Locally small

Small and large categories

A category ${\displaystyle {\mathcal {C}}}$ is called small if both ${\displaystyle \operatorname {ob} ({\mathcal {C}})}$ and ${\displaystyle \operatorname {mor} ({\mathcal {C}})}$ are actually sets and not proper classes, and large otherwise. A locally small category is a category such that for all objects ${\displaystyle a}$ and ${\displaystyle b}$, the hom-class ${\displaystyle \operatorname {hom} (a,b)}$ is a set, called a homset. Many important categories in mathematics (such as the category of sets), although not small, are at least locally small. Since, in small categories, the objects form a set, a small category can be viewed as an algebraic structure similar to a monoid but without requiring closure properties. Large categories on the other hand can be used to create "structures" of algebraic structures.