Commutative monads as a theory of distributions (Q2884475)

\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)].