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THE UNITY AND IDENTITY OF DECIDABLE OBJECTS AND DOUBLE-NEGATION SHEAVES
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- Journal:
- The Journal of Symbolic Logic / Volume 83 / Issue 4 / December 2018
- Published online by Cambridge University Press:
- 21 December 2018, pp. 1667-1679
- Print publication:
- December 2018
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- Article
- Export citation
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Let
${\cal E}$ be a topos, 
${\rm{Dec}}\left( {\cal E} \right) \to {\cal E}$ be the full subcategory of decidable objects, and 
${{\cal E}_{\neg \,\,\neg }} \to {\cal E}$ be the full subcategory of double-negation sheaves. We give sufficient conditions for the existence of a Unity and Identity 
${\cal E} \to {\cal S}$ for the two subcategories of 
${\cal E}$ above, making them Adjointly Opposite. Typical examples of such 
${\cal E}$ 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.
