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Convenient antiderivatives for differential linear categories

Published online by Cambridge University Press:  03 July 2020

Jean-Simon Pacaud Lemay Open the ORCID record for Jean-Simon Pacaud Lemay [Opens in a new window]
Jean-Simon Pacaud Lemay*
Affiliation:
Department of Computer Science, University of Oxford, Oxford, UK
*
*Corresponding author. Email: jean-simon.lemay@kellogg.ox.ac.uk
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Abstract

Differential categories axiomatize the basics of differentiation and provide categorical models of differential linear logic. A differential category is said to have antiderivatives if a natural transformation , which all differential categories have, is a natural isomorphism. Differential categories with antiderivatives come equipped with a canonical integration operator such that generalizations of the Fundamental Theorems of Calculus hold. In this paper, we show that Blute, Ehrhard, and Tasson's differential category of convenient vector spaces has antiderivatives. To help prove this result, we show that a differential linear category – which is a differential category with a monoidal coalgebra modality – has antiderivatives if and only if one can integrate over the monoidal unit and such that the Fundamental Theorems of Calculus hold. We also show that generalizations of the relational model (which are biproduct completions of complete semirings) are also differential linear categories with antiderivatives.

Keywords

Differential categoriesantiderivativesconvenient vector spaces
Type
Paper
Information
Mathematical Structures in Computer Science , Volume 30 , Issue 5 , May 2020 , pp. 545 - 569
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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