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The Discrete Objects in the Effective Topos
 Proc. London Math. Soc
, 1990
"... The original aim of this paper was to give a rather quick and undemanding proof that the effective topos contains two nontrivial small (i.e. internal) full subcategories which are closed under all small limits in the topos (and hence in particular are internally complete). The interest in such subc ..."
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The original aim of this paper was to give a rather quick and undemanding proof that the effective topos contains two nontrivial small (i.e. internal) full subcategories which are closed under all small limits in the topos (and hence in particular are internally complete). The interest in such subcategories arises from
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).
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.
TIGHTLY BOUNDED COMPLETIONS
"... Abstract. By a ‘completion ’ on a 2category K we mean here an idempotent pseudomonad on K. We are particularly interested in pseudomonads that arise from KZdoctrines. Motivated by a question of Lawvere, we compare the Cauchy completion [23], defined in the setting of VCat for V a symmetric monoida ..."
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Abstract. By a ‘completion ’ on a 2category K we mean here an idempotent pseudomonad on K. We are particularly interested in pseudomonads that arise from KZdoctrines. Motivated by a question of Lawvere, we compare the Cauchy completion [23], defined in the setting of VCat for V a symmetric monoidal closed category, with the Grothendieck completion [7], defined in the setting of SIndexed Cat for S a topos. To this end we introduce a unified setting (‘indexed enriched category theory’) in which to formulate and study certain properties of KZdoctrines. We find that, whereas all of the KZdoctrines that are relevant to this discussion (Karoubi, Cauchy, Stack, Grothendieck) may be regarded as ‘bounded’, only the Cauchy and the Grothendieck completions are ‘tightly bounded ’ – two notions that we introduce and study in this paper. Tightly bounded KZdoctrines are shown to be idempotent. We also show, in a different approach to answering the motivating question, that the Cauchy completion (defined using ‘distributors ’ [2]) and the Grothendieck completion (defined using ‘generalized functors’ [21]) are actually equivalent constructions1.