Notion of path in a topos

Question

Given a theory $T$, we can form a syntactic site $(C_T,J)$ and get the classifying topos $\mathcal E = Sh(C_T,J)$ of $T$. The points of $\mathcal E$ are defined to be geometric morphisms $Sets\to \mathcal E$, and these points correspond to models of the theory $T$.

My question is then whether there is a way to geometrically interpret homomorphisms of models of $T$ in the classifying topos $\mathcal E$. In particular, given points $P_1,P_2:Sets\to \mathcal E$ with corresponding models $M_1,M_2$ and a homomorphism $\alpha:M_1\to M_2$, is there a notion of a "path" between the points $P_1$ and $P_2$? If so, are these simply the natural transformations between $P_1$ and $P_2$?

Answer 1

A great reference for this is Sheaves in Geometry and Logic by Saunders Mac Lane and Ieke Moerdijk. Chapters VIII and X are completely dedicated to classifying topoi. Any further references in this answer will also be to that book. Also, whenever I write "topos" I mean "Grothendieck topos".

Given any theory $T$, and any topos $\mathcal{E}$, we can form it the category of models of $T$ in $\mathcal{E}$. The arrows in this category are the homomorphisms of models you mentioned. We denote this category by $\mathbf{Mod}(T; \mathcal{E})$. So if we take $\mathcal{E} = \mathbf{Set}$ then we get ordinary models and homomorphisms. That is, $\mathbf{Mod}(T; \mathbf{Set})$ becomes the category of models and homomorphisms as we know it from model theory.

You ask about the connection between these homomorphisms and the classifying topos. First, to talk about the classifying topos, we have to restrict ourselves to geometric logic (this was not in your question, so I wanted to add this point). Let me first introduce some notation: $\mathbf{Topos}$ denotes the category of topoi with geometric morphisms as arrows, and for topoi $\mathcal{E}$ and $\mathcal{F}$ we have a category $\mathbf{Topos}(\mathcal{E}, \mathcal{F})$ where the objects are geometric morphisms $\mathcal{E} \to \mathcal{F}$ and arrows are natural transformations (either between the inverse image parts or the direct image parts, doesn't really matter which, as long as you pick one). Now we can state the definition of a classifying topos (which can also be found on page 433 of Mac Lane and Moerdijk).

Definition. Given a geometric theory $T$, the classifying topos $\mathcal{B}(T)$ for $T$ is then a topos such that for every other topos $\mathcal{E}$ we have an equivalence of categories $$ \mathbf{Mod}(T; \mathcal{E}) \simeq \mathbf{Topos}(\mathcal{E}, \mathcal{B}(T)), $$ that is natural in $\mathcal{E}$.

So, filling in $\mathcal{E} = \mathbf{Set}$ this tells us that $$ \mathbf{Mod}(T; \mathbf{Set}) \simeq \mathbf{Topos}(\mathbf{Set}, \mathcal{B}(T)). $$ Thus homomorphisms of models in $\mathbf{Set}$ do indeed correspond to natural transformations of points on $\mathcal{B}(T)$,

In all of this I did not mention the syntactic site $(C_T, J)$. That is because this is how a classifying topos is defined. The syntactic site is a way to construct such a classifying topos, and indeed it shows us that every geometric theory has a classifying topos. This construction goes essentially as follows:

1. It is not hard to see that a certain type of functors $C_T \to \mathcal{E}$ correspond to models of $T$ in $\mathcal{E}$. Here by "certain type" I mean left exact, regular epi preserving, and preserving joins up to a big enough cardinality. So essentially precisely that what is necessary to preserve geometric logic.
2. Then writing out definitions, one can also fairly easily see that natural transformations between such functors $C_T \to \mathcal{E}$ correspond to homomorphisms of models.
3. By Diaconescu's theorem (Theorem VII.7.2) we have an equivalence of categories $$ \mathbf{FlatCon}((C_T, J), \mathcal{E}) \simeq \mathbf{Topos}(\mathcal{E}, Sh(C_T, J)), $$ where $\mathbf{FlatCon}((C_T, J), \mathcal{E})$ denotes the flat continuous functors $C_T \to \mathcal{E}$ (continuinity depends on the topology $J$) with natural transformations. These flat continuous functors turn out to be precisely those from point 1.
4. Putting everything together, you can conclude that $Sh(C_T, J)$ is the classifying topos for $T$.

So in step 3 a lot of the magic comes from Diaconescu's theorem, and the proof of that theorem is not very easy. But if you accept that theorem, you might be able to see how homomorphisms correspond to natural transformations. They correspond to natural transformations of functors $C_T \to \mathcal{E}$, and those correspond to natural transformations of geometric morphisms.

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Finally, I should mention something about the geometric logic part. There is a very good article from Carsten Butz and Peter Johnstone, named Classifying toposes for first-order theories (from 1998). As the title suggests, they investigate classifying toposes where we allow $T$ to be a first-order theory. The idea is that instead of geomtric morphisms, one looks at open geometric morphisms because these are precisely the ones that preserve full first-order logic (or well, their inverse image part does). Then it turns out that we cannot always build a first-order classifying topos in that sense, but they characterise precisely those theories that do admit such a first-order classifying topos. So there is definitely something to be said there.