum R with a unit
map j: S R and a commutative and associative product : R.
L
RR such
that the following unit diagram is commutative
S .
L
R
).id

M
M
M
M
M
M
M
M
M
M
M
R.
L
R

R.
L
S
id .)

zr
.q
q
q
q
q
q
q
q
q
q
q
R
The only difference from an honest commutative monoid is that the diagonal unit
arrows are weak equivalences rather than isomorphisms. Thinking of the unit maps
e: S R of E
o
ring spectra as preassigned, we can specify a product + in the
category LS|
e
of Lspectra under S that gives that category a symmetric monoidal
structure, and then an E
o
ring spectrum is an honest commutative monoid in that
category [13, XIII.1.16]. We prefer to keep to the perspective of Denition 11.3, and
[13, II.4.6] gives the following result.
Theorem 11.4 The category of commutative monoids in LS| is isomorphic to the
category of E
o
ring spectra.
Obviously, we have not lost the connection with E
o
ring spaces. Since we are used to
working in symmetric monoidal categories and want to work in a category of spectra
rather than of spectra under S, we take one further step. The unit map z: S.
L
EE
is often an isomorphism. This holds when E =
o
X and when E =S .
L
E
t
for
another Lspectrum E
t
. We dene an Smodule to be an Lspectrum E for which
z is an isomorphism, and we let M
S
be the category of Smodules. It is symmetric
monoidal with unit S under the smash product E.
S
E
t
=E.
L
E
t
that is inherited
from LS| , and it also inherits a natural isomorphism of Smodules

o
(X .Y )
o
X .
S

o
Y.
Commutative monoids in M
S
are called commutative Salgebras. They are those
commutative monoids in LS| whose unit maps are isomorphisms. Thus they are espe-
cially nice E
o
ring spectra. For any E
o
ring spectrum R, S .
L
R is a commutative
Geometry & Topology Monographs, Volume 16 (2009)
What precisely are E
o
ring spaces and E
o
ring spectra? 265
Salgebra and the unit equivalence S .
L
RR is a map of E
o
ring spectra. Thus
there is no real loss of generality in restricting attention to the commutative Salgebras.
Their 0th spaces are still E
o
ring spaces.
However the 0th space functor C
o
: M
S
T is not a right adjoint. The functor
S .
L
(): LS| M
S
is right adjoint to the inclusion : M
S
LS| , and it
has a right adjoint F
L
(S. ). Thus, for D LS| and E M
S
, we have
M
S
(E. S .
L
D) LS|(E. D)
and
M
S
(S .
L
D. E) LS|(D. F
L
(S. E)).
Letting the weak equivalences and brations in M
S
be created by F
L
(S. ), the
second adjunction gives a Quillen equivalence between LS| and M
S
. Since there
is a natural weak equivalence

z: E F
L
(S. E), the weak equivalences, but not
the brations, are also created by . On the 0th space level,

z induces a natural weak
equivalence
C
o
E .C
o
F
L
(S. E).
We conclude that we have lost no 0th space information beyond that which would lead
to a contradiction to Theorem 11.1 in our passage from S to M
S
.
As explained in [13, Section II.2], there is actually a mirror image category M
S
that
is equivalent to M
S
and whose 0th space functor is equivalent, rather than just weakly
equivalent, to the right adjoint of a functor S M
S
. It is the subcategory of
objects in LS| whose counit maps

z: E F
L
(S. E) are isomorphisms. It has
adjunctions that are mirror image to those of M
S
, switching left and right. Writing
r: M
S
LS| for the inclusion and taking D LS| and E M
S
, we have
M
S
(F
L
(S. D). E) LS|(D. rE)
and
M
S
(E. F
L
(S. D)) LS|((S .
L
rE). D).
The following theorem from [13, Section VII.4] is more central to our story and should
be compared with Remark 11.2.
Theorem 11.5 The category of E
o
ring spectra is a Quillen model category with
brations and weak equivalences created by the forgetful functor to LS| . The category
of commutative Salgebras is a Quillen model category with brations and weak
equivalences created by the forgetful functor to M
S
.
Geometry & Topology Monographs, Volume 16 (2009)
266 J P May
12 The comparison with commutative diagram ring spectra
For purposes of comparison and to give some completeness to this survey, we copy the
following schematic diagram of Quillen equivalences
25
from [23].
'& %$
! "#
P
P

