nLab hom-object

Idea

An ordinary locally small category $\mathcal{C}$ has for any ordered pair of objects $x,y$ a hom-set $\mathcal{C}(x,y)$—an object in the category Set.

For $\mathcal{C}$ more generally an enriched category over a closed monoidal category $\mathcal{V}$, there is – by definition – for all $x,y$ an object $\mathcal{C}(x,y) \in obj V$ that plays the role of the “collection of morphisms” from $x$ to $y$.

Accordingly, if here $\mathcal{V}$ is a category of “spaces” of sorts, then one also speaks of a hom space. For instance if $\mathcal{V}$ =vector spaces such that $\mathcal{C}$ is a linear category, then hom-spaces are vector spaces.

Or if $\mathcal{V}$=topological spaces such that $\mathcal{C}$ 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 “hom spaces”, but see at derived hom space.

Examples

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

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

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

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

Related concepts

• enriched hom-functor

• hom-object

• hom-set, hom-group, hom-category

• hom-functor

• derived hom-space

• hom-functor

• enriched hom-functor, internal hom-functor

• derived hom-functor

	homotopy	cohomology	homology
	$[S^n,-]$	$[-,A]$	$(-) \otimes A$
category theory	covariant hom	contravariant hom	tensor product
homological algebra	Ext	Ext	Tor
enriched category theory	end	end	coend
homotopy theory	derived hom space $\mathbb{R}Hom(S^n,-)$	cocycles $\mathbb{R}Hom(-,A)$	derived tensor product $(-) \otimes^{\mathbb{L}} A$