## Franklin's Notes

### Skeleton category

A category $\mathsf C$ is called skeletal if $x\cong y$ implies $x=y$ for any objects $x,y\in\mathsf C$. That is, $\mathsf C$ is skeletal iff all of its isomorphisms are automorphisms . A skeleton of a category is a subcategory with the property that every object of $\mathsf C$ is isomorphic to precisely one object in the subcategory. We may also define "the" skeleton of a category, denoted $\mathrm{sk}(\mathsf C)$, to be a category whose objects are equivalence classes of isomorphic objects of $\mathsf C$.

An alternative definition of categorical equivalence is to say that two categories are equivalent if and only if they have isomorphic skeletons. We may also say that $\mathrm{sk}(\mathsf C)$ is a terminal object in the category of categories that are equivalent to $\mathsf C$.