A functorial approach to differential calculus

TL;DR

This paper introduces a functorial framework for differential calculus that unifies classical and topological variants, enabling calculus over arbitrary commutative rings and potential extensions to graded rings.

Contribution

It provides a new symmetric presentation of differential calculus emphasizing the anchor map and embeds calculus into a functor category, broadening its applicability.

Findings

Differential calculus can be fully embedded into a functor category.

A symmetric presentation of differential calculus is developed.

The approach allows calculus over any commutative ring, including finite rings.

Abstract

We show that differential calculus (in its usual form, or in the general form of topological differential calculus) can be fully imdedded into a functor category (functors from a small category of anchord tangent algebras to anchored sets). To prepare this approach, we define a new, symmetric, presentation of differential calculus, whose main feature is the central r{\^o}le played by the anchor map, which we study in detail. Our aim for developing this theory is twofold: (1) define a setting for calculus over any commutative ring, including finite rings; (2) define a setting that can be generalized to categories of graded rings (super differential calculus).