| Definition: Let $\mathcal C$ be a category and let $c\in \mathcal C$ be an object. | |
| A $\textbf{regular subobject}$ of $c$ is a pair $(x,i)$ where $i$ is a regular monomorphism. | |
| Definition: Let $\mathcal C$ be a category. | |
| $\mathcal C$ is called $\textbf{regular wellpowered}$ if no object in $\mathcal C$ has a proper class of pairwise non-isomorphic regular subobjects. | |
| Definition: A category $\mathcal C$ is called $\textbf{concretizable}$ over a category $\mathcal B$ if there exists a faithful functor $U:\mathcal C\to \mathcal B$. | |
| Theorem: Let $\mathcal{C}$ be a category that admits finite limits. | |
| Then $\mathcal{C}$ is concretizable over $\mathcal{S}\mathrm{et}$ if and only if $\mathcal{C}$ is regular wellpowered. |