THE HARE AND THE TORTOISE
J.P. MAY
It is a pleasure to be here to help celebrate Mike Boardman's 60th birthday.1
I have just finished writing a history of stable algebraic topology from the *
*end
of World War II through 1966 [18]. The starting point was natural enough. The
paper of Eilenberg and Mac Lane [6] that introduced the categorical language we
now all speak appeared in 1945, and so did the paper of Eilenberg and Steenrod *
*[7]
that announced the axiomatic treatment of homology and cohomology.
The ending point was more artificial, at first dictated by constraints of time
and energy and the fact that Steenrod's compendium of Mathematical Reviews in
topology contained all reviews published through 1967 and thus all papers publi*
*shed
through 1966. It also made it easy for me to be modest and impersonal. Although
I got my PhD in 1964, I only plugged into the circuit and began to know what was
going on when I arrived at Chicago, at the end of 1966.
Mike also got his PhD in 1964. Since he is two years older than I am, I guess
he was a little slow. But then, his thesis was a lot more important than mine w*
*as,
although people at the time didn't seem to understand that. Its results became
available in a mimeographed summary in 1966. So maybe 1966 wasn't such a bad
stopping point mathematically. It is amazing how much we didn't know then, how
many familiar names had not yet made their mark.
In fact, a complete list of the people who made sustained and important contr*
*i-
butions to the development of stable algebraic topology in the years 1945 throu*
*gh
1966 would have no more than around 40 names on it. On the other hand, the
caliber of the people working in the field was extraordinary.
The towering figures were Adams, Atiyah, Borel, Bott, Cartan, Eilenberg, Hirz*
*e-
bruch, Mac Lane, Milnor, Serre, Steenrod, Thom, and J.H.C. Whitehead. Others
who made major contributions were Adem, Araki, Barratt, Brown, Conner, Dold,
Dyer, Eckmann, Floyd, Heller, Hilton, James, Kan, Kudo, Lashof, Liulevicius,
Moore, Peterson, Pontryagin, Postnikov, Puppe, Spanier, Stong, Swan, Toda, Wall,
G.W. Whitehead, and Wu.
Several people attempted to build good stable homotopy categories. In 1953,
Spanier and J.H.C. Whitehead [22] gave a start by introducing the "S-category",
which we now understand to be the full subcategory of finite CW complexes in the
stable homotopy category. In 1962, G.W. Whitehead [24] gave the definitive cate-
gory of -spectra equivalent to the category of cohomology theories on spaces. I*
*t is
definitely not triangulated, however, and therefore cannot be the stable homoto*
*py
category as we know it today.
Adams [1] and Puppe [19, 20] gave constructions with roughly the same starting
point in the early 1960's. They consider CW prespectra, which are sequences of
CW complexes Tn and inclusions of subcomplexes Tn ! Tn+1, subject to certain
____________
1This is a revised and shortened version of my talk at Baltimore
1
2 J.P. MAY
connectivity restrictions. The term "prespectrum" rather than "spectrum" agrees
with Mike's terminology; this distinction was first made by Dan Kan [11]. There*
* is
an analogy with presheaves that was made precise later.
Adams [1] described his point of view in an amusing analogy, well worth recal*
*ling.
"The hare is an idealist: his preferred position is one of elegant and all em*
*bracing
generality. He wants to build a new heaven and a new earth and no half-measures.
The tortoise, on the other hand, takes a much more restrictive view. He says th*
*at
his modest aim is to make a cleaner statement of known theorems, and he'd like
to put a lot of restrictions on his stable objects so as to be sure that his ca*
*tegory
has all the good properties he may need. Of course, the tortoise tends to put on
more restrictions than are necessary, but the truth is that the restrictions gi*
*ve him
confidence.
You can decide which side you're on by contemplating the Spanier-Whitehead
dual of an Eilenberg-Mac Lane object. This is a "complex" with "cells" in all s*
*table
dimensions from -1 to -n. According to the hare, Eilenberg-Mac Lane objects
are good, Spanier-Whitehead duality is good, therefore this is a good object: A*
*nd
if the negative dimensions worry you, he leaves you to decide whether you are a
tortoise or a chicken. According to the tortoise, on the other hand, the first *
*theorem
in stable homotopy theory is the Hurewicz isomorphism theorem, and this object
has no dimension at all where that theorem is applicable, and he doesn't mind t*
*he
hare introducing this object as long as he is allowed to exclude it. Take the n*
*asty
thing away!"
