nLab hom-set

Definition

For locally small categories

Given objects $x$ and $y$ in a locally small category, the hom-set $hom(x,y)$ is the collection of all morphisms from $x$ to $y$. In a closed category, the hom-set may also be called the external hom to distinguish it from the internal hom.

• We say a category is locally small if this collection is always actually a set instead of a proper class, or in the context of a Grothendieck universe: if this set is a $U$-small set. This holds for many examples, but not for all. For example, if $C$ and $D$ are not small categories, then the functor category $C^D$ (whose morphisms are natural transformations) will not be locally small.

For enriched categories

For a category $C$ enriched over a category $V$, the “hom-set” $C(x,y)$ is an object of $V$, the hom-object.

For internal categories

For $C = (C_0, C_1, s,t,e, c)$ an internal category, the generalized objects of $C$ are morphisms $x: X \to C_0$ and $y: Y \to C_0$, and the “hom-set” becomes the pullback $C(x,y)$ in

$$\array{ C(x,y) & \to & Y \\ \downarrow & \searrow & & \searrow^{y} \\ X & & C_1 & \stackrel{t}\to & C_0 \\ & \searrow^{x} & \downarrow_s \\ & & C_0 }$$

In particular, in a category with a terminal generator $*$, we may identify morphisms $x,y: * \to C_0$ with global objects of $C$ and form $C(x,y)$ as above.

Related concepts

• hom-object

• hom-set, hom-group

  heteromorphism

• hom-category

• hom-space

• derived hom-space

• hom-functor

• enriched hom-functor, internal hom-functor

• derived hom-functor

References

Textbook accounts:

• Saunders MacLane, §I.8 of: Categories for the Working Mathematician, Graduate Texts in Mathematics 5 Springer (1971, second ed. 1997) [doi:10.1007/978-1-4757-4721-8]