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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 Butz 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
up vote 5 down vote accepted

Perhaps these slides 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.

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Thanks for this. It will take me time to sort this out. – Benjamin Steinberg Nov 30 '15 at 11:54

Ben, I cannot fit a description of etale completeness in the comments, so I will have to do so here:

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 $Sh(G_1).$

This is equivalent to the following:

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 $G_1$.

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