Langlands correspondence describes an equivalence between Galois representations and automorphic representations under some conditions.
Laurent Lafforgue applying Olivia Caramello thesis described in this answer asked a similar question of whether Langland correspondence is a Morita equivalence of classifying topos $\mathcal{E}_{G} \cong \mathcal{E}_{A}$. Topos are sheaves on a Grothendieck site. There is Galois descent described by Keith Conrad here and automorphic descent described in "The descent map from automorphic representations of $GL(n)$ to classical groups", World Scientific Press (2011) by D. Ginzburg, S. Rallis, and D. Soudry. Both have descent theories which suggests that there should exist a Galois sheaf on some site and an automorphic sheaf on some site.
- What would be the Galois topos $\mathcal{E}_{G}$?
- What would be the automorphic topos $\mathcal{E}_{A}$?
Following this answer and related "An Extension of the Galois theory of Grothendieck" by Joyal and Tierney describes an equivalence between the fundamental group $\pi_1(X,x_0)$ and Galois groups $Gal(L/K)$ using topoi. Following an "Introduction to automorphic representations" by Jayce R. Getz in 15.2 automorphic representations are defined as sheaves of sections $Sh^K$ using a local system and in 15.11 a connection between automorphic representations and Galois representations using a Shimura variety $Sh(G,X)$ is mentioned.
If Galois topoi are the fundamental group $\pi_1(X, x_0)$ and automorphic topoi are local systems, then work has already been done on that problem. See this answer noting that a local system can be derived from a locally constant sheaf. I think this equivalence between the fundamental group and local systems is restating the geometric Langlands correspondence formulated in 3.1 and 3.2 of "Lectures on the Langlands Program and Conformal Field Theory" by Frenkel. That is say this construction using topoi may just be the geometric case.
At least for the Galois side I have some idea of what the topos may be. Groups are examples of topoi. Consider the classical derivation of group cohomology by an (aspherical) Eilenberg-MacLane space $K(\pi_1(X) = G, 1)$ for some group $G$. The Eilenberg-MacLane space $K(G,1)$ is a classifying space $BG$ for the fundamental group $G$. Corollary 1.28 of "Algebraic Topology" by Hatcher provides an explicit construction of a group $G$ being the fundamental group of a 2d cell complex $X_G$ with $G = \pi_1(X_G)$. I think the Galois topos $\mathcal{E}_{G}$ would be $K(G,1)$ for some Galois group $G = Gal(L/K)$.
