Spectrum (homotopy theory)

Spectrum (Homotopy Theory)

Spectrum (homotopy theory)

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In algebraic topology, a branch of mathematics, a spectrum is an object representing a generalized cohomology theory. There are several different constructions of categories of spectra, any of which gives a context for the same stable homotopy theory.

The definition of a spectrum

One definition of a spectrum is that a spectrum is a sequence <math>E:= _ </math> of CW complexes together with cellular inclusions <math> Sigma E_n to E_ </math>.There are many variations of this definition.

Functions, maps, and morphisms of spectra

There are three natural categories whose objects are spectra, whose morphisms are the functions, or maps, or morphisms defined below.

A function between two spectra E and F is a sequence of maps from E<sub>n</sub> to F<sub>n</sub> that commute with themaps &Sigma;E<sub>n</sub>&rarr;E<sub>n+1</sub> and &Sigma;F<sub>n</sub>&rarr;F<sub>n+1</sub>.

Given a spectrum <math>E_n</math>, a subspectrum <math>F_n</math> is a sequence of subcomplexes that is also a spectrum. Noting that each i-cell in <math>E_j</math> becomes an (i+1)-cell in <math>E_</math>, 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...
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