If $G$ is, for example, a finite directed graph, one can attach to it a topos $T_G$ whose objects are "$G$-sheaves". A $G$-sheave $F$ is the data of:
For each verticies $x$ a set $F(x)$, for each arrow $v:x \rightarrow y$ an arrow $F(v) : F(y) \rightarrow F(x)$ and such that for each verticies $x$, the natural map:
$$ F(x) \rightarrow \prod_{y \rightarrow x} F(y)$$ is an isomorphism.
One can check that this topos $T_G$ is in fact an etendu and corresponds to the étale groupoid of infinite paths in the graph, which can be used to describe the Leavitt path algebra or the graph $C^*$-algebra. Its a nice example where a simple but non sub-canonical site give rise to a more subtle structure.
I have known this construction for a long time, but I can't remember where I learned it !
Now I would like to attribute it correctly... Does someone know where it first appears or at least a paper mentioning it ?
(I found a reference to a 1989 paper of W.Lawvere "Qualitative Distinctions between some Toposes of Generalized Graphs" which mention something about small toposes of certain graph but unless I missed something in that paper I don't think that is what I'm looking for)
Edit: As I couldn't find any reference to this in the literature, I gave a brief overview of this construction in paragraph 5.6 of my last preprint, but I still have the impression it was known before and I learned it from somewhere...
