A separator, or separating family, in a category is a full subcategory $\mathcal{S} \hookrightarrow \mathcal{E}$ of a category $\mathcal{E}$ which satisfies the following:
- For any parallel pair of arrows $f, g : A \rightrightarrows B$ in $\mathcal{E}$, if for all $S \in \mathcal{S}$ and $s : S \to A$ we have $fs = gs$, then $f = g$.
I’m wondering what the right 2-categorical analogue of this notion is; for example, one which works well for the 2-category of categories internal to a topos. Has this been worked out, and does it appear in the literature?
Thanks!
The question reduces to that or what a faithful 2-functor is: you want the Yoneda embedding, restricted to the separator, to remain faithful.
The strictest possible notion, that a separator separates 1-morphisms in the usual sense and 2-morphisms in the analogous sense, is probably fine for some purposes. However, it's perverse: we've proposed that a 2-functor $F:K\to L$ is faithful when it induces faithful, injective-on-objects functors $K(a,b)\to L(Fa,Fb)$ for every $a,b$. But this is not invariant under equivalence. In fact, up to equivalence, a faithful injective-on-objects functor is no different than a faithful functor.
So you can consider adding additional conditions on $K(a,b)\to L(Fa,Fb)$. Natural options are that it be conservative, or that it be injective on isomorphism classes of objects. But these are independent conditions, which are again both independent of being injective-on-objects! I think the lesson of this all is that "faithful", and thus "separator", are not words that generalize naturally to 2-category theory, in essence because there isn't a single obvious notion of monomorphism of categories.