finite integer . For a CW complex there is an inverse construction which takes a spectrum and forms a spacecalled the infinite loop space of the spectrum. For a CW complex and this construction comes with an inclusion for every , hence gives a mapwhich is injective. Unfortunately, these two structures, with the addition of the smash product, lead to significant complexity in the theory of spectra because there cannot exist a single category of spectra which satisfies a list of five axioms relating these structures. The above adjunction is valid only in the homotopy categories of spaces and spectra, but not always with a specific category of spectra (not the homotopy category). Ω-spectrum An Ω-spectrum is a spectrum such that the adjoint of the structure map (i.e., the map) is a weak equivalence. The K-theory spectrum of a ring is an example of an Ω-spectrum. Ring spectrum A ring spectrum is a spectrum X such that the diagrams that describe ring axioms in terms of smash products commute "up to homotopy" ( corresponds to the identity.) For example, the spectrum of topological K-theory is a ring spectrum. A module spectrum may be defined analogously. For many more examples, see the list of cohomology theories. 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