C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
P
{
{
{
{
{
{
{
{
{
{
{
{
{
{
{
{
{
'& %$
! "#
S
P

Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
U

{
{
{
{
{
{
{
{
{
{
{
{
{
{
{
{
{
'& %$
! "#
IS
U

U
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
'& %$
! "#
FT
P
'& %$
! "#
W T
U

U
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
We have a lexicon:
(i) P is the category of N spectra, or (coordinatized) prespectra.
(ii) S is the category of spectra, or symmetric spectra.
(iii) IS is the category of Ispectra, or orthogonal spectra.
(iv) FT is the category of Fspaces, or Ispaces.
(v) W T is the category of Wspaces.
These categories all start with some small (topological) category D and the category
DT of (continuous) covariant functors D T , which are called Dspaces. The
domain categories have inclusions among them, as indicated in the following diagram
of domain categories D.
'& %$
! "#
N

C
C
C
C
C
C
C
C
{
{
{
{
{
{
{
{
'& %$
! "#

Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
'& %$
! "#
I

'& %$
! "#
F

'& %$
! "#
W
25
There is a caveat in that FT only models connective spectra.
Geometry & Topology Monographs, Volume 16 (2009)
What precisely are E
o
ring spaces and E
o
ring spectra? 267
To go from Dspaces to Dspectra, we start with a sphere space functor S: D T
with smash products. It makes sense to dene a module over S, and the Smodules
are the Dspectra. Alternatively but equivalently, one can use S to build a new
(topological) domain category D
S
such that a D
S
space is a Dspectrum. Either way,
we obtain the category DS of Dspectra. When D = F or D = W , there is no
distinction between Dspaces and Dspectra and DT =DS .
In the previous diagram, N is the category of nonnegative integers, is the category
of symmetric groups, I is the category of linear isometric isomorphisms as before, F
is the category of nite based sets, which is the opposite category of Segals category I,
and W is the category of based spaces that are homeomorphic to nite CW complexes.
The functors U in the rst diagram are forgetful functors associated to these inclusions
of domain categories, and the functors P are prolongation functors left adjoint to the U.
All of these categories except P are symmetric monoidal. The reason is that the functor
S: D T is symmetric monoidal in the other cases, but not in the case of N . The
functors U between symmetric monoidal categories are lax symmetric monoidal, the
functors P between symmetric monoidal categories are strong symmetric monoidal,
and the functors P and U restrict to adjoint pairs relating the various categories of
rings, commutative rings, and modules over rings.
We are working with spaces but, except that orthogonal spectra should be omitted, we
have an analogous diagram of categories of spectra that are based on simplicial sets
(see Bouseld and Friedlander [9], Hovey, Shipley and Smith [15], Lydakis [20; 19]
and Schwede [37]). That diagram compares to ours via the usual adjunction between
simplicial sets and topological spaces. Each of these categories of spectra has intrinsic
interest, and they have various advantages and disadvantages. We focus implicitly
on symmetric and orthogonal spectra in what follows; up to a point, Wspaces and
Fspaces work similarly. Full details are in Mandell et al [23] and the references just
cited.
We emphasize that no nontrivial symmetric (or orthogonal) spectrum E can also be
an LMS spectrum. If it were, its 0th space E
0
, with trivial
2
action, would be
homeomorphic as a
2
space to the nontrivial
2
space C
2
E
2
.
We recall briey how smash products are dened in diagram categories. There are two
equivalent ways. Fix a symmetric monoidal domain category D with product denoted
. For Dspaces T and T
t
, there is an external smash product T ZT
t
, which is a
D Dspace. It is specied by
(T ZT
t
)(d. e) =Td .T
t
e.
Geometry & Topology Monographs, Volume 16 (2009)
268 J P May
Applying left Kan extension along , one obtains a Dspace T .T
t
. This construction
is characterized by an adjunction
((D D)T )(T ZT
t
. V ) DT (T .T
t
. V )
for Dspaces V . When T and T
t
are Smodules, one can construct a tensor product
T .
S
T
t
by mimicking the coequalizer description of the tensor product of modules
over a commutative ring. That gives the required internal smash product of Smodules.
Alternatively and equivalently, one can observe that D
S
is a symmetric monoidal
category when S: D T is a symmetric monoidal functor, and one can then apply
left Kan extension directly, with D replaced by D
S
. Either way, DS becomes a
symmetric monoidal category with unit S.
In view of the use of left Kan extension, monoids R in DS have an external equivalent
dened in terms of maps R(d) .R(e) R(d e). These are called DFSPs. As
we have already recalled, Thom spectra give naturally occurring examples of IFSPs.
We also recall briey how the model structures are dened. We begin with the evident
level model structure. Its weak equivalences and brations are dened levelwise. We
then dene stable weak equivalences and use them and the cobrations of the level
