Definition:Hom Class

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Definition

Let $\mathbf C$ be a metacategory.

Let $C$ and $D$ be objects of $\mathbf C$.

The collection of morphisms $f: C \to D$ is called a hom class and is denoted $\operatorname{Hom}_{\mathbf C} \left({C, D}\right)$.

Also known as

If $\operatorname{Hom}_{\mathbf C} \left({C, D}\right)$ is a set, then it is also called a hom set.

Some authors hyphenate, resulting in hom-class and hom-set.

Also denoted as

When the category $\mathbf C$ is clear, it is mostly dropped from the notation, yielding $\operatorname{Hom} \left({C, D}\right)$.

The hom class is also denoted $\mathbf C \left({C, D}\right)$, or in the case of a functor category, $\operatorname{Nat}(C, D)$.

Also see

• Definition:Hom Functor

Sources

• 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 1.8$: Definition $1.12$
• 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 2.7$