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The hyperconnected maps that are local. (English) Zbl 1462.18002

The author contributes to the theory of axiomatic cohesion, that is the study of properties of a geometric morphism \(p \colon \mathcal{E} \to \mathcal{S}\) that are typical of \(\mathcal{E}\) as a topos of spaces over the base topos \(\mathcal{S}.\) Such a geometric morphism is called pre-cohesive if it is hyperconnected (the inverse image \(p^*\) is fully faithful and the counit of \(p^* \dashv p_* \) is monic), essential (\(p^*\) has a further left adjoint \(p_!\)) with \(p_!\) preserving finite products, and local (the direct image \(p_*\) has a fully faithful right adjoint \(p^!\)).
In view of results in [P. T. Johnstone, Theory Appl. Categ. 25, 51–63 (2011; Zbl 1239.18004)], the author concludes that if \(p\) is bounded, essential, hyperconnected with \(p^*\) cartesian closed, then \(p\) is pre-cohesive if and only if \(p_*\) preserves coequalizers. He naturally poses the question whether there are (unbounded) geometric morphisms which are essential, hyperconnected, with \(p^*\) cartesian closed and \(p_*\) preserving coequalizers, yet not pre-cohesive. The paper is devoted to answering this question in the negative. In particular, the main result (Corollary 6.1) shows that a hyperconnected \(p\) is local if and only if \(p_*\) preserves coequalizers. The answer is obtained via an analysis of interconnections between principal topologies on a topos (meaning topologies \(j\) such that each object has a minimal \(j\)-dense subobject), the mono-coreflective subcategories of a topos \(\mathcal{E}\) associated with such topologies and the idempotent comonads induced by such subcategories.

MSC:

18F99 Categories in geometry and topology
18F10 Grothendieck topologies and Grothendieck topoi
14A99 Foundations of algebraic geometry

Citations:

Zbl 1239.18004
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References:

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