model structure to construct the stable model structure. The resulting brant objects
are the Cspectra. In all of these categories except that of symmetric spectra, the
stable weak equivalences are the maps whose underlying maps of prespectra induce
isomorphisms of stabilized homotopy groups.
26
It turns out that a map of symmetric
spectra is a stable weak equivalence if and only if P is a stable weak equivalence
of orthogonal spectra in the sense just dened. There are other model structures here,
as we shall see. The thing to notice is that, in the model structures just specied, the
sphere spectra S are cobrant. Compare Remark 11.2 and Theorem 11.5.
These model structures are compared in [23]. Later work of Schwede and Shipley
[38; 39; 41] gives S a privileged role. Given any other sufciently good stable
model category whose homotopy category is correct, in the sense that it is equivalent
to Boardmans original stable homotopy category, there is a left Quillen equivalence
from S to that category.
However, this is not always the best way to compare two models for the stable homotopy
category. If one has models S
1
and S
2
and compares both to S , then S
1
and S
2
are compared by a zigzag of Quillen equivalences. It is preferable to avoid composing
left and right Quillen adjoints, since such composites do not preserve structure. For
example, using a necessary modication of the model structure on S to be explained
26
The underlying prespectrum of an Fspace is obtained by rst prolonging it to a W space and then
taking the underlying prespectrum of that, and we are suppressing some details.
Geometry & Topology Monographs, Volume 16 (2009)
What precisely are E
o
ring spaces and E
o
ring spectra? 269
shortly, Schwede gives a left Quillen equivalence S M
S
[38], and [23] shows
that the prolongation functor P: S IS is a left Quillen equivalence. This
gives a zigzag of Quillen equivalences between M
S
and IS . These categories are
both dened using I , albeit in quite different ways, and it is more natural and useful
to construct a left Quillen equivalence N: IS M
S
. Using a similar necessary
modication of the model structure on IS , this is done in Mandell and May [22,
Chapter I]; Schwedes left Quillen equivalence is the composite N P .
In any case, there is a web of explicit Quillen equivalences relating all good known
models for the stable homotopy category, and these equivalences even preserve the
symmetric monoidal structure and so preserve rings, modules, and algebras [22; 23; 38;
41]. Thus, as long as one focuses on stable homotopy theory, any convenient model can
be used, and information can easily be transferred from one to another. More precisely,
if one focuses on criteria (i) and (iii) of Theorem 11.1, one encounters no problems.
However, our focus is on (ii), the relationship between spectra and spaces, and here
there are signicant problems.
For a start, it is clear that we cannot have a symmetric monoidal Quillen left adjoint
from S or IS to M
S
with the model structures that we have specied since the
sphere spectra in S and IS are cobrant and the sphere spectrum in M
S
is not.
For the comparison, one must use different model structures on S and IS , namely
the positive stable model structures. These are obtained just as above but starting with
the level model structures whose weak equivalences and brations are dened using
only the positive levels, not the 0th space level. This does not change the stable weak
equivalences, and the resulting positive stable model structures are Quillen equivalent
to the original stable model structures.
However, the brant spectra are now the positive Cspectra, for which the structure
maps o: T
n
CT
n1
of the underlying prespectrum are weak equivalences only
for n > 0. This in principle throws away all information about the 0th space, even
after brant approximation. The analogue of Theorem 11.5 reads as follows. Actually,
a signicant technical improvement of the positive stable model structure has been
obtained by Shipley [42], but her improvement does not effect the discussion here: one
still must use the positive variant.
Theorem 12.1 The categories of commutative symmetric ring spectra and commuta-
tive orthogonal ring spectra have Quillen model structures whose weak equivalences
and brations are created by the forgetful functors to the categories of symmetric spectra
and orthogonal spectra with their positive stable model structures.
Parenthetically, as far as I know it is unclear whether or not there is an analogue of
this result for Wspaces. The results of [22; 23; 38; 41] already referred to give
Geometry & Topology Monographs, Volume 16 (2009)
270 J P May
the following comparisons, provided that we use the positive model structures on the
diagram spectrum level.
Theorem 12.2 There are Quillen equivalences from the category of commutative
symmetric ring spectra to the category of commutative orthogonal ring spectra and
from the latter to the category of commutative Salgebras.
Thus we have comparison functors
Commutative symmetric ring spectra
P

