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Galois Groupoids and Covering Morphisms in Topos Theory
"... The goals of this paper are (1) to compare the Galois groupoid that appears naturally in the construction of the fundamental groupoid of a topos E bounded over an arbitrary base topos S given by Bunge (1992), with the formal Galois groupoid defined by Janelidze (1990) in a very general setting given ..."
Abstract

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The goals of this paper are (1) to compare the Galois groupoid that appears naturally in the construction of the fundamental groupoid of a topos E bounded over an arbitrary base topos S given by Bunge (1992), with the formal Galois groupoid defined by Janelidze (1990) in a very general setting given by a pair of adjoint functors, and (2) to discuss a good notion of covering morphism of a topos E over S which is general enough to include, in addition to the covering projections determined by the locally constant objects, also the unramified morphisms of topos theory given by those local homeomorphisms which are at the same time complete spreads in the sense of BungeFunk (1996, 1998).
Constructive Theory of Galois Toposes
"... Galois toposes were considered by Grothendieck in connection with the fundamental progroup of a topos. They were subsequently shown by Moerdijk to correspond to the classifying toposes of prodiscrete localic groups. Over an arbitrary base topos S, the (coverings) fundamental group constructed by Bun ..."
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Galois toposes were considered by Grothendieck in connection with the fundamental progroup of a topos. They were subsequently shown by Moerdijk to correspond to the classifying toposes of prodiscrete localic groups. Over an arbitrary base topos S, the (coverings) fundamental group constructed by Bunge is (the classifying topos of) a prodiscrete localic groupoid. We introduce a notion of Galois topos relative to an arbritrary base topos S and show that it reduces to the usual when the base topos is Set. The constructive theory of SGalois toposes requires some nontrivial modi cations. Locally constant coverings must be replaced by the locally componentwise constant coverings. Whereas the former satisfy the unique pathlifting property, the latter only have the pathlifting property stated in terms of open surjections. This, however, suffices for establishing the existence of a comparison map between the paths and the coverings fundamental groups in the general case.