An étendue is a topos \({\mathcal E}\) which has a covering \(U\to 1\) of the terminal object such that \({\mathcal E}/U\) is a localic topos. It has been a long standing conjecture of F. W. Lawvere that étendues are precisely those topoi which can be presented by a site, all of whose maps are monic. Work was first done on the connections between étendues and categories with monic maps by the reviewer [J. Pure Appl. Algebra 22, 193-212 (1981; Zbl 0473.18004)].
In the paper under review, the authors verify Lawvere’s conjecture by showing that if \({\mathcal E}\) is an étendue, then \({\mathcal E}\cong \text{sh}({\mathcal M}({\mathcal E}))\), where \({\mathcal M}({\mathcal E})\) is a site with monic maps. The construction of \({\mathcal M}({\mathcal E})\) involves consideration of the notions of locally monic map and torsion free object in a topos. In the second part of the paper, a characterization is given as to when the classifying topos \({\mathcal B}G\) of a localic groupoid \(G\) is an étendue. This characterization is given in terms of the existence of what is called an étale, fully supported principal \(G\)-bundle.