Spectrum (homotopy theory)

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In homotopy theory, a branch of mathematics, a spectrum is an object in a category constructed for the purposes of stable homotopy theory, starting with the category of CW complexes and aiming to make the suspension functor S invertible.

The objects of the category of spectra are sequences

E_n

of CW complexes as pointed spaces, such that

SE_n

is homeomorphic to a subcomplex of E_{n + 1}.

Morphisms in the category of spectra are defined in a non-obvious way, as a type of partial function, subject to an equivalence relation: essentially from the minimum mapping information that is possible, allowing S to be applied to bring any given cell into the domain.

The construction is related, on a conceptual level at least, to that of the derived category, but using spaces rather than algebra.

[edit] History

A version of the concept of a spectrum was introduced in the 1958 doctoral dissertation of Lima. His advisor Edwin Spanier wrote further on the subject in 1959. There was development of the topic by J. M. Boardman, amongst others. The now-standard setting came together during the mid-1960s.