Maps with Discrete Fibers and the Origin of Basepoints

Abstract

Let p: E→ S be a hyperconnected geometric morphism. For each X in the ‘gros’ topos E, there is a hyperconnected geometric morphism pX: E/ X→ S(X) from the slice over X to the ‘petit’ topos of maps (over X) with discrete fibers. We show that if p is essential then pX is essential for every X. The proof involves the idea of collapsing a connected subspace to a ‘basepoint’, as in Algebraic Topology, but formulated in topos-theoretic terms. In case p is local, we characterize when pX is local for every X. This is a very restrictive property, typical of toposes of spaces of dimension ≤ 1.