Functions, Maps, and Homotopies of Spectra
There are three natural categories whose objects are spectra, whose morphisms are the functions, or maps, or homotopy classes defined below.
A function between two spectra E and F is a sequence of maps from En to Fn that commute with the maps ΣEn → En+1 and ΣFn → Fn+1.
Given a spectrum, a subspectrum is a sequence of subcomplexes that is also a spectrum. As each i-cell in suspends to an (i + 1)-cell in, a cofinal subspectrum is a subspectrum for which each cell of the parent spectrum is eventually contained in the subspectrum after a finite number of suspensions. Spectra can then be turned into a category by defining a map of spectra to be a function from a cofinal subspectrum of to, where two such functions represent the same map if they coincide on some cofinal subspectrum. Intuitively such a map of spectra does not need to be everywhere defined, just eventually become defined, and two maps that coincide on a cofinal subspectrum are said to be equivalent. This gives the category of spectra (and maps), which is a major tool. There is a natural embedding of the category of pointed CW complexes into this category: it takes to the suspension spectrum in which the nth complex is .
The smash product of a spectrum and a pointed complex is a spectrum given by (associativity of the smash product yields immediately that this is indeed a spectrum). A homotopy of maps between spectra corresponds to a map, where is the disjoint union with * taken to be the basepoint.
The stable homotopy category, or homotopy category of (CW) spectra is defined to be the category whose objects are spectra and whose morphisms are homotopy classes of maps between spectra. Many other definitions of spectrum, some appearing very different, lead to equivalent stable homotopy categories.
Finally, we can define the suspension of a spectrum by . This translation suspension is invertible, as we can desuspend too, by setting .
Read more about this topic: Spectrum (homotopy Theory)