There are standard notions of "surjections" and "embeddings" of toposes. However, not every surjection is an epimorphism, and not every regular monomorphism is an embedding. (EDIT: as Alexander Campbell points out in the comments, in this higher context, regular mono does not imply mono. So perhaps embeddings are not as strange as I make them out to be here. I still find it strange that most surjections are not epimorphisms, though.)
Let $f: \mathcal Y \to \mathcal X$ be a geometric morphism. Recall that $f$ is said to be
surjective if $f^\ast$ is conservative (or equivalently, $f^\ast$ is comonadic.)
an embedding if $f_\ast$ is fully faithful (equivalently, $f^\ast$ is a localization).
I'd say the correct notion of monomorphism / epimorphism is the $(\infty,1)$-categorical one: $f$ is a monomorphism iff the canonical square $f \circ 1 = f \circ 1$ is a pullback, and dually for epimorphisms. Since $(\infty,1)$-colimits of topoi are computed by taking $(\infty,1)$-limits of the inverse image functors between the underlying categories, $f$ is an epimorphism iff $f^\ast$ is a monomorphism of $(\infty,1)$-categories. That is,
- $f$ is an epimorphism iff $f^\ast$ is a replete subcategory inclusion, i.e. $f^\ast$ reflects the property of being isomorphic and is an inclusion of path components on hom-spaces.
(Note that the coalgebras for any accessible left exact comonad on $\mathcal Y$ is an $\infty$-topos $\mathcal X$ which admits a canonical surjection from $\mathcal Y$ which will typically not be an epimorphism.)
As for monomorphisms, clearly if $f$ is an embedding, then it is a monomorphism. But not even every "regular monomorphism" is an embedding (EDIT: which is not to say that not every monomorphism is an embedding -- see Alexander Campbell's comment below!). For example, if $F,G: C \to D$ are functors, then the $(\infty,1)$-equalizer of the induced geometric morphisms $Psh(C) \to Psh(D)$ is presheaves on the iso-inserter of $F$ and $G$. The canonical map $Psh(IsoIns(F,G)) \to Psh(C)$ typically fails to be an embedding. Anyway, this leaves me with the question:
Question: What are the monomorphisms of topoi, or of $\infty$-topoi?
I expect this may be complicated, given how complicated monomorphisms of affine schemes are (a category which behaves in some ways similarly to the category of toposes).
Note that because every embedding is a monomorphism, by the surjection / embedding factorization system it suffices to determine which surjections are monomorphisms.