Table Of Content- i -
O. Introduction(1)
These notes are a revised version of several talks held at the E.T.H.
Forschungsinstitut fur Mathematik in ZUrich during the spring of 1967. My purpose in
giving these talks, and in writing the notes, was to try to understand, and clarify
somewhat in the process, the categorical constructions used by Boardman in his
definition of CW-spectra 2. Also,I wanted to see in what relation this category of
CW-spectra stood to the category of simplicial spectra introduced by Kan ii. I hope
these aims are at least partially realized in the following pages.
w and w are concerned with two categorical constructions necessary to define
Boardman's category of CW-spectra starting with the category of finite CW-complexes
and its endofunctor suspension. In both cases it is shown that these are solutions of
universal problems in certain categories of categories. In fact, the first freely
inverts an endofunctor, and the second freely adds certain colimits to the category in
question. This first "stabilization" construction is well known, appearing, for
example, in Freyd 7 and Heller 8. The "completion" process is more subtle, and is
discussed in some detail in w
In w the category of simplicial spectra in the sense of Kan is introduced. The
main result here is that this category is obtained by applying the constructions of
w and w to the category of simplicial sets with finitely many non-degenerate elements
together with its endofunctor suspension. One consequence of this is that there
exists a stable geometric realization functor from Kan's category of simplicial
spectra to Boardman's category of CW-spectra.
w treats the abelian case of w One result of the structure theorem obtained
here is that the stable Dold-Kan theorem, which asserts the equivalence of the cate-
gories of FD-spectra and unbounded chain complexes, is seen to be a consequence of the
ordinary unstable version.
)I( This research was partially supported by the NSF under Grant GP6783.
- 2 -
I would like to express my gratitude to Prof. Beno Eckmann for making possible
my visit to the Forschungsinstitut, and for his interest in these notes.
I. Invertinq an Endomorphism
In this section we consider the problem of constructing for a given category
with endomorphism E ~ A > A a "best" category ~E in which E becomes an auto-
morphism. Two solutions to this will be given. The first is based on consideration of
triples in Cat - the category of categories - and the second is given by a direct
construction. The two are, of course, naturally equivalent.
For the triples construction, let ~ denote the discrete category of non-
negative natural numbers O,1,2, Let ~ denote the terminal object of Cat -
the category with one object o and one morphism 1 . ~ is a monoid in Cat under
o
the operations
defined respectively, by (m,n)~--> m + n and o~-9o. Thus, ( ) x ~is a triple
in Cat - a "doctrine" in the sense of Lawvere - and the algebras over this triple-
Cat ~ - are categories with endomorphism. Algebra morphisms are functors compatible
with the given endomorphisms. Following Heller, we shall call such functors "stable".
In fact, an algebra over ( )x ~ is a category A equipped with a functor
such that the diagrams
* For basic facts concerning triples and algebras see Eilenberg-Moore 5. For state-
ments on the existence of coadjoints to functors on algebras induced by morphisms of
triples see Linton 13. Foundational questions concerning Cat are treated in 12.
- 2 -
I would like to express my gratitude to Prof. Beno Eckmann for making possible
my visit to the Forschungsinstitut, and for his interest in these notes.
I. Invertinq an Endomorphism
In this section we consider the problem of constructing for a given category
with endomorphism E ~ A > A a "best" category ~E in which E becomes an auto-
morphism. Two solutions to this will be given. The first is based on consideration of
triples in Cat - the category of categories - and the second is given by a direct
construction. The two are, of course, naturally equivalent.
For the triples construction, let ~ denote the discrete category of non-
negative natural numbers O,1,2, Let ~ denote the terminal object of Cat -
the category with one object o and one morphism 1 . ~ is a monoid in Cat under
o
the operations
defined respectively, by (m,n)~--> m + n and o~-9o. Thus, ( ) x ~is a triple
in Cat - a "doctrine" in the sense of Lawvere - and the algebras over this triple-
Cat ~ - are categories with endomorphism. Algebra morphisms are functors compatible
with the given endomorphisms. Following Heller, we shall call such functors "stable".
In fact, an algebra over ( )x ~ is a category A equipped with a functor
such that the diagrams
* For basic facts concerning triples and algebras see Eilenberg-Moore 5. For state-
ments on the existence of coadjoints to functors on algebras induced by morphisms of
triples see Linton 13. Foundational questions concerning Cat are treated in 12.
