Mathematics – Category Theory
Scientific paper
2010-12-29
Mathematics
Category Theory
19 pages, results partially presented at McGill University in the fall 2010
Scientific paper
In "The fundamental progroupoid of a general topos" (Journal of Pure and Applied Algebra 212) we have introduced the notion of covering projection on a general topos. These are locally constant objects with an additional property. We show there that the category of covering projections trivialized by a fix cover is an atomic topos with points. This determines a progroupoid of localic groupoids suitable indexed by a filtered poset of covers, which generalize the known results on the fundamental progroupoid of a locally connected topos to general topoi. In this paper we consider simplicial families, that is, simplicial objects in indexed by a simplicial set. We show that covering projections can be defined as objects constructed from a descent datum of a simplicial set on a family of sets. The simplicial set is the index of a hypercover refinement of the cover. In particular, we show that any locally constant object in a locally connected topos is constructed by descent from a descent datum on a family of sets. We construct a groupoid such that the category of covering projections trivialized by a fix hypercover is its classifying topos. This determines a progroupoid of ordinary groupoids, this time suitable indexed by a filtered poset of hypercovers. Thus, by switching from covers to hypercovers we construct the fundamental progroupoid of a general topos as a progroupoid of ordinary groupoids. This construction is novel also in the case of locally connected topoi. The salient feature that distinguishes these topoi is that the progroupoid is strict, which is not the case in general.
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