Spans and simplicial families

Abstract

In this paper we consider simplicial families, that is, simplicial objects indexed by a simplicial set. We develop a method to construct family hypercover refinements of a cover family based on the notion of \emph{n-spans} that we introduce here. In [The fundamental progroupoid of a general topos, Journal of Pure and Applied Algebra 212 (2008)] we had introduced the notion of \emph{covering projection} in a topos. They are locally constant objects satisfying an additional condition which is valid in all locally constant objects when the topos is locally connected, and developed the theory of the fundamental groupoid of a general topos. Here we show that covering projections can be obtained as objects constructed from a descent datum of a simplicial set on a family of sets. We construct a groupoid $\nn{G}_\cc{H}$ such that the category $\cc{G}_\cc{H}$ of covering projections trivialized by $\cc{H}$ is its classifying topos. This determines a protopos $\{\cc{G} \cc{H}\}_\cc{H}$ and a progroupoid $\{\nn{G}_\cc{H}\}_\cc{H}$, suitable indexed by a filtered poset of hypercovers. Then we show that this progroupoid classifies torsors. This construction is novel also in the case of locally connected topoi, showing that locally constant object in a locally connected topos are constructed by descent from a descent datum on a family of sets. The salient feature that distinguishes locally connected topoi is that the progroupoid is \emph{strict}, that is, the transition morphisms are surjective on triangles, or, equivalently, the transition inverse image functors in the underlying indcategory are full and faithful.