Commutative orthogonal ring spectra

N

Commutative Salgebras
E
o
ring spectra
S.
L
(-)

D
1

E
o
ring spaces
The functors P , N, and S .
L
() are left Quillen equivalences. The functor C
o
is
a right adjoint. The composite is not homotopically meaningful since, after brant
approximation, commutative symmetric ring spectra do not have meaningful 0th spaces;
in fact, their 0th spaces are then just S
0
. If one only uses diagram spectra, the original
E
o
ring theory relating spaces and spectra is lost.
13 Naive E
1
ring spectra
Again recall that we can dene operads and operad actions in any symmetric monoidal
category. If a symmetric monoidal category W is tensored over a symmetric monoidal
category V , then we can just as well dene actions of operads in V on objects of W .
All good modern categories of spectra are tensored over based spaces (or simplicial
sets). We can therefore dene an action of an operad O

in T on a spectrum in any
such category. Continuing to write . for the tensor of a space and a spectrum, such an
action on a spectrum R is given by maps of spectra
O(j )

.R
(j)
R.
Geometry & Topology Monographs, Volume 16 (2009)
What precisely are E
o
ring spaces and E
o
ring spectra? 271
Taking O to be an E
o
operad, we call such Ospectra naive E
o
ring spectra.
27
They are dened in terms of the already constructed internal smash product and thus
have nothing to do with the internalization of an external smash product that is intrinsic
to the original denition of E
o
ring spectra.
They are of interest because some natural constructions land in naive E
o
spectra (of
one kind or another). In some cases, such as Wspaces, where we do not know of
a model structure on commutative ring spectra, naive E
o
ring spectra provide an
adequate stopgap. In other cases, including symmetric spectra, orthogonal spectra, and
Smodules, we can convert naive E
o
ring spectra to equivalent commutative ring
spectra, as we noted without proof in [23, 0.14]. The reason is the following result,
which deserves considerable emphasis. See [13, III.5.5], [23, 15.5], and, more recently
and efciently, [42, 3.3] for the proof.
Proposition 13.1 For a positive cobrant symmetric or orthogonal spectrum or for a
cobrant Smodule E, the projection
: (E
j
)

j
E
(j)
E
(j)
,
j
(induced by E
j
+) is a weak equivalence.
This is analogous to something that is only true in characteristic zero in the setting of
differential graded modules over a eld. In that context, it implies that E
o
DGAs
can be approximated functorially by quasiisomorphic commutative DGAs (see Kr z
and May [16, II.1.5]). The following result is precisely analogous to the cited result
and can be proven in much the same way. That way, it is another exercise in the
use of the two-sided monadic bar construction. The cobrancy issues can be handled
with the methods of [41], and I have no doubt that in all cases the following result
can be upgraded to a Quillen equivalence. The more general result in Elmendorf and
Mandell [14, 1.4] shows this to be true for simplicial symmetric spectra and suitable
simplicial operads, so I will be purposefully vague and leave details to the interested
reader. Let O be an E
o
operad and work in one of the categories of spectra cited in
Proposition 13.1.
Proposition 13.2 There is a functor that assigns a weakly equivalent commutative
ring spectrum to a (suitably cobrant) naive Ospectrum. The homotopy categories of
naive Ospectra and commutative ring spectra are equivalent.
27
In recent e-mails, Tyler Lawson has jokingly called these MIT E
o
ring spectra, to contrast them
with the original Chicago variety. He alerted me to the fact that some people working in the area may be
unaware of or indifferent to the distinction.
Geometry & Topology Monographs, Volume 16 (2009)
272 J P May
It is immediately clear from this result and the discussion in the previous section that
naive E
o
ring spectra in S and IS have nothing to do with E
o
ring spaces,
whereas the 0th spaces of naive E
o
ring spectra in M
S
are weakly equivalent to E
o
ring spaces.
14 Appendix A. Monadicity of functors and comparisons of
monads
Change of monad results are well-known to category theorists, but perhaps not as readily
accessible in the categorical literature as they might be, so we give some elementary
details here.
28
We rst make precise two notions of a map relating monads (C. j. |)
and (D. v. ) in different categories V and W . We have used both, relying on context
to determine which one is intended.
Denition 14.1 Let (C. j. |) and (D. v. ) be monads on categories V and W . An
op-lax map (F. ) from C to D is a functor F: V W and a natural transformation
: FC DF such that the following diagrams commute.
FCC
F

