Oct 27, 1973 - of ¡J-modules, and ® the usual tensor product. If C also admits a suspension, we may form the corresponding. Spanier-. Whitehead [9] category.
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 49, Number 1, May 1975
STABILIZING TENSOR PRODUCTS HAROLD M. HASTINGS1 ABSTRACT. Let C be a symmetric monoidal category with a suspension, and let SC be the resulting stable category. We shall give necessary and sufficient conditions for extending the symmetric monoidal structure to a monoidal structure on SC. These imply that the usual smash product on finite pointed CW complexes cannot be extended to a smash product (with S as unit) a conjecture
on finite of Alex
1. Introduction.
(Eilenberg
spectra, Heller.
Let
(C, ®,
® is an associative,
C x C —• C. The standard C the category If C also Whitehead
admits
are functors
monoidal
known fact:
product metric
and
has
U a commutative
structure
a category
stable
necessary
to SWC. This
of modules
monoidal
cannot
structure
continuous
be extended
category
ring,
with identity,
product.
over
(here
to SWF; hence
is motivated
F of finite
not to SF.
the symby the well-
R admits a tensor
As a consequence,
the usual
pointed
X®y=XAy=Xx
[6]).
See §2.
to extend
a ring with identity
Spanier-
SC (Heller
a full inclusion.
condition
and
functor
category
condition
on the category maps
confirms
we may form the corresponding
if and only if R is commutative.
and pointed
tensor
C —> SWC — SC, the latter
a simple
monoidal
and "unitary"
® the usual
SWC and universal
give
This
® is the tensor product,
commutative
a suspension,
[9] category
spectra.
U) be a symmetric
example
of ¡J-modules,
In §3 we shall metric
not on Boardman
and Kelly [4, pp. 472, 512]). Here
U the unit;
There
hence
sym-
CW complexes
Y/X V Y, (7=5°)
This
confirms
a conjecture
of Alex Heller. In §4 we shall necessary
structure
condition
show that is also
to SWC. Extension
Presented
to the Society,
if X = ? ® S
sufficient
for some
to extend
to SC will then October
ary 28, 1974. AMS (MOS) subject classifications
27,
1973;
(1970).
S
in C, then
the symmetric
our
monoidal
be automatic. received
by the editors
Janu-
Primary 18D10, 55B20; Secondary
55D99, 55E10, 55E15, 55J99. Key words and phrases. Smash product, stable smash product, SpanierWhitehead category, stable category, Boardman spectra, monoidal category, metric monoidal category, category with suspension. 1 Partially supported by a NSF Institutional Grant.
sym-
Copyright © ]l>75. American Mathematical Society License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
1
2
H. M. HASTINGS Two applications,
extend
the usual
category)
the second
symmetric
structure
given
We shall
In 'S6, we shall of chain
stable
due to Heller Call
compare
that
In ^5 we shall
(H denotes
the usual
category
category
these
homotopy
B is the c-comple-
monoidal
suspension
structure
on the
can be extended
to
(§6). Alex
Heller
for helpful
discussions
work.
categories.
[6], except
as noted.
C with
HB.
monoidal
constructions.
symmetric
to thank
of stable
2 generically
of a symmetric
with translation
of this
a category
use
homotopy
We wish
the preparation
2. Review
stable
show
Acknowledgement.
stable
will be given. on HF
constructions
briefly
complexes
the corresponding
shall
ad hoc
on Boardman's
tion [6] of SF.
during
known,
structure
to SHF (ss HSF). Boardman [2], [lO], Adams [l], Puppe [8], May
[7], and we [5] have
category
well
monoidal
We shall
endofunctor
to denote
if 2 is an automorphism.
need
the following
£ a category
suspensions. A functor
definitions,
with suspension.
Such a category
T between
We
is called
categories
with
sus-
pension is called stable if T2 = 2T. To each
category
stable
category
where
X is an object
C with suspension,
[6, Proposition
l.l]
of C, and
Heller
SC.
Objects
m is an integer.
associates
a universal
of SC are pairs
Morphisms
(X, m),
are given
by
SC((X, m), (y, »)) = colim C(2m+kX, ln+kY), where
k ranges
over any unbounded
e.g., the proof of Theorem SC admits
functor
monotone
sequence
of integers.
See,
4.
a suspension
given
S: C -> SC; on objects
by S(X,
m) = (X, m + l).
