# Toposes (topoi) as classifying toposes of groupoids

A famous theorem of Joyal and Tierney says that each Grothendieck topos is equivalent to the classifying topos of a localic groupoid. I believe that Buntz and Moerdijk have shown that if the topos has enough points then one can use a topological groupoid.

I am not a category theorist and I have trouble following the proof in Johnstone's Elephant. I was wondering if someone can explain what is going on for the specific case of presheaves on a monoid. This is a very special kind of topos with enough points and so I am hoping somebody can describe a groupoid explicitly.

If the monoid is cancellative, the topos is an etendue and I know how to get an etale groupoid via inverse semigroup theory so I am interested in the non-cancellative case.

Motivation: I study monoids and so it is interesting to try and understand how a Morita class of groupoids can be associated to a monoid.

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In general, you'll get more than a Morita class. There are non-Morita equivalent localic (or topological) groupoids which have equivalent classifying topoi. –  David Carchedi Dec 24 '11 at 5:29
I thought Moerdijk proved that two localic groupoids with open domain and range maps are Morita equivalent iff they have the same classifying topos (you can always assume the open range). Or maybe I am too used to the etale case and one needs this etale completeness in Johnstone. –  Benjamin Steinberg Dec 24 '11 at 12:53
Indeed, you need to assume etale completeness. –  David Carchedi Dec 24 '11 at 17:09
Is there a nice way to understand etale completeness? The definition was hard for me to parse. –  Benjamin Steinberg Dec 24 '11 at 18:56

The easiest way to say that a groupoid $G$ is etale complete is that the pullback (in the 2-category of topoi) of the canonical geometric morphism $p:Sh(G_0) \to BG$ against itself is equivalent to $Sh(G_1).$
Given $x,y$ points of $G_0$, consider the associated geometric morphisms $\hat x:Set \to Sh(G_0),$ $\hat y:Set \to Sh(G_0).$ Then $G$ is etale complete if and only if natural isomorphisms $$\alpha:\hat x^* \circ p^* \Rightarrow \hat y^* \circ p^*$$ are in bijection with arrows $g:x \to y$ in $G_1$.