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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 ..."
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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).
Distribution Algebras and Duality
, 2000
"... INTRODUCTION By an Svalued distribution on a topos E bounded over a base topos S it is meant here a cocontinuous Sindexed functor : E ! S. Since introduced by F. W. Lawvere in 1983, considerable progress has been made in the study of distributions on toposes from a variety of viewpoints [19, 15, ..."
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INTRODUCTION By an Svalued distribution on a topos E bounded over a base topos S it is meant here a cocontinuous Sindexed functor : E ! S. Since introduced by F. W. Lawvere in 1983, considerable progress has been made in the study of distributions on toposes from a variety of viewpoints [19, 15, 24, 5, 6, 12, 7, 8, 9]. However, much work still remains to be done in this area. The purpose of this paper is to deepen our understanding of topos distributions by exploring a (dual) latticetheoretic notion of distribution algebra. We characterize the distribution algebras in E relative to S as the Sbicomplete Satomic Heyting algebras in E . As an illustration, we employ distribution algebras explicitly in order to give an alternative description of the display locale (complete spread) of a distribution [10, 12, 7].
The Michael Completion of a Topos Spread
"... We continue the investigation of the extension into the topos realm of the concepts introduced by R.H. Fox [10] and E. Michael [22] in connection with topological singular coverings. In particular, we construct an analogue of the Michael completion of a spread and compare it with the analogue of the ..."
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We continue the investigation of the extension into the topos realm of the concepts introduced by R.H. Fox [10] and E. Michael [22] in connection with topological singular coverings. In particular, we construct an analogue of the Michael completion of a spread and compare it with the analogue of the Fox completion obtained earlier by the first two named authors [4]. Two ingredients are present in our analysis of geometric morphisms ': F ! E between toposes bounded over a base topos S. The first is the nature of the domain of ', which need only be assumed to be a "definable dominance" over S, a condition that is trivially satisfied if S is a Boolean topos. The Heyting algebras arising from the object S of truth values in the base topos play a special role in that they classify the de nable monomorhisms in those toposes. The geometric morphisms F ! F 0 over E which preserve these Heyting algebras (and that are not typically complete) are said to be strongly pure. The second is the nature of ' itself, which is assumed to be some kind of a spread. Applied to a spread, the (strongly pure, weakly entire) factorization obtained here gives what we call the "Michael completion" of the given spread. Whereas the Fox complete spreads over a topos E correspond to the Svalued Lawvere distributions on E [21] and relate to the distribution algebras [7], the Michael complete spreads seem to correspond to some sort of "Sadditive measures" on E whose analysis we do not pursue here. We close the paper with several other open questions and directions for future work.