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The unity and identity of decidable objects and double-negation sheaves. (English) Zbl 1409.18002

Summary: Let \({\mathcal E}\) be a topos, \({\mathrm{Dec}}\left( {\mathcal E} \right) \to {\mathcal E}\) be the full subcategory of decidable objects, and \({{\mathcal E}_{\neg\neg}} \to {\mathcal E}\) be the full subcategory of double-negation sheaves. We give sufficient conditions for the existence of a Unity and Identity \({\mathcal E} \to {\mathcal S}\) for the two subcategories of \({\mathcal E}\) above, making them Adjointly Opposite. Typical examples of such \({\mathcal E}\) include many ‘gros’ toposes in Algebraic Geometry, simplicial sets and other toposes of ‘combinatorial’ spaces in Algebraic Topology, and certain models of Synthetic Differential Geometry.

MSC:

18B25 Topoi
18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects)
03E99 Set theory
03G30 Categorical logic, topoi
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