Deconstructing span categories for profinite groups

TL;DR

This paper explores how the span categories of profinite groups can be constructed as colimits of their quotients' span categories, with implications for equivariant higher algebra.

Contribution

It provides a concrete case study on (co)limits of $ abla$-categories in the context of profinite groups, illustrating their behavior and applications.

Findings

Span category of a profinite group is a colimit of quotient span categories.

Demonstrates the behavior of (co)limits of $ abla$-categories in a specific setting.

Provides applications to equivariant higher algebra.

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.