# Spectrum

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