Spectrum (Homotopy Theory)

All Updates

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 Σ*E*<sub>*n*</sub>→*E*<sub>*n*+1</sub> and Σ*F*<sub>*n*</sub>→*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...

Read More

A

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

Read More

No messages found

about this page

for companies, colleges, celebrities or anything you like.Get updates on MyPage.

Create a new Page