Topological rigidity as a monoidal equivalence

TL;DR

This paper investigates the concept of topological rigidity in commutative rings, establishing a duality framework that connects free modules, topologically-free modules, and generalized topological algebras with coalgebras.

Contribution

It introduces a duality-based approach to topological rigidity, linking monoidal categories of generalized topological algebras and coalgebras.

Findings

Rigidity leads to a dual equivalence between categories of modules.

A suitable topological tensor product lifts this to an equivalence of monoids.

The work relates topological duality to finite duality between algebras and coalgebras.

Abstract

A topological commutative ring is said to be rigid when for every set $X$, the topological dual of the $X$-fold topological product of the ring is isomorphic to the free module over $X$. Examples are fields with a ring topology, discrete rings, and normed algebras. Rigidity translates into a dual equivalence between categories of free modules and of "topologically-free" modules and, with a suitable topological tensor product for the latter, one proves that it lifts to an equivalence between monoids in this category (some suitably generalized topological algebras) and coalgebras. In particular, we provide a description of its relationship with the standard duality between algebras and coalgebras, namely finite duality.