Frank was a great mathematician and my closest friend, but to my mind this
is not the best attitude to take towards matters of foundations: that was not h*
*is
strongest suit. His category had only connective spectra in it, so excluded eve*
*n the
periodic Bott spectra, and his spectra therefore could not be desuspended. Pupp*
*e's
spectra were bounded below, still excluding the Bott spectra, but at least they*
* could
be desuspended.
Mike's introduction [4] gave persuasive propaganda for his version of the sta*
*ble
homotopy category.
"This introduction attempts to give some criteria for a stable category. It *
*is
addressed without compromise to the experts. The novice has the advantage of
not having been misled by previous theories. The place of our theory on Adams'
tortoise-hare scale is obvious."
Mike was the prototypical "hare". He was operating at a level of categorical
sophistication that, while commonplace and even fashionable now, was well ahead
of its time. Unfortunately for the subject, this prototypical hare was and is a
tortoise when it comes to publication. His treatment of the stable category has
never been published. Mike, let me again ask you, as I think I have done every *
*year
for the last decade: please publish your paper on conditionally convergent spec*
*tral
sequences. For those not in the know, the cited paper gives the definitive trea*
*tment
of convergence of spectral sequences.
The introduction goes on: "In this advertisement we compare our category S of
CW spectra, or rather its homotopy category Sh, with competing products. We find
the comparison quite conclusive, because the more good properties the competito*
*rs
have, the closer they are to Sh. Findings subject to verification by an indepen*
*dent
consumer agency."
All consumers are now in agreement: Mike's stable homotopy category is defini-
THE HARE AND THE TORTOISE 3
tively the right one, up to equivalence. However, the really fanatical hare dem*
*ands
a good category even before passage to homotopy, with all of the modern bells
and whistles. The ideal category of spectra should be a complete and cocomplete
Quillen model category, tensored and cotensored over the category of based spac*
*es
(or simplicial sets), and closed symmetric monoidal under the smash product. Its
homotopy category (obtained by inverting the weak equivalences) should be equiv-
alent to Mike's original stable homotopy category.
We now have such a category, namely the category of S-modules of [9]; an
introductory account is given in [8]. Indeed, we have several such categories a*
*nd
know how to compare them [10, 14, 17, 13, 21].
Mike was brought up in the English tradition of J.H.C. Whitehead, whose stu-
dents were reared on beer and CW complexes. I have no objection to beer, but I
early came to the conclusion that to restrict to CW complexes, as Adams, Puppe,
and Boardman all did, is to err on the side of the tortoise. At a conference in
1968 [15], I advertised the idea that spectra really should be defined as seque*
*nces
of based spaces En and homeomorphisms En ~=En+1. Many constructions, such
as products, limits, and function spectra, that are impossible with any kind of*
* CW
spectra become easy with spectra of that sort.
The S-modules of [9] are spectra with additional structure. It took a number
of developments over nearly thirty years to arrive at the category of S-modules.
One key idea was to introduce coordinate-free spectra [16], indexing them on fi*
*nite
dimensional inner product spaces rather than on the non-negative integers: those
just give the canonical inner product spaces Rn. This led to a formal developme*
*nt
of the theory of smash products of coordinate-free spectra that is closely anal*
*ogous
to Mike's original treatment of smash products of CW spectra, despite a great
difference between the definitions of the two kinds of spectra.
A brief summary of Mike's theory may be of interest, since younger algebraic
topologists are unlikely to have seen it. Details are in Mike's preprints [3, 4*
*] and
in an exposition by his student Rainer Vogt [23]. For each countably infinite d*
*i-
mensional real inner product U, Mike constructs a category S(U) of CW spectra.
He starts with a copy FA of the category of finite CW complexes for each finite
dimensional subspace A of U. He first constructs a category F(U) from the FA
by a certain colimit of categories construction. Intuitively, F(U) is a copy of*
* the
category of finite CW spectra, constructed by stabilization from the category of
finite CW complexes. Then S(U) is constructed from F(U) by another categorical
colimit construction. Intuitively, the objects of S(U) are all of the colimits *
*of dia-
grams of inclusions between objects of F(U). The stable homotopy category Sh(U)
is obtained from S(U) by passage to homotopy classes of maps. The canonical
stable homotopy category is Sh = Sh(R1 ).
There is an obvious external smash product Z : S(U) x S(U0) ! S(U U0). For
a linear isometry f : U ! U0, there is a functor f* : S(U) ! S(U0). In particul*
*ar,
each linear isometry f : R1 R1 ! R1 can be composed with the external smash
product Z to give an internal smash product S(R1 ) x S(R1 ) ! S(R1 ). More
generally, for any finite CW complex K and any map k : K ! I(U; U0), where
I(U; U0) is the space of linear isometries U ! U0, there is a functor k* : S(U)*
* !
