Kähler Categories

by Richard Blute, J.R.B. Cockett, Timothy Porter, & R.A.G. Seely


This paper establishes a relation between the recently introduced notion of differential category and the more classic theory of Kähler differentials in commutative algebra. A codifferential category is an additive symmetric monoidal category with a monad T, which is furthermore an algebra modality, i.e. a natural assignment of an associative algebra structure to each object of the form T(C). Finally, a codifferential category comes equipped with a differential combinator satisfying typical differentiation axioms, expressed algebraically.

The traditional notion of Kähler differentials defines the notion of a module of A-differential forms with respect to A, where A is a commutative k-algebra. This module is equipped with a universal A-derivation. A Kähler category is an additive monoidal category with an algebra modality and an object of differential forms associated to every object. Under the assumption that the free algebra monad exists and that the canonical map to T is epimorphic, codifferential categories are Kähler.