A famous theorem of [Joyal and Tierney](https://books.google.se/books?id=NDbUCQAAQBAJ&lpg=PR8&ots=A6hV2iaYxQ&dq=joyal%20tierney%20an%20extension%20of%20the%20galois%20theory%20of%20grothendieck&pg=PP1#v=onepage&q&f=false) says that each Grothendieck topos is equivalent to the classifying topos of a localic groupoid. I believe that [Butz and Moerdijk have shown](http://www.sciencedirect.com/science/article/pii/S0022404997001072) 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. 



Toposes (topoi) as classifying toposes of groupoids
Perhaps [these slides](https://www.dpmms.cam.ac.uk/~zll22/slides/2013-06-19-ToposesAsGroupoids.pdf) will be helpful. I'll try to explain what happens in your special case.


Let $M$ be a monoid and let $\mathcal{B} M$ be the topos of right $M$-sets. The points of $\mathcal{B} M$ are the *left* $M$-sets $P$ that satisfy the following conditions:


* $P$ is inhabited.
* Given elements $p\_0$ and $p\_1$ of $P$, there exist an element $p$ of $P$ and elements $m\_0$ and $m\_1$ of $M$ such that $m\_0 \cdot p = p\_0$ and $m\_1 \cdot p = p\_1$.
* Given an element $p$ of $P$ and elements $m\_0$ and $m\_1$ of $M$ such that $m\_0 \cdot p = m\_1 \cdot p$, there exist an element $p'$ of $P$ and an element $m'$ of $M$ such that $m' \cdot p' = p$ and $m\_0 m' = m\_1 m'$.


For example, the left regular action of $M$ on itself is a point of $\mathcal{B} M$, and as it happens, this point covers all of $\mathcal{B} M$. However, what we need to find is an *open* cover of $\mathcal{B} M$. The Butz–Moerdijk construction yields such a thing. 


Let $K$ be a fixed set of cardinality $\ge \left| M \right|$. An **enumeration** of $M$ is a partial surjection $K \rightharpoonup M$ with infinite fibres. An **isomorphism** of enumerations of $M$ is an isomorphism of left $M$-sets making the following diagram commute:
$$\require{AMScd}
\begin{CD}
K @= K \\
@VVV @VVV \\
M @>>> M
\end{CD}$$
Choose a representative in each isomorphism class of enumerations of $M$. We define a groupoid $\mathbb{G}$ as follows:


* The objects are the chosen representatives.
* The morphisms $\alpha \to \beta$ are tuples $(\alpha, \beta, m)$ where $m$ is an *invertible* element of $M$. (We do *not* require any compatibility with the partial surjections here.)
* Composition is given by $(\beta, \gamma, m) \circ (\alpha, \beta, n) = (\alpha, \gamma, m n)$.


Write $G\_0$ (resp. $G\_1$) for the set of objects (resp. morphisms) in $\mathbb{G}$. There is a Galois topology on these making $\mathbb{G}$ a topological groupoid:


* The basic open subsets of $G\_0$ are the subsets
$$U\_{\vec{i}, C} = \{ \alpha \in G\_0 : \alpha (\vec{i}) \in C \}$$
where $\vec{i}$ is an $n$-tuple of elements of $K$ and $C$ is a *right* $M$-subset of $M^n$.
* The basic open subsets of $G\_1$ are the subsets
$$W\_{\vec{i}, C, \vec{j}, D} = \{ (\alpha, \beta, m) \in G\_1 : \alpha (\vec{i}) \in C, \beta (\vec{j}) \in D, \alpha (\vec{i}) \cdot m = \beta (\vec{j}) \}$$
where $\vec{i}$ and $\vec{j}$ are $n$-tuples of elements of $K$ and $C$ and $D$ are *right* $M$-subsets of $M^n$.


The theorem of Butz and Moerdijk is that $\mathcal{B} M$ is equivalent to the topos of equivariant sheaves on this topological groupoid $\mathbb{G}$. Note that the domain and codomain maps $G\_1 \to G\_0$ are locally connected, hence open *a fortiori*.


