Categories of sets with infinite addition

TL;DR

This paper explores the structure of sets with infinite addition, called $\\Sigma$-monoids, establishing their relationships with existing categories, their properties, and their tensor products, which differ from those in topological abelian groups.

Contribution

It introduces a unified definition of $\Sigma$-monoids, relates them to existing structures, and demonstrates their tensor product existence, expanding the understanding of infinite addition in algebraic contexts.

Findings

Every Hausdorff commutative monoid is a $\Sigma$-monoid

There exists a free Hausdorff commutative monoid for each $\Sigma$-monoid

$\Sigma$-monoids have well-defined tensor products

Abstract

We consider sets with infinite addition, called $\Sigma$-monoids, and contribute to their literature in three ways. First, our definition subsumes those from previous works and allows us to relate them in terms of adjuctions between their categories. In particular, we discuss $\Sigma$-monoids with additive inverses. Second, we show that every Hausdorff commutative monoid is a $\Sigma$-monoid, and that there is a free Hausdorff commutative monoid for each $\Sigma$-monoid. Third, we prove that $\Sigma$-monoids have well-defined tensor products, unlike topological abelian groups.