nLab function spectrum




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 XX and EE, their function spectrum F(X,E)F(X,E) is the internal hom in a suitable category of spectra.

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


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

For symmetric spectra:

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