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.)
Keywords: topos, Frobenius, dimension rigs
2000 MSC: 12F99, 18F10, 14A99
Theory and Applications of Categories,
Vol. 20, 2008,
No. 14, pp 497-503.