Multivariate functorial difference

Robert Paré
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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}.
多元函数差
引入了一大类预子范畴之间的函数的部分差分算子,把我们的差分算子从 \cite{Par24} 扩展到多变量情况。它们结合成雅各布剖分器,为宽松的链式规则提供了环境。我们引入了多变量牛顿数列的函子版本,其目的是从迭代差分中恢复函子。并不是所有的函数都能复原,但我们得到了一个左邻接形式的最佳近似值,而且诱导的逗点是幂等的。它的定点就是我们所说的软解析函子,是菲奥雷等人的多变量解析函子的广义化。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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