Categories of sets with infinite addition

TL;DR

This work develops a unifying, extensible theory of Σ-monoids that generalizes infinite summation beyond topological or convergent contexts. It introduces weak Σ-monoids, alongside strong and group variants, and shows these categories are locally presentable with robust limit/colimit behavior and adjunctions between flavors. It proves the existence of tensor products and a symmetric monoidal closed structure on Σ-monoids, and demonstrates that every Hausdorff commutative monoid naturally yields a Σ-monoid with a corresponding left adjoint to the forgetful functor, enabling a free construction of Hausdorff commutative monoids on Σ-monoids. The results bridge algebra, topology, and category theory, providing a foundation for enriched categories and semantic models—e.g., in quantum programming—where infinite linear combinations and tensorial composition are essential. Overall, the paper shows how to systematically extend summation structures to a categorical setting with adjunctions, tensor products, and free constructions across several flavors of Σ-monoids.

Abstract

We consider sets with infinite addition, called $Σ$-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 $Σ$-monoids with additive inverses. Second, we show that every Hausdorff commutative monoid is a $Σ$-monoid, and that there is a free Hausdorff commutative monoid for each $Σ$-monoid. Third, we prove that $Σ$-monoids have well-defined tensor products, unlike topological abelian groups.

Theorems & Definitions (85)

• Definition 2.1
• Proposition 2.2
• proof
• Definition 2.4
• Definition 2.6
• Definition 2.7
• Definition 2.8
• Proposition 2.9
• Definition 3.1
• Proposition 3.2