David Ellerman: The Logical Theory of Canonicity: The Elements & Distinctions Analysis of Morphisms, Duality, Canonicity, and Universal Constructions in Sets
Abstract: Category theory gives a mathematical characterization of naturality but leaves canonicity as an intuitive concept. The purpose of this paper is to develop a logical theory of canonical maps. This theory is based on the broader demonstration that the dual notions of elements & distinctions (its and dits) are the basic analytical concepts needed to unpack and analyze morphisms, duality, canonicity, and universal constructions in Sets, the category of sets and functions. The analysis extends directly to other Sets-based concrete categories (groups, rings, vector spaces, etc.). Elements and distinctions are the building blocks of the two dual logics, the Boolean logic of subsets and the logic of partitions. The partial orders (inclusion and refinement) in the two lattices for the dual logics define morphisms. The thesis is that the maps that are canonical in Sets are the ones that are defined (given the data of the situation) by these two logical partial orders and by the compositions of those maps.
