# Kähler Categories

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

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.