The Lefschetz fixed-point theorem is a nice theorem in topology which counts the number of fixed points (counted with multiplicities) of a continuous map $f\colon X\to X$ using the traces of the homomorphisms $$H^n(X,\mathbb Q)\to H^n(X, \mathbb Q)$$ induced by $f$.
Proving an analogue of that statement for schemes instead of topological spaces was a huge step towards a proof of the Weil conjectures.
The cohomology of schemes is a special case of the cohomology of topoi (consider the étale topos of a scheme), which makes me wonder:
Question: Can one formulate and prove a Lefschetz fixed-point theorem for all Grothendieck topoi and geometric morphisms? If yes, does that theorem imply both the topological Lefschetz fixed-point theorem and its analogue for schemes?
This question is motivated by Grothendieck's philosophy stating that topoi are the real objects of "topology". They are a very general type of "space" which "subsumes" (in a sense) all other spaces (topological spaces can be regarded as sheaf topoi, schemes can be regarded as étale topoi, ...). So conceptually it would be nice if one could generalize some of the theorems in topology and algebraic geometry to topoi.
