nLab
hom-object

Idea

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

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

Examples

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

Related concepts

• hom-object

• hom-set, hom-functor

• hom-category

• hom-space, derived hom-space

	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 $ℝ\mathrm{Hom}(S^n,-)$	cocycles $ℝ\mathrm{Hom}(-,A)$	derived tensor product $(-){\otimes }^{𝕃}A$