nLab function spectrum

Idea

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 symmetric 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 $[\Sigma_\infty X, E]$.

Related concepts

• smash product of spectra, symmetric monoidal smash product of spectra

• stable splitting of mapping spaces

References

• 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)