S(U0). If L is a subcomplex and deformation retract of K and ` is the restricti*
*on of
k to L, there is a natural equivalence `* ! k*. Using this and the contractibil*
*ity of
the spaces I(U; U0), it follows formally that the internal smash products all b*
*ecome
4 J.P. MAY
canonically equivalent on passage to the stable homotopy category Sh and Sh is a
symmetric monoidal category under the internal smash product.
The theory that Gaunce Lewis and I developed later [12] looks formally the sa*
*me.
We also have categories S(U), external smash products Z, functors f* and k*, and
so on; we call the functors k* twisted half-smash products. However, although
the formal structure of our theory closely follows Mike's blueprint, the realiz*
*ation
of the formal structure is entirely different: whereas Mike's categories S(U) *
*are
constructed categorically out of finite CW complexes, the objects of our catego*
*ries
S(U) are the coordinate-free spectra indexed on the finite dimensional subspace*
*s of
U, and CW complexes play no role in the basic definitions. The generality allows
us to define twisted half-smash products k* for maps k : K ! I(U; U0) defined on
arbitrary spaces K. In particular, we can take k to be the identity map of I(U;*
* U0),
in which case we write k*E = I(U; U0) n E.
Let L(j) = I((R1 )j; R1 ). As Boardman and Vogt observed and exploited in [5],
the spaces L(j) give an E1 operad. More precisely, since I had not yet introduc*
*ed
operads, they observed that the L(j) are some of the morphism spaces of a PROP
(product and permutation category). In [16], the linear isometries operad L was
used to define E1 -ring spaces and E1 -ring spectra. We did not then have the
general twisted half-smash products. However, once we had them, we could form
the extended powers L(j) n Ej of a spectrum E, where Ej is the j-fold external
smash power of E. An E1 -ring spectrum is just a spectrum E together with
an action of L given by maps L(j) n Ej ! E such that the appropriate unit,
asssociativity, and equivariance diagrams commute.
The twisted half-smash product later became the starting point of the theory *
*of
S-modules. We have the canonical j-fold internal smash product L(j)nE1Z. .Z.Ej.
The j-fold internal smash products determined by the linear isometries f 2 L(j)*
* all
map into this canonical j-fold smash power. Imposing extra structure on spectra,
defined in terms of maps L(1) n E ! E, we can construct a quotient E ^L E0 of
L(2)nE^E0. The smash product ^L is commutative and associative. A slight vari-
ant gives the notion of an S-module E and a commutative, associative, and unital
smash product E ^S E0 of S-modules. Up to canonical weak equivalence, E1 ring
spectra are exactly the commutative monoids in the symmetric monoidal category
of S-modules. This, roughly, is the starting point of the theory of Elmendorf, *
*Kriz,
Mandell, and myself [8, 9].
As this brief sketch indicates, the theory of S-modules owes a great deal to
ideas and insights due to Boardman. It is typical of Mike's generosity to the i*
*deas
of others that he began his review of [8] with the sentence "This is the paper
that promises to revolutionize stable homotopy theory" and did not mention the
influence or relevance of his original approach to the stable homotopy category*
*. I
am happy to have had this occasion to rectify the omission.
References
1.J.F. Adams. Stable homotopy theory. (Lecture Notes, Berkeley 1961.) Lecture *
*Notes in Math-
ematics Vol. 3. Springer-Verlag. 1964. (Second edition 1966; third edition 1*
*969.)
2.J.M. Boardman. PhD Thesis. Cambridge. 1964.
3.J.M. Boardman. Stable homotopy theory. (Summary). Mimeographed notes. Warwic*
*k Uni-
versity, 1965-66.
THE HARE AND THE TORTOISE 5
4.J.M. Boardman. Stable homotopy theory. Mimeographed notes. Johns Hopkins Uni*
*versity,
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5.J.M. Boardman and R.M. Vogt. Homotopy invariant algebraic structures on topo*
*logical spaces.
Lecture Notes in Mathematics Vol. 347. Springer-Verlag. 1973.
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* Amer. Math.
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* Nat. Acad.
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*stable homo-
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and FSP's. Preprint, 1998.
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* Vol. 99. Springer-
Verlag 1969, 448-479.
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*ed by I.M. James).
Elsevier. To appear.
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*lgebraic topol-
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*Aarhus Univer-
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