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Applications of sup-lattice enriched category theory to sheaf theory. (English) Zbl 0619.18005

Grothendieck toposes are studied via the process of taking the associated sup-lattice enriched category of relations. It is shown that this process is adjoint to that of taking the topos of sheaves of an abstract category of relations. As a result pullback and comma toposes are calculated in a new way. The calculations are used to give a new characterization of localic morphisms and to derive interpolation and conceptual completeness properties for a certain class of interpretations between geometric theories. A simple characterization of internal sup-lattices in terms of external \({\mathcal S}l\)-enriched category theory is given.

MSC:

18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects)
18D20 Enriched categories (over closed or monoidal categories)
06B23 Complete lattices, completions
18F10 Grothendieck topologies and Grothendieck topoi
18B25 Topoi
03C75 Other infinitary logic
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