Grothendieck topoi and geometric morphisms generalise topological spaces and continuous maps.  Loosely speaking, a topos can be viewed as a space where the points have been endowed with the extra structure of a space of isomorphisms between points, that is, a topological groupoid.

Moerdijk showed that, when these spaces are taken to be point-free (i.e. locales), every geometric morphism is induced by a continuous map that preserves the isomorphisms of these points, albeit up to a weak equivalence.  However, for concrete applications, it is often preferable to work with point-set spaces, not locales.

This presentation exposits a point-set parallel to the Moerdijk result.  We establish a bi-equivalence between Grothendieck topoi (with enough points) and a category of fractions on topological groupoids.  A potential application of our approach is an extension of a result of Ahlbrandt-Ziegler/Coquand in model theory, which relates bi-interpretability of structures to their topological automorphism groups.