Maps with Discrete Fibers and the Origin of Basepoints

Abstract

Let ${p : \mathcal {E}\rightarrow \mathcal S}$ be a hyperconnected geometric morphism. For each X in the ‘gros’ topos $\mathcal {E}$, there is a hyperconnected geometric morphism ${p_X : \mathcal {E}/X \rightarrow \mathcal S(X)}$ from the slice over X to the ‘petit’ topos of maps (over X) with discrete fibers. We show that if p is essential then $p_X$ 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 ${p_X}$ is local for every X. This is a very restrictive property, typical of toposes of spaces of dimension ${\le 1}$.