Definition:Hom Class

Definition
Let $\mathbf C$ be a metacategory.

Let $X$ and $Y$ be objects of $\mathbf C$.

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

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

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

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

Another common notation is $\mathbf C \left({X, Y}\right)$.