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.