I have a longstanding interest in topos theory. (See this MSE search of my questions about topos theory.) I am studying for a postgraduate research degree in linear algebraic group theory. Naturally, I wonder what connections there are between the two.
A quick Google search produces this page, in which it states
In 1973, Grothendieck gave three lectures series at the Department of Mathematics of SUNY at Buffalo, the first on ‘Algebraic Geometry’, the second on ‘The Theory of Algebraic Groups’ and the third one on ‘Topos Theory’.
If, my reasoning goes, Grothendieck worked on algebraic groups around the time he worked on topos theory, then perhaps there's a connection between the two!
My Question:
What, if anything, is the connection between algebraic groups and topoi?
Further Context:
It has been too long since I sat down & did any topos theory, so please pitch answers at an advanced undergraduate level, if possible.
A Small Request:
Please let me know where I can find out more about the connection(s), if any exist.
Here's a nontrivial connection between topoi and algebraic groups:
More details: First of all, one might ask the question: what are and why do we care about group schemes? Here's an explanation for those who are not (yet) familiar with schemes: Very roughly, a group scheme is like an algebraic group minus some finiteness (and other) conditions that can have any ring (or even scheme) as its base. The category of group schemes over a field $k$ contains the category of algebraic groups over $k$ as a full subcategory. Even if one only cares about the things that happen over fields, having the more general theory as an "intermediate step" can help.
Here's an example from algebraic groups (but alas, not linear algebraic groups): Suppose you have some abelian variety $A$ over a field $K$ which is the quotient field of a Dedekind domain $D$. It can be very useful to consider the "reduction of $A$ mod $\mathfrak{p}$" for a prime ideal $\mathfrak{p} \subset D$. But this does not make immediate sense: There's in general no field homomorphism from $K$ to $D/\mathfrak{p}$ (they can have different characteristics, for example). What one does to solve the problem is to observe that $D$ admits ring homomophisms to both $K$ and $A/\mathfrak{p}$, so one tries to find an object defined over $D$ that interpolates between $A$ and the reduction of $A$ mod $\mathfrak{p}$ that we're trying to define. It turns out in our situation, that this is always possible, and even in a "universal", i.e. best possible way: there exists a Néron model of $A$, let's call it $\mathcal A$ that is a smooth group scheme over $D$ such that the generic fiber (that is, the base change along the inclusion $D \hookrightarrow K$) of $\mathcal A$ is $A$. Having a ring homomorphism $D \to D/\mathfrak{p}$ we can just base change $\mathcal A$ to get something that deserves to be called "the reduction of $A$ mod $\mathfrak{p}$" (You don't need the full strength of the Néron model to do this, but you need some kind of interpolating object, if you want to be formal.)
Having established that group schemes over a more general base can be useful, let's look at how topos theory can automatically extend theorems about algebraic groups over a field (or more generally, group schemes over a field) to theorems about groups schemes over a more general base:
There's an internal logic to every topos and the internal logic of the small Zariski topos has been systematically used by Ingo Blechschmidt (see his PhD thesis for an introduction) to give alternative, sometimes much simpler proofs of theorems in algebraic geometry. One can translate between internal logic and external logic and often objects appear simpler in the internal logic. For instance, let $S$ be an reduced scheme, then the structure sheaf $\mathcal O_S$ is in the internal logic just a field. And sheaves of modules over $(S,\mathcal O_S)$ are just vector spaces in the internal logic. Along the same lines, we can reason about group schemes internally. The simplest case is that $S$ is reduces and the group scheme $G$ is affine over $S$. Then internally, $G$ corresponds to a commutative Hopf algebra over the field $\mathcal O_S$.
Now if we take a result that can be proven in intuitionistic logic, then one can infer a result on group schemes (over a reduced scheme). Sometimes, the reduced condition can be removed (by a "classical", i.e. external argument).
Take for instance the case of tori. Over a field $k$ with a separable closure $k^s$, one can show constructively that the assignment: $T \mapsto \operatorname{Hom}_{k^s\textrm{-}\mathbf{Grp}}(T_{k^s},\Bbb G_{m,k^s})$ which takes a torus to the abelian group of characters equipped with an action from the absolute Galois group $\mathrm{Gal}(k^s/k)$. Defines a duality between the category of tori over $k$ and the category of free abelian groups of finite rank with a continuous $\mathrm{Gal}(k^s/k)$-action.
Now what happens over a general base? For this question, one can actually always reduce to the reduced case (pun intended). So let $S$ be a scheme. Let's also assume some technical conditions so that the following is actually the correct definition of a torus over $S$: assume that $S$ is connected, locally noetherian and geometrically unibranch. Then a torus over $S$ may be defined as a group scheme $G$ over $S$ such that there exists a finite étale morphism $T \to S$ such that $T\times_S G$ is isomorphic to $\Bbb G_{m,T}^n$ for some $n$. Choose a geometric point $\overline{x}=\mathrm{Spec}(\Omega)$ of $S$, then it turns out that the functor $T \mapsto \mathrm{Hom}_{\Omega\textrm{-}\mathbf{Grp}}(\overline{x} \times_S T,\Bbb G_{m,x})$ defines a duality between tori over $S$ and free abelian groups of finite rank with a continuous action of $\pi_1^{ét}(S,\overline{x})$. With some effort, this can be proven from the version over fields by means of topos theory, but as far as I am aware, this is not written down anywhere.