Hom-set

Definition

The collection of all morphisms from ${\displaystyle X}$ to ${\displaystyle Y}$ is denoted ${\displaystyle \mathrm {Hom} _{\mathcal {C}}(X,Y)}$ or simply ${\displaystyle \mathrm {Hom} (X,Y)}$ and called the hom-set between ${\displaystyle X}$ and ${\displaystyle Y}$. Some authors write ${\displaystyle \mathrm {Mor} _{\mathcal {C}}(X,Y)}$, ${\displaystyle \mathrm {Mor} (X,Y)}$ or ${\displaystyle {\mathcal {C}}(X,Y)}$. The term hom-set is something of a misnomer, as the collection of morphisms is not required to be a set; a category where ${\displaystyle \mathrm {Hom} (X,Y)}$ is a set for all objects ${\displaystyle X}$ and ${\displaystyle Y}$ is called locally small. Because hom-sets may not be sets, some people prefer to use the term "hom-class".