SX = (X, 0). We shall
There
sometimes
is a
identify
X
with (X, 0). Note that Si £ IS. Finally,
the Spanier-Whitehead
of SC whose
objects
category
are in C (precisely,
[9] SWC is the full subcategory in the image
of S). S factors
through SWC. 3. The necessary a (not necessarily
condition
symmetric)
and some consequences. monoidal
category.
That
Let is,
(D, ®, U) be
® need
not be
commutative. Theorem Proof.
1.
D(il,
We shall
U) is a commutative use
the following
part
monoid. of the monoidal
p. 472], to show that any two maps in D(U, U) commute. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
structure
For any
[4,
X in D,
3
STABILIZING TENSOR PRODUCTS there
are natural
right
and left unit isomorphisms
rx: X ® U —>X, U X = U, ru = /y;
call
this
map
lx: U ® X -> X.
u. Hence
any map
/ in D(il,
U) factors
as
/= u{f ® U)u~l = u(U ® /)«_1
Let /, g £ D(U, U). Then
fg = u(f ® Li)«- lu(U ® g)«~ ' = u(f ® (7)((7 ® g)a~ ' = «(/ ® g)«_ ' = «(«7 ® g)(/ ® U)u~ : = z/((7 ® g)u~ lu(f ® t/)w" l = g/, as required.
□
Corollary extended
2.
T/)e following
from the indicated
or universal
(a)
stable
(symmetric)
categories
monoidal
to their
structures
cannot
Spanier-Whitehead
be
categories,
categories.
C = F, the category
of finite
pointed
CW complexes
are continuous
pointed maps, ® = A, U = S°, 2X = X AS1. (b)
C is the category
phisms
(or any larger
of finite
category
usual direct sum ©,
dimensional
of vector
vector
spaces)
spaces
over a field
® is the
U = 0, 2X = X © F.
(c)
C as in (b),
For
(a), let the symmetric
® is the usual
tensor
group
S
product,
U = F,
52 A ••• A S2. Define inclusions
&n C S +, by "leaving is an induced
o^ = colim
o
There
2X = X © F.
act on S2n by permuting
fixed."
Let
and isomorF,
factors
of
the last letter
inclusion
of monoids
S^ C SWF(S°, S°) = colim F(S2n, S2n), hence
SWF(S , 5°) Proofs
topological
of (b) and reduced
4. Sufficient
gory.
is not abelian.
conditions.
Fix some
object give
monoidal
(c) are similar.
"signs"
These
are not involved.
examples
are motivated
by
Kp theory.
We can then metric
Note that
Let
(C, ®,
S1 in C, and define a rough converse
structure
to SWC and
U) be a symmetric
a suspension
to Theorem
= S"-1 ® Sl.
cate-
by 2X = X ® S .
1 on extending
SC.
Let S° = U. For n> 1, let Sn = IS"-1
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
monoidal
the sym-
4
H. M. HASTINGS Let
group
Cl
denote
of ö
"leaving
mutator
See Corollary
the last
Definition
Theorem
3.
The standard
4.
suspension
acts
Regard dn;
dn
then
sub-
C &n + j by
(3^
is the com-
of hx
S°),
trivially
) on S , a monoid of the homomorphisms
of S" = S1 ® • • • ® S1.
on a symmetric
monoidal
category
monoid.
structure
on C can be extended
to SC
monoidal functor [4, pp. 473, 512]).
monoidal
U^
structure
of choice. Choose
can be extended
is the commutator
® on objects
® on maps
in SC.
factors
conditions
monoidal
is independent
S
on S •
(b) => (c), define
define
(and
is the colimit
2 = ? ® S1 are equivalent.
(a) =» (b) since
(y , n ) be maps
Let
= colim
action
The following
with
The symmetric
We shall inition
of (a).
d^
(3^ C S^ —> SWC(S°,
The symmetric
Proof, For
Let
the commutator
of ö ,.
(S: C —' SC is a symmetric (d)
on n letters,
2, verification
fixed."
SWC(S , S ) is a commutative
(b) (ix (c)
group
— C(S", Sn) induced by permuting
(C, ®, S°)
(a)
letter
subgroup
homomorphism
Ö C§
the alternating
"up Let
X ® Sm + 2k -> X' ® Sm' + 2k,
(/
® g ),,, be the composite
and then
of S^.
n) = (X ® y, m + n).
show
/: (X, m) —• (X , m ) and
representatives
/':
subgroup
by (X, m) ® (y,
to choice"
to SWC.
that
the def-
g: (y,
n) —>
of the form
g': Y ® Sn + 21 — Y' ® Sn'+2!.
Sm*n + 2k+2t Si X
