Spectrum (homotopy theory)
From Wikipedia, the free encyclopedia
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
- En
of CW complexes as pointed spaces, such that
- SEn
is homeomorphic to a subcomplex of En + 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 Elon Lages Lima. His advisor Edwin Spanier wrote further on the subject in 1959. There was development of the topic by J. Michael Boardman, amongst others. The above setting came together during the mid-1960s, and is still used for many purposes. Important theoretical advances have however again been made since 1990, so that some recent technical literature uses modified definitions of spectra.
