Introduction

The idea (or, perhaps better, the need) for this volume became clear to us when we were working on our monograph, Varieties of continua: from regions to points and back. 1 We deveoped an interest in various contemporary accounts of continuity: the prevailing Dedekind–Cantor account, smooth infinitesimal analysis (or synthetic differential geometry), and intuitionisic analysis. Each of these theories sanctions some long-standing properties that have been attributed to the continuous, at the expense of other properties so attributed. The intuitionistic theories violate the intermediate value theorem, while the Dedekind–Cantor one gives up the thesis that continua are unified wholes, and cannot be divided cleanly. The slogan is that continua are viscous, or sticky....

THE UNITY AND IDENTITY OF DECIDABLE OBJECTS AND DOUBLE-NEGATION SHEAVES

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.