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Is Chemistry a Branch of
 Physics’, Journal for General Philosophy of Science
, 1982
"... coverings of quasi locally connected toposes ..."
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.
FUNDAMENTAL PUSHOUT TOPOSES
"... Abstract. The author [2, 5] introduced and employed certain ‘fundamental pushout toposes ’ in the construction of the coverings fundamental groupoid of a locally connected topos. Our main purpose in this paper is to generalize this construction without the local connectedness assumption. In the spir ..."
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Abstract. The author [2, 5] introduced and employed certain ‘fundamental pushout toposes ’ in the construction of the coverings fundamental groupoid of a locally connected topos. Our main purpose in this paper is to generalize this construction without the local connectedness assumption. In the spirit of [16, 10, 8] we replace connected components by constructively complemented, or definable, monomorphisms [1]. Unlike the locally connected case, where the fundamental groupoid is localic prodiscrete and its classifying topos is a Galois topos, in the general case our version of the fundamental groupoid is a locally discrete progroupoid and there is no intrinsic Galois theory in the sense of [19]. We also discuss covering projections, locally trivial, and branched coverings without local connectedness by analogy with, but also necessarily departing from, the locally connected case [13, 11, 7]. Throughout, we work abstractly in a setting given axiomatically by a category V of locally discrete locales that has as examples the categories D of discrete locales, and Z of zerodimensional locales [9]. In this fashion we are led to give unified and often simpler proofs of old theorems in the locally connected case, as well as new ones without that assumption.