2. EXTERNAL SMASH PRODUCTS AND TWISTED HALF-SMASH PRODUCTS 11

isomorphisms

T^S71

=

srn+n

for m 0 and integers n and there are canonical

isomorphisms E ^ 5

n

=

Sn~m

for m 0 and n 0. Sphere spectra are used

to define the homotopy groups of spectra, 7rn(J5) =

hyU(Sn,E),

and a map

of spectra is a weak equivalence if and only if it induces an isomorphism of

spectrum-level homotopy groups.

Although we shall not introduce different notations for space level and spec-

trum level spheres, we shall generally write S for the zero sphere spectrum,

reserving the notation 5° for the two-point space.

The theory of cell and CW spectra is developed by taking sphere spectra as

the domains of attaching maps [38, I§5]. The stable homotopy category hSflJ is

equivalent to the homotopy category of CW spectra. It is important to remember

that homotopy-preserving functors on spectra that do not preserve weak equiv-

alences are transported to the stable category by first replacing their variables

by weakly equivalent CW spectra.

2. External smash products and twisted half-smash products

The construction of our new smash product will start from the external smash

product of spectra. This is an associative and commutative pairing

S?U x S?U' —• S*(U 0 U')

for any pair of universes U and U'. It is constructed by starting with the pre-

spectrum level definition

(E A E')(V © V) = EV A E'V.

The structure maps fail to be homeomorphisms when E and E' are spectra, and

we apply the spectrification functor L to obtain the desired spectrum level smash

product. This external smash product is the one used in [20].

There is an associated function spectrum functor

F : (yUf)op x y(U 0 U') —- S?U

and an adjunction

y(U 0 U')(E A E\ E") ^ yU{E, F{E\ E"))

for E e SU, E' e SU', and E" e ^{U 0 U')\ see [38, p. 69].

Now let J denote the category whose objects are universes U and whose

morphisms are linear isometries. Universes are topologized as the unions of their

finite dimensional subspaces, and the set *?(U,

U1)

of linear isometries U — U'

is given the function space topology; it is a contractible space [38, II. 1.5]. The

category S? constructed in [20] augments to the category J. Since J fails to

have limits and colimits (it even fails to have coproducts), 5? suffers from the

same defects.

In order to obtain smash products internal to a single universe U, we shall

exploit the "twisted half-smash product". The input data for this functor consist

of two universes U and

Uf

(which may be the same), an unbased space A with a

given structure map a : A — ^(U, [/'), and a spectrum E indexed on U. The

output is the spectrum A K E, which is indexed on V. It must be remembered