for m 0 and integers n and there are canonical
isomorphisms E ^ 5
for m 0 and n 0. Sphere spectra are used
to define the homotopy groups of spectra, 7rn(J5) =
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 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,
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
(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
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