DFC
D

DDF
F

FC

DF
and
F
Ft
{
{
{
{
{
{
{
{
F

C
C
C
C
C
C
C
C
FC

DF
A lax map (F. ) from C to D is a functor F: V W and a natural transformation
: DF FC such that the following diagrams commute.
DDF
F

DFC
D

FCC
F

DF

FC
and
F
F
{
{
{
{
{
{
{
{
Ft

C
C
C
C
C
C
C
C
DF

FC
If : FC DF is a natural isomorphism, then (F. ) is an op-lax map C D if
and only if (F.
-1
) is a lax map D C . When this holds, we say that and
-1
are monadic natural isomorphisms.
These notions are most familiar when V =W and F =Id. In this case, a lax map
DC coincides with an op-lax map C D, and this is the usual notion of a map
of monads from C to D. The map : C Q used in the approximation theorem
28
This appendix is written jointly with Michael Shulman.
Geometry & Topology Monographs, Volume 16 (2009)
What precisely are E
o
ring spaces and E
o
ring spectra? 273
and the recognition principle is an example. As we have used extensively, maps (Id. )
lead to pullback of action functors.
Lemma 14.2 If (Id. ) is a map of monads C D on a category V , then a left
or right action of D on a functor induces a left or right action of C by pullback of
the action along . In particular, if (Y. y) is a Dalgebra in V , then (Y. y ) is a
Calgebra in V .
We have also used pushforward actions when V and W vary, and for that we need lax
maps.
Lemma 14.3 If (F. ) is a lax map from a monad C on V to a monad D on W ,
then a left or right action of C on a functor induces a left or right action of D by
pushforward of the action along (F. ). In particular, if (X. ) is a Calgebra in V ,
then (FX. F ) is a Dalgebra in W .
Now let F have a right adjoint U . Let j: Id UF and c: FU Id be the unit
and counit of the adjunction. We have encountered several examples of monadic natural
isomorphisms (F. ) relating a monad C in V to a monad D in W , where F has a
left adjoint U . Thus is a natural isomorphism DF FC . In this situation, we
have a natural map : CU UD, namely the composite
CU
)CU

UFCU
U
1
U

UDFU
UDt

UD.
It is usually not an isomorphism, and in particular is not an isomorphism in our examples.
Implicitly or explicitly, we have several times used the following result.
Proposition 14.4 The pair (U. ) is a lax map from the monad D in W to the monad
C in V . Via pushforward along (F. ) and (U. ), the adjoint pair (F. U) induces an
adjoint pair of functors between the categories CV | and DW | of Calgebras in V
and Dalgebras in W :
DW |(FX. Y ) CV |(X. UY ).
Sketch proof The arguments are straightforward diagram chases. The essential point
is that, for a Calgebra (X. ) and a Dalgebra (Y. y), the map j: X UFX is a
map of Calgebras and the map c: FUY Y is a map of Dalgebras.
These observations are closely related to the categorical study of monadicity. A functor
U: W V is said to be monadic if it has a left adjoint F such that U induces an
Geometry & Topology Monographs, Volume 16 (2009)
274 J P May
equivalence from V to the category of algebras over the monad UF. This is a property
of the functor U . If U is monadic, then its left adjoint F and the induced monad UF
such that V is equivalent to the category of UFalgebras are uniquely determined by
V , W , and the functor U .
This discussion illuminates the comparisons of monads in Section 4 and Section 6. For
the rst it is helpful to consider the following diagram of forgetful functors.
T
e

Ospaces with zero

o
o
o
o
o
o
o
o
o
o
o
o
o
T

Ospaces
o
o
o
o
o
o
o
o
o
o
o
o
o

U
Note that the operadic unit point is 1 in Ospaces with zero but 0 in Ospaces; the
diagonal arrows are obtained by forgetting the respective operadic unit points. These
forgetful functors are all monadic. If we abuse notation by using the same name for
each left adjoint and for the monad induced by the corresponding adjunction, then we
have the following diagram of left adjoints.
T
e
O
T

