Enriched category theory
Stable homotopy theory
An ordinary locally small category has for any ordered pair of objects a hom-set —an object in the category Set.
For more generally an enriched category over a closed monoidal category , there is – by definition – for all an object that plays the role of the “collection of morphisms” from to .
Accordingly, if here is a category of “spaces” of sorts, then one also speaks of a hom space. For instance if =vector spaces such that is a linear category, then hom-spaces are vector spaces.
Or if =topological spaces such that is topologically enriched category, then hom-spaces are topological spaces. If this last example is regarded in the context of homotopy theory, then one may consider just the homotopy type of these topological spaces, which is equivalently modeled as a simplicial set (Kan complex) and thought of as an infinity-groupoid. In all these cases people still often speak of “hom spaces”, but see at derived hom space.
Revised on November 4, 2016 09:18:24
by Urs Schreiber