What are the monomorphisms of ($\infty$-)toposes?

Question

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.

Answer 1

Edit: My original answer contained a big mistake, that I can't fix. A long time I had thought ago about monomorphisms of locales, and I wrongly convince myself that everything would generalizes to toposes, but I now realize things are more complicated than this... All my apologies about this.

I still can use what I said before to give an example of monomorphisms that are not embeddings, which is what I will explain now.

Given a locale $L$, there is an other locale $DL$, called the dissolution of $L$, endowed with a geometric morphism $ DL \rightarrow L$, which is universal for geometric morphism $p:K \rightarrow L$ such that for each open subspace $u \subset L$ (I.e. elements $u \in \mathcal{O}(L)$ of the frame corresponding to $L$), $p^* u$ is a complemented open subspace in $K$. $DL$ is explicitly constructed as the frames of nuclei of $L$, see for example Sketches of an elephant section C.1 for this construction.

Claim: For any non-boolean locale $L$, the geometric morphism $Sh(DL) \rightarrow Sh(L)$ is a monomorphisms of topos (or $\infty$-topos) that is not an embedding.

Indeed, $DL \rightarrow L$ is always a surjection, so if it is an embeddings it is an isomorphism, which only happen when $L$ is boolean.

Moreover, the map $DL \rightarrow L$ is also clearly a monomorphism in the $1$-category of locale: maps to $DL$ are a subset of maps to $L$ (those that send every open to a complemented element)

But as the functor $L \mapsto Sh(L)$ from the category of locales to the category of topos/$\infty$-topos is a right adjoint, it preserves finite limits and monomorphism. So $Sh(DL) \rightarrow Sh(L)$ is a monomorphism in the category of topos/ $\infty$-topos as well.