Functors of Differential Calculus in Diolic Algebras

TL;DR

This paper introduces a new algebraic framework for differential calculus using functors over graded commutative algebras, applied to Diolic algebras, unifying various geometric notions and revealing novel features.

Contribution

It develops a formalism based on functors of differential calculus over graded algebras and introduces Diolic algebras, connecting and extending concepts in differential, symplectic, and Poisson geometry.

Findings

Recovers classical differential geometric objects

Provides a unified algebraic language for geometry

Identifies unique properties of Diolic algebras

Abstract

We pose a new algebraic formalism for studying differential calculus in vector bundles. This is achieved by studying various functors of differential calculus over arbitrary graded commutative algebras (DCGCA) and applying this language to a particularly simple class of two-component graded objects introduced in this work, that we call Diolic algebras. A salient feature of this conceptual approach to calculus is that it recovers many well-known objects and notions from ordinary differential, symplectic and Poisson geometry but also provides some unique aspects, which are of their own independent interest.