J. -S. P. Lemay | Searcharxiv

arXiv2021

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

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.

math.CT↗

arXiv2019

Integral and differential structure on the free $C^{\infty}$-ring modality

Integral categories were recently developed as a counterpart to differential categories. In particular, integral categories come equipped with an integration operator, known as an integral transformation, whose axioms generalize the basic integration identities from calculus such as integration by parts. However, the literature on integral categories contains no example that captures integration of arbitrary smooth functions: the closest are examples involving integration of polynomial functions. This paper fills in this gap by developing an example of an integral category whose integral transformation operates on smooth 1-forms. We also provide an alternative viewpoint on the differential structure of this key example, investigate derivations and coderelictions in this context, and prove that free $C^{\infty}$-rings are Rota-Baxter algebras.

math.CT↗