David Ellerman: A fundamental duality in the exact sciences – the application to category theory duality and canonicity

Abstract: At the logical level, there are two dual logics, the Boolean logic of subsets and the logic of partitions. At a more basic level, there is a duality between elements (Its) of subsets and distinctions (Dits) of a partition (ordered pairs of elements in different blocks). The quantitative version of the two logics gives the duality between probability theory and information theory, i.e., the logical probability of a subset is the normalized number of Its in the subset and the logical entropy of a partition is the normalized number of Dits in the partition. When the morphisms (set functions) in the ur-category of $\mathsf{Sets}$ are defined in terms of Its & Dits, then interchanging Its & Dits defines the (concrete) morphisms ("cofunctions") in the dual category $\mathsf{Sets}^{\mathsf{op}}$, so Its & Dits duality is the basis for the reverse-the-morphisms duality in $\mathsf{Sets}$ which is abstracted in the reverse-the-arrows duality of category theory. Subsets and partitions each form a partial order with joins, meets, and implication operations to form a Boolean algebra and a partition algebra respectively. The partial orders of inclusion between subsets and refinement between partitions each induce a canonical injection and surjection respectively that provide a definition of canonicity in category theory.