Ospaces with zero

T
S
0
-

O
U

O
T
C

o
o
o
o
o
o
o
o
o
o
o
o
o
Ospaces
U
(-)
C

O
U
C

o
o
o
o
o
o
o
o
o
o
o
o
o
Since the original diagram of forgetful functors obviously commutes, so does the
corresponding diagram of left adjoints. This formally implies the relations
O
U
(X

) O
U

(X) and O
T
(S
0
X) O
T

(X)
of (4-3) and (4-7). The explicit descriptions of the four monads O
U

, O
U
, O
T

, and
O
T
are, of course, necessary to the applications, but it is helpful conceptually to
remember that their denitions are forced on us by knowledge of the corresponding
forgetful functors. As an incidental point, it is also important to remember that, unlike
the case of adjunctions, the composite of two monadic functors need not be monadic,
although it is in many examples, such as those above.
Geometry & Topology Monographs, Volume 16 (2009)
What precisely are E
o
ring spaces and E
o
ring spectra? 275
Similarly, for Section 6, it is helpful to consider two commutative diagrams of forgetful
functors. In both, all functors other than the C
o
are monadic. The rst is
S
e

Lspectra under S

D
1

o
o
o
o
o
o
o
o
o
o
o
o
o
S
D
1

Lspaces with zero

n
n
n
n
n
n
n
n
n
n
n
n
n
T
The lower diagonal arrow forgets the action of L and remembers the basepoint 0. The
corresponding diagram of left adjoints is
S
e
L

Lspectra under S
S
S-

L
C

o
o
o
o
o
o
o
o
o
o
o
o
o
Lspaces with zero

L
C

n
n
n
n
n
n
n
n
n
n
n
n
n
Its commutativity implies the relations
L(S E) L

(E) and L

o
X
o
L

X
of (6-1) and (6-4). The second diagram of forgetful functors is
S
e
D
1

Lspectra under S

D
1

T
e

Lspaces with zero

T
1
Lspaces

Here T
1
denotes the category of based spaces with basepoint 1. The lower two vertical
arrows forget the basepoint 0 and remember the operadic unit 1 as basepoint. The
Geometry & Topology Monographs, Volume 16 (2009)
276 J P May
corresponding diagram of left adjoints is
S
e
L

Lspectra under S
T
e

Lspaces with zero

T
1
(-)
C

Lspaces
(-)
C

This implies the relation L

o
(X

)
o
(LX)

of (6-7), which came as a compu-

tational surprise when it was rst discovered.
15 Appendix B. Loop spaces of E
1
spaces and the recogni-
tion principle
Let X be an Ospace, where O is an E
o
operad. Either replacing X by an equivalent
C space or using the additive innite loop space machine on C Ospaces, we
construct a spectrum EX as in Theorem 9.3. For deniteness, we use notations
corresponding to the rst choice. As promised in Remark 9.10, we shall reprove the
following result. The proof will give more precise information than the statement, and
we will recall a consequence that will be relevant to our discussion of orientation theory
in the second sequel [33] after giving the proof.
Theorem 15.1 The space CX is an Ospace and there is a natural map of spectra
o: ECX EX that is a weak equivalence if X is connected. Therefore its adjoint
o: ECX CEX is also a weak equivalence when X is connected.
We begin with a general result on monads, but stated with notations that suggest our
application. It is an elaboration of [25, 5.3]. The proof is easy diagram chasing.
Lemma 15.2 Let T be any category, let C be a monad on T , and let (. C) be
an adjoint pair of endofunctors on T . Let : C C be a monadic natural
isomorphism, so that the following diagrams commute.
CC

CC

CC

C
and

)
{
{
{
{
{
{
{
{
)

C
C
C
C
C
C
C
C
C

C
Geometry & Topology Monographs, Volume 16 (2009)
What precisely are E
o
ring spaces and E
o
ring spectra? 277
(i) The functor CC is a monad on T with unit and product the composites
Id
)

C
D)

CC
and
CCCC
DCtC

CCC
D

CC.
Moreover, the adjoint

: C CC of is a map of monads on T .
(ii) If (X. 0) is a Calgebra, then CX is an CCalgebra with action map
CCCX
DCt

CCX
D0

CX.
hence CX is a Calgebra by pull back along

.
(iii) If (F. v) is a C functor (F: T V for some category V ), then F is an
CCfunctor with action transformation
FCC
FtC

