Often in topos theory, one starts with a geometric morphism $f: \mathcal Y \to \mathcal X$, but quickly passes to the Grothendieck fibration $U_f: \mathcal Y \downarrow f^\ast \to \mathcal X$, which is "$\mathcal Y$ regarded as an $\mathcal X$-topos". This maneuver takes one outside of the category of toposes and geometric morphisms, and thus tends to remove me from thinking of toposes geometrically, as generalized spaces.

It seems there should be some geometric interpretation lurking here -- after all, Grothendieck fibrations are "categories varying over a base", and a topos is "a base that can be varied over". But although $\mathcal Y \downarrow f^\ast$ is a topos, the fibration $U_f: \mathcal Y \downarrow f^\ast \to \mathcal X$ is not (the direct image of) a geometric morphism! In fact it has a right adjoint $F^\ast$ which has a further right adjoint $F_\ast: \mathcal Y \downarrow f^\ast \to \mathcal X$. We in fact have a totally connected geometric morphism $F: \mathcal Y \downarrow f^\ast \to \mathcal X$.

Moreover, the fibers of $U_f$ are toposes, but the reindexing functors are most naturally viewed as the *inverse* images of étale geometric morphisms. So we don't naturally have a "topos fibered in toposes" it seems.

For one more perspective, $\mathcal Y \downarrow f^\ast$ is a cocomma object in the 2-category of toposes. As such, $\mathcal Y \downarrow f^\ast \leftarrow \mathcal Y$ is the free co-fibration on $f$ in the 2-category of toposes. I think this is the point of departure for Rosebrugh and Wood. But I have no geometric intuition for what a co-Grothendieck fibration is. And anyway, the functor $U_f$ doesn't even live in the category of toposes.

I'm not really sure what to make of this. So here are some

**Questions:** Let $f: \mathcal Y \to \mathcal X$ be a geometric morphism.

Is there a geometric interpretation of the fibration $U_f: \mathcal Y \downarrow f^\ast \to \mathcal X$?

Does the totally connected geometric morphism $F: \mathcal Y \downarrow f^\ast \to \mathcal X$ enjoy some sort of universal property with respect to $f$? (for example, does the inclusion of the category of toposes and totally connected geometric morphisms into toposes and all geometric morphisms have a left adjoint which is being applied here?)

Is there a fibration in the 2-category of toposes lurking around here somewhere (see the Elephant B4.4 for some discussion of these)?

Is there a general geometric interpretation of an opfibration over a topos whose fibers are toposes, with étale geometric morphisms for reindexing?

**Addendum:** Here are a few places where "indexed" notions come up where the viewpoint that "everything is indexed over the codomain" seems kind of inflexible to me:

A proper geometric morphism $f: \mathcal Y \to \mathcal X$ is one such that $f_\ast$ preserves $\mathcal X$-indexed filtered colimits of subterminal objects. It's a bit awkward, for example, to even discuss the composition of proper morphisms if you're tied to the view that $\mathcal Y$ is internal to $\mathcal X$.

A separated geometric morphism is one such that the diagonal is proper; in particular, a Hausdorff topos is one such that $\mathcal X \to \mathcal X \times \mathcal X$ is proper. It seems very unnatural to me in this context to think of $\mathcal X$ as being primarily an object "internal to $\mathcal X \times \mathcal X$".

Locally connected geometric morphisms also use indexing in one form of their definition. So do local geometric morphisms.

Let me also point out there there is at least one place in *Sketches of an Elephant* where $\mathcal Y \downarrow f^\ast$ plays a role *qua* topos -- in Ch C3.6 on local geometric morphisms, ~~ where it's referred to as a "scone". See in particular Example 3.6.3(f), Lemma 3.6.4, and Corollary 3.6.13~~ The scone is the dual construction $f^\ast \downarrow \mathcal Y$. But in the end of Ch C3.6, Johnstone does consider $\mathcal Y \downarrow f^\ast$ *qua* topos, and shows that it is related to totally connected geometric morphisms in the same way that the scone is related to local geometric morphisms.