Core varieties,extensivity, and rig geometry

Abstract

The role of the Frobenius operations in analyzing finite spaces, as well as the extended algebraic geometry over rigs, depend partly on varieties (Birkhoffian inclusions of algebraic categories) that have coreflections as well as reflections and whose dual category of affine spaces is extensive. Even within the category of those rigs where 1+1 = 1, not only distributive lattices but also the function algebras of tropical geometry (where x + 1 = 1) and the dimension rigs of separable prextensive categories (where x + x 2 = x 2) enjoy those features. (Talk given at CT08, Calais.) Algebraic geometry, analytic geometry, smooth geometry, and also simplicial topology, all enjoy the axiomatic cohesion described in my recent article [4]. The cohesion theory aims to assist the development of those subjects by revealing characteristic ways in which their categories differ from others. (Such considerations will be important in order to carry out Grothendieck's 1973 program [2] for simplifying the foundations of algebraic geometry.) An axiomatic theory often captures more examples than originally intended. In the present case not only the smooth generalization (SDG) but also some semi-combinatorial ones are of interest. Some of these can be approached via sites of definition that can be handled in ways very closely analogous to Grothendieck's algebraic geometry constructions. Even ultra-basic properties of cohesive categories, such as extensivity, may not be true in the sites themselves; a modest step toward the treatment of that problem is the recognition of some algebraic categories as core varieties within others. But first I recall another remarkable feature of many cohesive toposes that, although discovered through its relevance to differential equations, nevertheless points toward the power of map spaces in building combinatorial objects from simple ingredients. 1. Euler's principle Focussing on the role of map spaces in generating spaces reveals a distinct feature of algebraic geometry over K-rigs (a feature shared by smooth geometry where the category of algebras consists of C ∞ rings) that I call Euler's principle. He used the fact that macro-quantities are representable as ratios of infinitesimals. We can objectify this by exploiting the fact that the category of spaces has exponentiation. Considering the theory