Exponential Functions in Cartesian Differential Categories
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Exponential Functions in Cartesian Differential Categories Jean-Simon Pacaud Lemay1 Received: 12 December 2019 / Accepted: 14 September 2020 © The Author(s) 2020
Abstract In this paper, we introduce differential exponential maps in Cartesian differential categories, which generalizes the exponential function e x from classical differential calculus. A differential exponential map is an endomorphism which is compatible with the differential combinator x x in such a way that generalizations of e0 = 1, e x+y = e x e y , and ∂e ∂ x = e all hold. Every differential exponential map induces a commutative rig, which we call a differential exponential rig, and conversely, every differential exponential rig induces a differential exponential map. In particular, differential exponential maps can be defined without the need of limits, converging power series, or unique solutions of certain differential equations—which most Cartesian differential categories do not necessarily have. That said, we do explain how every differential exponential map does provide solutions to certain differential equations, and conversely how in the presence of unique solutions, one can derivative a differential exponential map. Examples of differential exponential maps in the Cartesian differential category of real smooth functions include the exponential function, the complex exponential function, the split complex exponential function, and the dual numbers exponential function. As another source of interesting examples, we also study differential exponential maps in the coKleisli category of a differential category. Keywords Cartesian differential categories · Exponential functions · Differential exponential maps · Differential exponential rigs
1 Introduction Cartesian differential categories [3], introduced by Blute, Cockett, and Seely, come equipped with a differential combinator D which provides a categorical axiomatization of the differential from multivariable differential calculus. Important examples of Cartesian differential
Communicated by Stephen Lack. The author would like to thank the following for financial support regarding this paper: Kellogg College, the Department of Computer Science of the University of Oxford, the Clarendon Fund, the Oxford-Google DeepMind Graduate Scholarship, and the Oxford Travel Abroad Bursary.
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Jean-Simon Pacaud Lemay [email protected] Department of Computer Science, University of Oxford, Oxford, UK
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J.-S. P. Lemay
categories include the category of real smooth functions (Example 1), the coKleisli category of a differential category [5], the differential objects of a tangent category [12], and categorical models of Ehrhard and Regnier’s differential λ-calculus [16] (which are in fact called Cartesian closed differential categories [21]). Other interesting (and surprising) examples include abelian functor calculus [2] and cofree Cartesian differential categories [14,19]. Since their introduction, Cartesian differential categories have a rich literature and have been successful i
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