# A topological groupoid representing the topos of presheaves on a monoid

## TL;DR

This paper provides an algebraic construction of a topological groupoid representing the topos of presheaves on a monoid, extending the understanding of topos representations in algebraic and topological terms.

## Contribution

It offers a new algebraic approach to constructing topological groupoids for presheaf topoi on monoids, especially when the monoid embeds into a group.

## Key findings

- Constructs topological groupoids from monoids algebraically.
- Shows how to compute points of the topos for embeddable monoids.
- Relates the groupoid construction to action groupoids for groups.

## Abstract

Butz and Moerdijk famously showed that every (Grothendieck) topos with enough points is equivalent to the category of sheaves on some topological groupoid. We give an alternative, more algebraic construction in the special case of a topos of presheaves on an arbitrary monoid. If the monoid is embeddable in a group, the resulting topological groupoid is the action groupoid for a discrete group acting on a topological space. For these monoids, we show how to compute the points of the associated topos.