Coproducts Internal to Profinite Spaces

TL;DR

This paper provides a categorical framework explaining properties of profinite coproducts of groups, unifying previous case-by-case proofs and linking to internal category theory and Bass-Serre theory.

Contribution

It introduces a general categorical explanation for properties of profinite coproducts, simplifying and unifying prior results.

Findings

Categorical explanation for properties of profinite coproducts

Examples demonstrating the framework's applicability

Connections to internal category theory and Bass-Serre theory

Abstract

We give a categorical explanation for many properties of profinite coproducts of profinite groups, which were previously proven on a case-by-case basis. All of these properties take the form "certain functors preserve profinite coproducts". We give various examples to show how our framework can be applied. We also point out connections to internal category theory and profinite Bass-Serre theory.