Spectrum (homotopy theory)

In algebraic topology, a branch of mathematics, a spectrum is an object representing a generalized cohomology theory. There are several different categories of spectra, but they all determine the same homotopy category, known as the stable homotopy category.

There are many variations of the definition: in general, a "spectrum" is any sequence ${\displaystyle X_{n}}$ of pointed topological spaces or pointed simplicial sets together with the structure maps ${\displaystyle S^{1}\wedge X_{n}\to X_{n+1}}$.

The treatment here is due to Adams (1974): a spectrum (or CW-spectrum) is a sequence ${\displaystyle E:=\{E_{n}\}_{n\in \mathbb {N} }}$ of CW-complexes together with inclusions ${\displaystyle \Sigma E_{n}\to E_{n+1}}$ of the suspension ${\displaystyle \Sigma E_{n}}$ as a subcomplex of ${\displaystyle E_{n+1}}$.

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