What is the geometric significance of fibered category theory in topos theory?

Question

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.

Answer 1

My short answer, is that in the great majority of situation where this $\mathcal{Y}/f^*$ plays a role, it is a mistake to see it as a topos. There are exceptions, but most of the time it is not meant to be a topos, but rather a $\mathcal{X}$-indexed topos.

---

To clarify the discussion, I will follow Joyal's terminology:

• I'm calling "Logos" what we usually call a Grothendieck topos. Morphisms of logos are the continuous left exact functor, i.e. the $f^*$.

• Toposes are the object of the opposite category of the category of logos. If $\mathcal{X}$ is a topos I denote by $Sh(\mathcal{X})$ the corresponding logos, which I think of as the category of sheaves of sets over $\mathcal{X}$. If I say "$x \in \mathcal{X}$" I mean that $x$ is a point (or maybe a generalized point) of $\mathcal{X}$.

This is mean't to mimic the picture of the conection between frames and locales (where the frame corresponding to a locales is denoted by $\mathcal{O}(\mathcal{X})$.

---

Now, The topos corresponding to the logos $Sh(\mathcal{Y})/f^*$ is the following object:

$$ \left(\mathbb{S} \times \mathcal{Y} \right) \coprod_{\mathcal{Y}} \mathcal{X}$$

where $\mathbb{S}$ is the Sierpinski topos, i.e. corresponding to the topological space with one closed point and one open point. (this is easily seen using that colimits of toposes corresponds to limits of logos, which are justs limits of categories)

So you can think of it as a kind of "cone construction" where the Sierpinski space is used as an interval. This object can indeed be interesting, and you can somehow guess that it will be especially interesting for the theory of local morphisms as you mentioned in your post.

But, in my opinion, this topos as simply nothing to do with the idea of working with $\mathcal{Y}$ as "an object over $\mathcal{X}$", i.e. to try to think of the map $\mathcal{Y} \rightarrow \mathcal{X}$ by somehow looking the fiber $\mathcal{Y}_x$ for $x \in \mathcal{X}$ and how it varies when $x$ varies in $\mathcal{X}$. Which is what we want to do in all the case you mentioned:

• A proper map is a map whose fiber are compacts, in a "nicely locally uniforme way""

• A locally connected map is map whose fiber are locally connected in a nicely locally uniform way.

• A separated map... well you can actually also see it as a map whose fiber are separated in a locally uniforme way, but that is not quite what the definition you gave really says. The way I think about this definition is that when you have a class of map stable under composition and pullback then it is nice to consider the class of maps whose diagonals have this property, because you then get for free the lemma that "if $f \circ g$ is proper and $f$ is separated then $g$ is proper".

---

Let's now look at what happen when you want to work with $\mathcal{Y}$ as an object over $\mathcal{X}$ :

Essentially, instead of looking at $Sh(\mathcal{Y})$ you want to look at something like $Sh(\mathcal{Y}_x)$ for all $x \in \mathcal{X}$, which should give you some kind of family of logoses parametrized by $x \in \mathcal{X}$.

In the same way that the correct notion of "familly of set parametrized by $x \in \mathcal{X}$ is a sheaves over $\mathcal{X}$ the correct notion of such familly of logos is something like "a sheaves of logos". It is not quite just a "sheaves in the category of logos" because the definition of logos involved some infinitary operations (the infinite coproduct/colimits) which you want to replace by $Sh(\mathcal{X})$-indexed colimits/coproducts. So the correct notion is what I would call an "internal logos" and is a special kind of sheaf of logos (It is exactly a sheaf of logos which admits $Sh(\mathcal{X})$-indexed disjoint and universal coproducts).

Also note that these sheaf of logos in particular gives you the type of structure that you asked about in your last question. (they satisfies stronger property though, like Beck Cheavely conditions related to the fact that they have indexed colimits)

Then for technical reason, we tend to look at sheaves of categories rather as indexed categories or fibered categories, that is why you end up with a fibration over $Sh(\mathcal{X})$.

