## Franklin's Notes

### Category of categories

We have to be careful when defining a category of categories in order to avoid Russell's paradox. However, we are able to skirt this issue by placing size restrictions on the elements of our categories. For instance:

• $\mathsf{Cat}$ is defined as the category of small categories whose morphisms are functors .

• $\mathsf{CAT}$ is defined as the category of locally small categories whose morphisms are functors.

This suggests a natural way of defining isomorphism of categories. For two small categories, in $\mathsf{Cat}$ or $\mathsf{CAT}$, we say that they are isomorphic iff there exists an isomorphism, or an invertible functor, between them. If we wish to generalize beyond these size-restricted categories, we may say that two arbitrary categories $\mathsf{C}$ and $\mathsf{D}$ are isomorphic iff there exist functors $F:\mathsf{C}\to \mathsf{D}$ and $G:\mathsf{D}\to\mathsf{C}$ so that $FG$ and $GF$ are identity functors on $\mathsf{D}$ and $\mathsf{C}$ respectively.