FC

F.
hence F is a C functor by pull back along

.
If : C C
t
is a map of monads on T , then so is C: CCCC
t
.
The relevant examples start with the loop suspension adjunction (. C) on T .
Lemma 15.3 For any (reduced) operad C in U with associated monad C on T ,
there is a monadic natural transformation : C C. There is also a monadic
natural transformation ,: Q Q such that the following diagram commutes,
where C is the Steiner operad (or its product with any other operad).
C

Q
p

Q
Proof For c C(j ), x
i
X, and t I , we dene
((c: x
1
. . . . . x
j
) .t ) =(c: x
1
.t. . . . . x
j
.t )
and check monadicity by diagram chases. A point QX can be represented by a
map : S
n
X.S
n
for n sufciently large and a point of QX can be represented
by a map g: S
n
X .S
1
.S
n
. We dene
,( .t )(y) =x .t .z.
Geometry & Topology Monographs, Volume 16 (2009)
278 J P May
where y S
n
and (y) = x . z X . S
n
and check monadicity by somewhat
laborious diagram chases. For the diagram, recall that is the composite
CX
C)

CQX
0

QX
and expand the required diagram accordingly to get
C

C)

C
C)

C)
.r
r
r
r
r
r
r
r
r
r
CQX
Q

CQX
Cp

CQX
0

Q
p

Q
The top left trapezoid is a naturality diagram and the top right triangle is easily seen to
commute by checking before application of C . The bottom rectangle requires going
back to the denition of the action 0 , but it is easily checked from that.
For any operad C , the action

0 of C on CX induced via Lemma 15.2 (ii) from an
action 0 of C on X is given by the obvious pointwise formula

0(c:
1
. . . . .
j
)(t ) =0(c:
1
(t ). . . . .
j
(t ))
for c C(j ) and
i
CX. The conceptual description leads to the following proof.
Proof of Theorem 15.1 For a spectrum E, (CE)
0
=C(E
0
), and it follows that we
have a natural isomorphism of adjoints y:
o

o
. We claim that this is an
isomorphism of C functors, where the action of C on the right is given by Lemma
15.2 (iii). To see this, we recall that the action of C on
o
is the composite

o
C

1

o
Q=
o
C
o

o
t
1

o
and check that the following diagram commutes.

o
C
yC

o
C

o
C

o
Q
yQ

o
Q

1
p

o
Q
t
1

o
y

Geometry & Topology Monographs, Volume 16 (2009)

What precisely are E
o
ring spaces and E
o
ring spectra? 279
The top left square is a naturality diagram and the top right square is
o
applied to
the diagram of Lemma 15.3. The bottom rectangle is another chase. The functor on
spectra commutes with geometric realization, and there results an identication
(15-4) ECX =B(
o
. C. CX) B(
o
. C. CX).
The action of C on CX is given by Lemma 15.2 (ii), and we have a map
(15-5) B(id.

. id): B(
o
. C. CX) B(
o
. CC. CX).
The target is the geometric realization of a simplicial spectrum with qsimplices

o
(CC)
q
CX =
o
(CC)
q
CX.
Applying c: CId in the q 1 positions, we obtain maps

o
(CC)
q
CX
o
C
q
X.
By further diagram chases showing compatibility with faces and degeneracies, these
maps specify a map of simplicial spectra. Its geometric realization is a map
(15-6) B(
o
. CC. CX) B(
o
. C. X) =EX.
Composing (15-4), (15-5), and (15-6), we have the required map of spectra
o: ECX EX.
Passing to adjoints and 0th spaces, we nd that the following diagram commutes.
CX
)
.v
v
v
v
v
v
v
v
v
D)

I
I
I
I
I
I
I
I
I
E
0
CX
o
0

CE
0
X
Since j is a group completion in general, both j and Cj in the diagramare equivalences
when X is connected and therefore o is then an equivalence.
As was observed in [26, 3.4], if G is a monoid in OT | , then BG is an Ospace such
that G CBG is a map of Ospaces. Since (CEX)
1
=CE
1
X E
0
X, Theorem
15.1 has the following consequence [26, 3.7].
Corollary 15.7 If G is a monoid in OT | , Then EG, ECBG, and CEBG are
naturally equivalent spectra. Therefore the rst delooping E
1
G and the classical
classifying space BG .E
0
BG are equivalent as Ospaces.
Geometry & Topology Monographs, Volume 16 (2009)
280 J P May
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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)