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Glueing analysis for complemented subtoposes. (English) Zbl 0876.18001

Given an open subtopos \({\mathcal H}\) of a given topos \({\mathcal M}\) it has a complementary subtopos \({\mathcal K}\) (called closed) and \({\mathcal M}\) can be constructed from \({\mathcal H}\) and \({\mathcal K}\) via “glueing”, originally done by A. Grothendieck in SGA 4 [Lect. Notes Math. 269 (1972; Zbl 0234.00007)] and later for elementary toposes by Wraith [see P. T. Johnstone, “Topos theory”, Academic Press (1977; Zbl 0368.18001)]. The paper under review extends the glueing construction to any pair of complemented subtoposes, not just an open and closed pair. The article then studies locally closed subtoposes, which are always complemented. The property of a subtopos \({\mathcal H}\) of \({\mathcal M}\) being locally closed (that is, the intersection of an open and a closed subtopos) is characterized by its having a complement \({\mathcal K}\) together with a technical condition on the so-called “fringe” functors induced by the inclusions of \({\mathcal H}\) and \({\mathcal K}\) into \({\mathcal M}\). The classical notion of prolongation by zero for abelian groups is then studied in this setting and it is shown that a complemented subtopos \({\mathcal H}\) of \({\mathcal M}\) admits prolongation by zero for abelian groups iff it is locally closed.

MSC:

18B25 Topoi
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