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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 $\mathrm{Set}, \mathrm{Grp}, \mathrm{Top}$.

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Essentially, what you have in mind is a concrete category - a category which can be mapped to Set by a faithful functor.

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Often categories whose objects "having underlying sets", and so that the morphisms have "underlying set-maps", are called "concrete categories".

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    $\begingroup$ To clarify the emphasized "and" for the OP: just because a category has objects which are sets (+ additional "stuff") doesn't mean that the morphisms can be appropriately understood as functions in the usual sense. $\endgroup$ – Noah Schweber Sep 5 '19 at 19:47
  • $\begingroup$ Thanks, @NoahSchweber, for the probably-appropriate additional emphasis! :) $\endgroup$ – paul garrett Sep 5 '19 at 19:50
  • $\begingroup$ @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 $U \subset V$? $\endgroup$ – Al Jebr Sep 5 '19 at 20:12
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    $\begingroup$ @AlJebr Well, in that case we can think of the morphisms as inclusion maps, so that's fixable. Less fixable is, for example, the opposite category of such a category, or more interestingly a cobordism category (roughly: objects are manifolds and morphisms are cobordisms). $\endgroup$ – Noah Schweber Sep 5 '19 at 20:41
  • $\begingroup$ @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. $\endgroup$ – Pece Sep 6 '19 at 6:35
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Category theory doesn't really care what its objects are per se. So it's not common for terminology in the field to distinguish categories based on what the objects are. You can ask about categories who behave somewhat like the category of sets (or a subcategory thereof), but that won't tell you what the objects truly are.

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