---
abstract: 'We give characterizations, for various fragments of geometric logic, of the class of theories classified by a locally connected (resp. connected and locally connected, atomic, compact, presheaf) topos, and exploit the existence of multiple sites of definition for a given topos to establish some properties of quotients of theories of presheaf type.'
author:
- |
    [Olivia Caramello]{}\
    [DPMMS, University of Cambridge,]{}\
    [Wilberforce Road, Cambridge CB3 0WB, UK]{}\
    [O.Caramello@dpmms.cam.ac.uk]{}
date: 'July 14, 2009'
title: 'One topos, many sites'
---

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Introduction
============

Given the fact that Grothendieck toposes are ‘the same thing as’ Morita-equivalence classes of geometric theories, it is naturally of interest to investigate how classical topos-theoretic properties of toposes translate into logical properties of the theories they classify.

Characterizations of the class of geometric theories classified by a Boolean (resp. De Morgan) topos have been provided in [@OC3]. In the third section of this paper, we provide syntactic characterizations, for various fragments of geometric logic, of the class of theories classified by a locally connected (resp. connected and locally connected, atomic, 