Coproducts Internal to Profinite Spaces

TL;DR

The paper develops a general categorical framework to explain properties of profinite coproducts indexed by profinite spaces by viewing them as internal coproducts in $\mathbf{Pro}$. Using internal category theory and pro-completions, it shows that many naturally arising functors are left adjoints and thus preserve these internal coproducts, enabling a unified treatment of topological coproducts like $\coprod_{x\in X}G_x$ and $\widehat{\bigoplus}_{x\in X}M_x$. A central achievement is proving that $\mathbf{PGrp}$ has all colimits internal to $\mathbf{Pro}$ (Theorem mainthm) and demonstrating applications to profinite Bass--Serre theory, including infinite amalgamations and Mayer--Vietoris-type results, as well as induction, tensor products, Tor, abelianisation, and Pontryagin duality. The framework thus unifies prior case-by-case results and connects pro-objects with internal categorical constructions, facilitating the transfer of classical categorical phenomena into the profinite realm.

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.

Theorems & Definitions (25)

• Example 2.1
• Remark 2.2
• Theorem 2.3
• proof
• Remark 2.4
• Proposition 2.5
• Remark 2.6
• Remark 2.7
• Proposition 3.1
• proof