The Unity and Identity of decidable objects and double-negation sheaves

TL;DR

This paper establishes conditions under which the subcategories of decidable objects and double-negation sheaves in a topos are linked by an adjoint equivalence, unifying their structure in various mathematical contexts.

Contribution

It provides sufficient conditions for a topos to have a unity and identity linking decidable objects and double-negation sheaves via adjoint functors, applicable in multiple areas of mathematics.

Findings

Conditions for the existence of a unifying topos structure.

Application to algebraic geometry, algebraic topology, and differential geometry.

Examples include many 'gros' toposes and models of synthetic differential geometry.

Abstract

Let ${\cal E}$ be a topos, ${{\rm Dec}({\cal E}) \rightarrow {\cal E}}$ be the full subcategory of decidable objects, and ${{\cal E}_{\neg\neg} \rightarrow {\cal E}}$ be the full subcategory of double-negation sheaves. We give sufficient conditions for the existence of a Unity and Identity ${{\cal E} \rightarrow {\cal S}}$ for the two subcategories of 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.