Commutative monads as a theory of distributions
type: Item
instance of: scholarly article
zbMATH Open document ID: 1261.18004
author: Anders Kock
publication date: 2012-05-29T00:00:00Z
full work available at URL: https://arxiv.org/abs/1108.5952
full work available at URL: 26-04abs.html
Mathematics Subject Classification ID: 18C15
Mathematics Subject Classification ID: 18D25
Mathematics Subject Classification ID: 46T30
zbMATH DE Number: 6039020
zbMATH Keywords: monad
zbMATH Keywords: distribution
zbMATH Keywords: extensive quantity
MaRDI profile type: Publication
arXiv ID: 1108.5952
arXiv classification: math.CT
title: Commutative monads as a theory of distributions
published in: Theory and Applications of Categories
review text: \textit{L. Schwartz} [Méthodes mathématiques pour les sciences physiques. Paris: Hermann \& Cie. (1961; Zbl 0101.41301)] argues that ``the mathematical distributions constitute a correct mathematical definition of the distribution one meets in physics''. To him, a distribution is a functional on a space of functions. The aim of the present paper is to present another mathematical theory of distributions not based on the double dualization of functional analysis. Distributions of compact support constitute a significant paradigm of an extensive quantity in the sense of \textit{F. W. Lawvere} [in: The space of mathematics. Philosophical, epistemological, and historical explorations. Revised papers from a symposium on structures in mathematical theories, Donostia/San Sebastian, Basque Country, Spain, September 1990. Berlin: Walter de Gruyter. Grundlagen der Kommunikation und Kognition. 14--30 (1992; Zbl 0846.18001)], whose outstanding feature is the covariant functorial dependence on the space at issue. Mathematically speaking, Lawvere's extensive quantity is a covariant functor. The present author starts with an endofunctor \(T\) on a cartesian closed category \(\mathcal{E}\), which is indeed a monad. A linear structure on \(T(X)\) grows naturally out of the monad structure, as far as monads with a certain property are concerned. What makes the author's theory considerably simple is the universal property of the unit map \(\eta:X\rightarrow T(X)\) guaranteed by the general theory of monads in conjunction with the assumption of commutativity of \(T\) in the sense of the author [Arch. Math. 21, 1--10 (1970; Zbl 0196.03403)].
reviewed by: Hirokazu Nishimura
description: scientific article; zbMATH DE number 6039020