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Deconstructing span categories for profinite groups

David Barnes, Niall Taggart

TL;DR

The paper develops a concrete, example-driven framework for computing (co)limits of ∞-categories in the setting of profinite groups, showing that the span category of a profinite group can be constructed as a colimit over its open normal subgroups’ quotients. It provides a detailed analysis of how limits and colimits behave across variants of Cat_∞ (small, large, presentable) and establishes that Span functors commute with limits and filtered colimits, with Ind-completion linking finite structures to full categories of G-sets and G-spaces. The main results give explicit equivalences: Fin_G^δ is the colimit of Fin_{G/N}, Set_G^δ is the limit of Set_{G/N}, Span(Fin_G) arises as a colimit of Span(Fin_{G/N}), and Mackey functors for profinite G are computed as limits/colimits of Mackey functors for the finite quotients, leading to a continuous model for G-spectra. These findings deepen the understanding of equivariant (higher) algebra and provide a robust, scalable framework for studying equivariant stable homotopy theory via ∞-categorical (co)limits.

Abstract

One of the major advantages of $\infty$-category theory over classical $1$-category theory is its robust and homotopically meaningful framework for taking (co)limits of diagrams of $\infty$-categories. However, it is both subtle and crucial to specify which variant of the $\infty$-category of $\infty$-categories is being used when forming such (co)limits. In this article, we present a concrete case study illustrating how (co)limits of $\infty$-categories behave in a specific setting. We demonstrate that the span category of a profinite group can be realised as the colimit of the span categories of its quotients by open normal subgroups and provide a number of applications to the world of equivariant (higher) algebra.

Deconstructing span categories for profinite groups

TL;DR

The paper develops a concrete, example-driven framework for computing (co)limits of ∞-categories in the setting of profinite groups, showing that the span category of a profinite group can be constructed as a colimit over its open normal subgroups’ quotients. It provides a detailed analysis of how limits and colimits behave across variants of Cat_∞ (small, large, presentable) and establishes that Span functors commute with limits and filtered colimits, with Ind-completion linking finite structures to full categories of G-sets and G-spaces. The main results give explicit equivalences: Fin_G^δ is the colimit of Fin_{G/N}, Set_G^δ is the limit of Set_{G/N}, Span(Fin_G) arises as a colimit of Span(Fin_{G/N}), and Mackey functors for profinite G are computed as limits/colimits of Mackey functors for the finite quotients, leading to a continuous model for G-spectra. These findings deepen the understanding of equivariant (higher) algebra and provide a robust, scalable framework for studying equivariant stable homotopy theory via ∞-categorical (co)limits.

Abstract

One of the major advantages of -category theory over classical -category theory is its robust and homotopically meaningful framework for taking (co)limits of diagrams of -categories. However, it is both subtle and crucial to specify which variant of the -category of -categories is being used when forming such (co)limits. In this article, we present a concrete case study illustrating how (co)limits of -categories behave in a specific setting. We demonstrate that the span category of a profinite group can be realised as the colimit of the span categories of its quotients by open normal subgroups and provide a number of applications to the world of equivariant (higher) algebra.
Paper Structure (10 sections, 22 theorems, 90 equations)

This paper contains 10 sections, 22 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. (Co)Limits of
  3. Limits of
  4. Colimits of small
  5. (Co)limits of potentially large
  6. (Co)limits of presentable
  7. (Co)limits and functor categories
  8. Categories of
  9. span categories
  10. Mackey functors for profinite groups

Key Result

Theorem A

Let $G$ be a profinite group. The collection of inflation functorsThe $N$-inflation functor arises from the quotient of groups $G \to G/N$ giving a finite $G/N$-set an action of $G$ by precomposition. running over the set of open and normal subgroups of $G$ assembles into an equivalence of $\infty$- between the colimit (in $\sf{Cat}_\infty$) of the $\infty$-categories of finite $G/N$-sets and the

Theorems & Definitions (46)

  • Theorem A: \ref{['thm: GSet as colim']}
  • Theorem B: \ref{['cor: GSet as limit']}
  • Theorem C: \ref{['thm: span as limit']} and \ref{['thm: span as colim']}
  • Theorem D: \ref{['thm: Mackey functors as a limit']}
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Corollary 2.4
  • ...and 36 more