Elementary axioms for local maps of toposes (Q1861481)

The authors give a completely elementary axiomatization of local maps of toposes by finding conditions on a topos \({\mathcal E}\) and a topology \(j\) such that the inclusion of the full subcategory \(D_j{\mathcal E}\) of \({\mathcal E}\) on the discrete objects into \({\mathcal E}\) is left exact and has a right adjoint. They also show that a wider class of sentences than the geometric sentences is preserved under the inclusion of the discrete objects.