Logic of differential calculus and the zoo of geometric strujctures

TL;DR

This paper explores the foundational role of differential calculus in generating a wide variety of geometric structures, linking them through a formalized calculus of functors rooted in classical physics observability.

Contribution

It introduces a unifying framework connecting diverse geometric structures to the calculus of functors over commutative algebras, based on a formalization of physical observability.

Findings

Unified view of geometric structures via functor calculus

Connection between classical physics observability and geometry

Expansion of geometric structures including Lie algebroids and BV-brackets

Abstract

Since the discovery of differential calculus by Newton and Leibniz and the subsequent continuous growth of its applications to physics, mechanics, geometry, etc, it was observed that partial derivatives in the study of various natural problems are (self-)organized in certain structures usually called geometric. Tensors, connections, jets, etc, are commonly known examples of them. This list of classical geometrical structures is sporadically and continuously widening. For instance, Lie algebroids and BV-bracket are popular recent additions into it. Our goal is to show that the "zoo" of all geometrical structures has a common source in the calculus of functors of differential calculus over commutative algebras, which surprisingly comes from a due mathematical formalization of observability mechanism in classical physics. We also use this occasion for some critical remarks and discussion of some perspectives.