A spectrum
is (roughly) a
sequence of based spaces
, provided with maps
(where denotes
suspension). There are many different definitions of the category of
spectra, but they all yield the same homotopy category, known as the
stable homotopy category. The homotopy category of spectra forms
a triangulated category (with shifts given by suspension and looping);
if we associate to a space the suspesion spectrum
with -space
, the homotopy classes of maps between the suspension
spectra of and are the stable homotopy classes of maps between
and . There is a correspondence between generalized
(co)homology theories and spectra as follows. Given a generalized
cohomology theory , the Brown representability theorem gives a
(universal) space such that
; the
suspension axiom provides the required structure maps for to
form a spectrum. Conversely, for any spectrum
, the
functor
is a generalized
cohomology theory, and
is a
generalized homology theory.

Jeffrey Herschel Giansiracusa
2005-06-27