# Is there a name for categories whose objects are sets?

A category $$C$$ is small if $$\operatorname{Obj}(C)$$ and $$\operatorname{Hom}(C)$$ are sets.

A category is locally small if for all objects $$a,b$$, the morphisms $$\operatorname{Hom}(a,b)$$ is a set.

Is there a name for categories whose objects are sets? For example $$\mathrm{Set}, \mathrm{Grp}, \mathrm{Top}$$.

• @NoahSchweber For example, a category on a topological space where the category consists of open sets, and a morphism between $U$ and $V$ exists if $U \subset V$? – Al Jebr Sep 5 '19 at 20:12
• @AlJebr Or the archetypal example of a non-concrete category: the category whose objects are CW-complexes and the morphisms $X\to Y$ are the equivalence classes of continuous functions $X\to Y$ modulo homotopy. – Pece Sep 6 '19 at 6:35