{"title":"Multivariate functorial difference","authors":"Robert Paré","doi":"arxiv-2409.09494","DOIUrl":null,"url":null,"abstract":"Partial difference operators for a large class of functors between presheaf\ncategories are introduced, extending our difference operator from \\cite{Par24}\nto the multivariable case. These combine into the Jacobian profunctor which\nprovides the setting for a lax chain rule. We introduce a functorial version of\nmultivariable Newton series whose aim is to recover a functor from its iterated\ndifferences. Not all functors are recovered but we get a best approximation in\nthe form of a left adjoint, and the induced comonad is idempotent. Its fixed\npoints are what we call soft analytic functors, a generalization of the\nmultivariable analytic functors of Fiore et al.~\\cite{FioGamHylWin08}.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Category Theory","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.09494","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
Partial difference operators for a large class of functors between presheaf
categories are introduced, extending our difference operator from \cite{Par24}
to the multivariable case. These combine into the Jacobian profunctor which
provides the setting for a lax chain rule. We introduce a functorial version of
multivariable Newton series whose aim is to recover a functor from its iterated
differences. Not all functors are recovered but we get a best approximation in
the form of a left adjoint, and the induced comonad is idempotent. Its fixed
points are what we call soft analytic functors, a generalization of the
multivariable analytic functors of Fiore et al.~\cite{FioGamHylWin08}.