hom-set, hom-object, internal hom, exponential object, derived hom-space
loop space object, free loop space object, derived loop space
topology of mapping spaces
evaluation fibration of mapping spaces
loop space, free loop space
differential topology of mapping spaces
manifold structure of mapping spaces
smooth loop space
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stable homotopy theory
loop space object
stable (∞,1)-category of spectra
stable homotopy category
smash product of spectra
symmetric smash product of spectra
A function spectrum or mapping spectrum is the analog of a mapping space in the context of stable homotopy theory.
Given two spectra XX and EE, their function spectrum [X,E][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].