But if you somehow forget that it was mean't to be a "sheaf of logos" over $Sh(\mathcal{X})$ and see the total category of the fibration as a new logos, then you just get a completely different and new object that have very little to do with what I was describing.

---

I'm finishing with an informal discussion of why the $\mathcal{X}$-indexed logos corresponding to $\mathcal{Y} \rightarrow \mathcal{X}$ should be $Sh(\mathcal{Y})/f^*$. This is not a proof, just some kind of heuristic arguement. The proof is justs that there is a relatively deep theorem saying that the category of $Sh(\mathcal{X})$-indexed logoses is equivalent to the category of toposes over $\mathcal{X}$ and that the equivalence is given by this construction, and is compatible with pullbacks along maps $\mathcal{X'} \rightarrow \mathcal{X}$.

So What should be the "sheaf of logos" (or internal logos) corresponding to $\mathcal{Y} \rightarrow \mathcal{X}$. I need to think about "what should be its section over some étale cover $p: \mathcal{E} \rightarrow \mathcal{X}$".

I want something that, in a continuous way associate to each $e \in \mathcal{E}$ a sheaves over $\mathcal{Y}_p(e)$, i.e. something that continuously in $e \in \mathcal{E}$ and $y \in \mathcal{Y}_{p(e)}$ associate a set. So basically it is a sheaf of set over $\mathcal{Y} \times_{\mathcal{X}} \mathcal{E}$.

Now if $\mathcal{E}$ is the etale space of a sheaves $ E \in Sh(\mathcal{X})$, one has that $Sh(\mathcal{E}) = Sh(\mathcal{X})/E$, and $Sh(\mathcal{Y} \times_{\mathcal{X}} \mathcal{E}) = Sh(\mathcal{Y})/f^* E$

So in the end you do get the fibered category we are talking about.

Answer 2

One interesting perspective is given by Spencer Breiner's thesis. Here, the codomain fibration $\mathcal Y^{[1]} \to \mathcal Y$ of a topos $\mathcal Y$, viewed as a sheaf of "local" logoi on $\mathcal Y$, is regarded as the "structure sheaf" $\mathcal O_{\mathcal Y}$ of a "logical scheme" structure on $\mathcal Y$. (Breiner shows, for instance, that topoi embed fully faithfully into "locally loigoi-ed topoi" in this way).

Then for $f: \mathcal Y \to \mathcal X$ a geometric morphism, the fibration $U_f: \mathcal Y \downarrow f^\ast \to \mathcal X$, viewed as a sheaf of logoi on $\mathcal X$, is simply the pushforward of $\mathcal O_{\mathcal Y}$ to $\mathcal X$.

So in this analogy, considering a sheaf of logoi over a base topos is like considering a sheaf of rings on a scheme. Studying a geometric morphism fibrationally is analogous to studying a morphism of schemes via the associated sheaf of rings.

I think ultimately I'm still confused about how to think about a sheaf of rings on a scheme, but at least it's something that algebraic geometers do consider, so there's the potential to do something with the analogy to geometry.

This is also reminiscent of the notation $\mathcal O_{\mathcal C}$ which is sometimes used for the twisted arrow cateogry of $\mathcal C$.

It would be interesting to see to what extent this perspective illuminates definitions such as "proper geometric morphism"!

Answer 3

See my notes on fibered categories (arXiv:1801.02927) where in the second part I expose the fibered view of gemetric morphisms explaining that bounded geometric morphisms to a base topos SS are equivalent to Grothendieck toposes fibered over SS as proved in Jean-Luc Moens's Thesis from 1982 in Louvain-la-Neuve. In my opinion this is the real justification of geometric morphisms since the usual one as generalized continuous maps is a mere analogy. There is no systematic translation of notions from topology from spaces to locales and/or toposes! Moreover, developing analysis in this setting would be impossible or at least most inconvenient: just imagine to define exp or sin as locale morphisms!