- 3 -
1Ax'q
Exl
A.~.-.- > AxN Ax~Jx ~ i) Ax~
and
A A(cid:141) " E ) A
commute. (A x~ ~ A is the inverse of the canonical isomorphism A ~ (cid:12)9 A x~.)
From these one sees immediately that if we write E )n( for E restricted to A (cid:141) n,
and identify each A x n with A, then
E(~ = 1 A, E (n) = E (1) . . . . E (1) n-times
and E (1) is an arbitrary endomorphism. We write simply E again for E (I) and
remember this correspondence.
Now let Z denote the discrete category of all natural numbers. Under the
same operation Z is a group in Cat, so ( )x Z is again a triple. As above one
sees that the algebras - Cat Z - are categories A together with an automorphism
E : A > A - restriction to A x I gives E and restriction to A (cid:141) -I gives
its inverse. Morphisms of algebras are the same as above.
There is an obvious morphism of monoids in Ca t - a functor respecting
addition and units - given by the inclusion
I : ~ )Z
I induces a morphism ( )x ~ - ) ( )x Z of triples, which we again call I,
and this in turn induces a forgetful functor
Cat I : Cat z ) Cat ~
given as follows. If (A,E)~ Cat z we have the composite
IA E
Axe, ~ A(cid:141) , > A ,
- 4 -
and Cat ~ (A,E) = (A,E'IA). That is, interpreting E as an automorphism of A, we
simply forget that E has an inverse.
We can construct a coadjoint to Cat I in the following manner 13. Let
(A,E)~ ~, i.e. A is a category with endomorphism E, and define F(A,E) by
requiring that the diagram
E(cid:141)
A(cid:141)215 - Z ) A~Z, > F (A,E)
A(cid:141)215
be a coequalizer diagram in Cat. Since ( )(cid:141) z is an adjoint triple it preserves
coequalizers, and hence it is easy to see that there is a unique ( )x Z-algebra
structure
- F (A, E) xZ ~ F (A, E)
such that setting
~,~)
= (F(~,E),~)
provides a coadjoint to t~___aC I. Thus,~(A,E) is the free category with automorphism on
(A,E).As a coadjoint,~(A,E) satisfies the following universal property. There is a
stable functor
a'(_A,E) (cid:12)9 (_A,~) ;~(_A,~)
such that if H : (A,E) ~(B,D) is a stable functor where D is an automorphism,
then there is a unique stable functor H : ~(A,E) (B,D) such that
(A,E) (cid:12)9 ~' (A,E) > ~(A,E)
)D,B_(
- 5 -
commutes.
The previous procedure is fine as quick sketch of a proof that for any category
A with endomorphism E : A A there exists a free category with automorphism,
but practically it suffers from the obvious defect that there is no immediate,
workable description of F(A,E). To remedy this, let ~E be the category whose
objects are pairs (A,n> where A ~ A and n ~ Z. Morphisms in ~E are given by
AE((A,n>,<A',n'>)= i~ A(En+kA,En'+kA )' ,
the limit being taken over those k for which n + k and n' + k are ~o. Compo-
sition is defined in the obvious way by picking representatives.
I.I Theorem
There is an automorphism E : ~E ~E and a stable functor
a(A,E) : (A,E) ) (AE,E)
having the same universal property as ~'(A,E). In particular, A is canonically
-E
equivalent, as a category with automorphism, to F(A,E).
Proof: If (A,n),(A',n') ~ ~E' denote the injection
A(En+kA,En'+kA , ) > AE(~A,n>,<A',n'>)
of the direct limit by i k. Then, on objects is given by
E(A,n) = (A,n+l>
and on morphisms by the commutative diagram
- 6 -
A ( En+kA, E n ' +kA ' )
i k
(E_~/ ~A,n>, ~A' ,n' ~) > (EA_ ~A,n+l~, ~A' ,n'+l>)
has the obvious inverse. Define
,A_( ).~ (cid:12)9 ,A_( ).~ ~ ).~,.~A_(
by a(A,E) (A) = (A,O> if A is an object of A, and a(A,E) )f( =
= iof ~ AE(~A,O>,~A',O>) if f z A ) A ~ is a morphism of A. a(A,E) is clearly a
functor, and is stable - i.e.
