Differential equations in a tangent category I: Complete vector fields, flows, and exponentials

TL;DR

This paper develops an abstract framework for differential equations within tangent categories, introducing curve objects and conditions for completeness, and explores their implications for vector fields, flows, and exponential structures.

Contribution

It introduces the concept of curve objects in tangent categories, providing a new abstract foundation for differential equations and their solutions, including conditions for completeness and exponential structures.

Findings

Defined curve objects as preinitial objects for dynamical systems

Established abstract conditions for the completeness of dynamical systems

Introduced differential exponential rigs with actions on differential objects

Abstract

This paper describes how to define and work with differential equations in the abstract setting of tangent categories. The key notion is that of a curve object which is, for differential geometry, the structural analogue of a natural number object. A curve object is a preinitial object for dynamical systems; dynamical systems may, in turn, be viewed as determining systems of differential equations. The unique map from the curve object to a dynamical system is a solution of the system, and a dynamical system is said to be complete when for all initial conditions there is a solution. A subtle issue concerns the question of when a dynamical system is complete, and the paper provides abstract conditions for this. This abstract formulation also allows new perspectives on topics such as commutative vector fields and flows. In addition, the stronger notion of a differential curve object, which is the centrepiece of the last section of the paper, has exponential maps and forms a differential exponential rig. This rig then, somewhat surprisingly, has an action on every differential object and bundle in the setting. In this manner, in a very strong sense, such a curve object plays the role of the real numbers in standard differential geometry.