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Differential Equations in a Tangent Category I: Complete Vector Fields, Flows, and Exponentials

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.

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Notes

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    That is, the addition \(+\) is invertible in each slice; see [3, Section 3.3].

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Correspondence to G. S. H. Cruttwell.

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J. R. B. Cockett: Partially supported by NSERC, Canada. G. S. H. Cruttwell: Partially supported by NSERC, Canada. J.-S. P. Lemay: Research supported by Kellogg College, the Clarendon Fund, and the Oxford-Google DeepMind Graduate Scholarship..

Our thanks to Rory Lucyshyn-Wright, who provided valuable contributions to earlier versions of this work, as well as Matthew Burke, Jeff Egger, Ben MacAdam, and Bob Paré for useful discussions..

Communicated by Stephen Lack.

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Cockett, J.R.B., Cruttwell, G.S.H. & Lemay, J.S.P. Differential Equations in a Tangent Category I: Complete Vector Fields, Flows, and Exponentials. Appl Categor Struct 29, 773–825 (2021). https://doi.org/10.1007/s10485-021-09629-x

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Keywords

  • Tangent categories
  • Differential equations
  • Abstract differential geometry
  • Synthetic differential geometry