Geometric morphism whose counit is epic

Question

Is there a name for the class of geometric morphisms whose counits are epic (equivalently, whose direct images are faithful)?

This notably includes the class of inclusions/embeddings of toposes. By Example A4.6.2 in Johnstone's Sketches of an Elephant: A Topos Theory Compendium, all such morphisms are localic, but since Johnstone doesn't include a name there (and his cross-referencing is generally very thorough) I expect that they don't feature in the text in their own right.

If no one can dig up a reference, I would like to put forward injection as a name for this property, not only because faithfulness gives injectivity on hom-sets, but also because this is the conceptual dual of a geometric surjection, which has monic unit/faithful inverse image.

EDIT: In response to a requested example of an "injection" which fails to be an inclusion, the most obvious first place to search for non-trivial examples is to consider injections into Set, but these turn out to be disappointing. The domain must be a localic topos, and so the global sections functor sends a natural transformation $\alpha: F \to G$ to $\alpha(1)$; this is faithful if and only if every natural transformation of sheaves is determined uniquely by its value on the full space. Originally I thought discrete spaces would work, but if there exists any proper open subset, a disjoint union of two copies of the corresponding representable sheaf has a non-trivial automorphism which the global sections functor ignores, so the locale must in fact be trivial!

See Tim's comment for a more plausible example.

Answer 1

I originally posted the following as a bunch of comments, but maybe it's better to consolidate into a not-really-an-answer:

• Note that $f_*$ is faithful iff it is conservative: to see that conservative implies faithful, note that $u=v:c\to d$ if and only if $Eq(u,v)\to c$ is an isomorphism and use the fact that $f_∗$ preserves equalizers; to see that faithful implies conservative, use the fact that faithful functors reflect epimorphisms and monomorphisms, and any epic monic in a topos is an iso. (I'm not sure how much of this generalizes to ∞-topoi...). Anyway, that's another equivalent formulation to look for.

• I once asked a question about (pseudo)monomorphisms of (∞)-topoi. Alexander Campbell pointed out a paper of Bunge and Lack where pseudomonomorphisms of topoi are discussed; they apparently don't know an example which is not an embedding. Which begs the question: do you know an example of an "injection" in your sense which is not an embedding? I'd also be curious about any example where $f_∗$ is not injective on isomorphism classes

• Note that $f_∗$ is a pseudomonomorphism of underlying categories iff it is faithful and is a full subgroupoid inclusion on underlying groupoids -- so this might be a natural condition to add to faithfulness of $f_∗$ in case you find you need to tweak the definition in search of better properties.

• Finally, note that if f is an "injection" in your sense, then so is the surjective part of its surjection / embedding factorization. So it would be of interest to study the surjective "injections". If $f: \mathcal X \to \mathcal Y$ is surjective, then $f_∗:\mathcal Y \to \mathcal X$ is comonadic. So $f$ will be an "injective" surjection iff it exhibits $\mathcal Y$ as the coalgebras for a left-exact comonad $G: \mathcal X \to \mathcal X$ such that $G$ is faithful (equivalently, conservative). I keep meaning to learn more about left exact comonads... perhaps this is a common case to be in... Anwway, an geometric morphism is then an "injection" iff it factors as coalgebras for a conservative lex comonad followed by an embedding.

Regarding the terminology: I get the sense that because inverse and direct images play rather different roles in topos theory (after all, inverse images preserve finite limits in addition to being left adjoints), it's typically not the case that taking a definition in topos theory and reversing the roles of the inverse and direct image functors yields a "similarly useful" definition. So I would be hesitant to use the term "injection" simply because of the formal duality to "surjection". That being said, I don't have anything against this terminology, but perhaps the experts might.