function spectrum

[[!include mapping space - contents]]

[[!include stable homotopy theory - contents]]

A *function spectrum* or *mapping spectrum* is the analog of a mapping space in the context of stable homotopy theory. It makes the stable homotopy category into a closed category, and together with the smash product of spectra into a smmyetric closed monoidal category

Given two spectra $X$ and $E$, their *function spectrum* $F(X,E)$ is the internal hom in a suitable category of spectra.

In the context of generalized (Eilenberg-Steenrod) cohomology the generalized $E$-cohomology of a topological space $X$ is given by the homotopy groups of the mapping spectrum $[\Sigma_\infty X, E]$.

- Harold Hastings,
*On function spectra*, Proceedings of the AMS, volume 44, Number 1, May 1974 (pdf)

For symmetric spectra:

- Stefan Schwede, example 3.38 of
*Symmetric spectra*(2012)

Last revised on July 7, 2016 at 09:09:13. See the history of this page for a list of all contributions to it.