)E,A(a
_A > E_A
commutes, since for each A ~ A
(EA,O> = <A,I> (cid:12)9
)>I,A<,>I,A<(E_A
(AS sets, AE(<EA,O>,<EA,O>) = and I<EA,O> = l<A,l>).One readily
checks that a(A,E)" E and E (cid:12)9 a(A,E) also have identical effect on morphisms.
To show that a(A,E) has the same universal property as ~'(A,E), let
H : (A,E) ) )D,B_( be a stable functor where D is an automorphism. To define
(cid:12)9 (A~,~) ~ (_B,D) put
H(A,n> = DnHA
- 7 -
for an object of ~E' and for a morphism (A,n) ) (A',n') - i.e. an element of
l~m A(En+kA,Ent+kA )' - choose a representative
k
f : En+k A ) Ent+k At
-k
in A and apply D to
Hf
HEn+kA ~ HE n ' +k A,
I U
Dn+kHA Dnt+kHA ,
to give a morphism
H(A,n> >H(AI,nt>
in B.Since H is stable this is independent of the choice of f, and makes H the
unique functor with the required properties.
Having given in 1.1 an "external" characterization of
=(A_,E) - )~,A_( ) 'E-~( ~)
we give now an "internal" one, which is more useful for verifying that a given care-
gory with automorphism (B,D) is equivalent to (AE,E).
~.2 P~opo~ition
Let (A,E) be a category with endomorphism, (B,D) a category with automorphism,
and
H (_A,E) ) (_B,D)
:
a stable functor, Then in the diagram
- 8 -
,_A( E) /
(~ )E, ,~.)
\ A_( E
/
B( )D,
is an equivalence of categories iff
)a( If g : HA , )HA' is a morphism in B, then there is an n)o and an
f : EnA ) EnA ' in A such that
HEnA Hf (cid:12)9 HEnA I
JI ;J
DnHA ) DnHA I
Dng
f
)b( If A~ ~A' are morphisms in A and Hf = Hf ~ , then there is an m)o such
ft
Emf = Emft.
)c( If B E B, then there is an i~o such that
DIB ~ HA
for some A ~ A.
Proof: The proof follows in a straightforward manner from the definition of H, and
we have the details to the reader except to remark that )a( makes H full, )b( makes
it faithful, and (c) makes it representative, where representative means that H hits
(up to isomorphism) every object of B. The conditions are obviously necessary.
We remark that among various possible alternatives to 2.2 this one was
chosen because it is easy to prove, and because if H is full and faithful conditions
)a( and )b( are trivial.
- 9 -
Examples
(I.) Let A be, say, an abelian category and let C(A) denote the category of chain
complexes over A. That is, an object of C(A) is a sequence (An) of objects
A n ~ _A for n ~ Z,together with A-morphisms_ 0n z An > An_l such that
~n_l~n = o. Morphisms are chain maps of degree o. Write C~ for the full sub-
category of C(A)_ consisting of complexes (An) such that An = o for n<o, and
C+(A) for the full subcategory consisting of complexes (A n ) for which there exists
n o~ Z such that An = o for n(no. Denote by S : _)A(C - ~ C(A)_ the endomorphism
called "suspension" - given as follows z if C = (An)~ C(A) and
-
f = (fn) : (A n) (A' n) is a morphism of C(A), then (SC)n = An_ l ,
oSC C and
(Sf) n = fn-l" That is, S shifts everything one place to the right.
n = ~n-1'
We write also S for the endomorphisms induced by S on C ~ )A( and C+(A). Note that
S has the obvious inverse on C(A) and C+(A), but is not invertible on C~
The inclusion
~C( )S, , (C+(A),S)
trivially satisfies the conditions of 2.2, and hence (C~ s,S) is canonically
equivalent as a category with endomorphism to (C+(A) ,S).
(2.) Let F be the category of finite CW-complexes with basepoint, and continuous,
basepoint preserving maps. Let S : F F be the (reduced) suspension. Then ~S
is, by definition, the "stable category" of Freyd 7 and Heller 8, or the "suspen-
sion category" of Boardman 2. (Actually, Freyd uses cellular maps but this is
immaterial after passage to homotopy classes)
More meaty examples of the stabilization process will be treated in w and w
