nLab
hom-object

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Idea

An ordinary locally small category π’ž\mathcal{C} has for any ordered pair of objects x,yx,y a hom-set π’ž(x,y)\mathcal{C}(x,y)β€”an object in the category Set.

For π’ž\mathcal{C} more generally an enriched category over a closed monoidal category 𝒱\mathcal{V}, there is – by definition – for all x,yx,y an object π’ž(x,y)∈objV\mathcal{C}(x,y) \in obj V that plays the role of the β€œcollection of morphisms” from xx to yy.

Accordingly, if here 𝒱\mathcal{V} is a category of β€œspaces” of sorts, then one also speaks of a hom space. For instance if 𝒱\mathcal{V} =vector spaces such that π’ž\mathcal{C} is a linear category, then hom-spaces are vector spaces.

Or if 𝒱\mathcal{V}=topological spaces such that π’ž\mathcal{C} 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 β€œhom spaces”, but see at derived hom space.

Examples

  • The category Prop of propositions is a enriched over itself. Hence for any two proposition a,ba,b, there is a hom-proposition Prop(A,B)Prop(A,B). This is the implication proposition aβ‡’ba \implies b.

  • The category Set of sets is a enriched over itself. Hence for any two sets A,BA,B, there is a hom-set Set(A,B)Set(A,B). This is the function set Aβ†’BA \to B.

  • The category Grpd of groupoids is a enriched over itself. Hence for any two groupoids A,BA,B, there is a hom-groupoid Grpd(A,B)Grpd(A,B). This is the functor category Func(A,B)Func(A,B).

  • The category Pos of posets is a enriched over itself. Hence for any two posets A,BA,B, there is a hom-poset Pos(A,B)Pos(A,B), the poset of monotones.

homotopycohomologyhomology
[S n,βˆ’][S^n,-][βˆ’,A][-,A](βˆ’)βŠ—A(-) \otimes A
category theorycovariant homcontravariant homtensor product
homological algebraExtExtTor
enriched category theoryendendcoend
homotopy theoryderived hom space ℝHom(S n,βˆ’)\mathbb{R}Hom(S^n,-)cocycles ℝHom(βˆ’,A)\mathbb{R}Hom(-,A)derived tensor product (βˆ’)βŠ— 𝕃A(-) \otimes^{\mathbb{L}} A

Last revised on May 17, 2021 at 10:24:54. See the history of this page for a list of all contributions to it.