The Clowder Project

Let $\kappa $ be a regular cardinal. A category $\mathcal{C}$ is

1. Locally small if, for each $A,B\in \text{Obj}\webleft (\mathcal{C}\webright )$, the class $\textup{Hom}_{\mathcal{C}}\webleft (A,B\webright )$ is a set.
2. Locally essentially small if, for each $A,B\in \text{Obj}\webleft (\mathcal{C}\webright )$, the class
   \[ \textup{Hom}_{\mathcal{C}}\webleft (A,B\webright )/\webleft\{ \text{isomorphisms}\webright\} \]

   is a set.

3. Small if $\mathcal{C}$ is locally small and $\text{Obj}\webleft (\mathcal{C}\webright )$ is a set.
4. $\kappa $-Small if $\mathcal{C}$ is locally small, $\text{Obj}\webleft (\mathcal{C}\webright )$ is a set, and we have $\# {\text{Obj}\webleft (\mathcal{C}\webright )}<\kappa $.