# nLab condensed infinity-groupoid

Contents

### Context

Ingredients

Concepts

Constructions

Examples

Theorems

#### $(\infty,1)$-Category theory

(∞,1)-category theory

Background

Basic concepts

Universal constructions

Local presentation

Theorems

Extra stuff, structure, properties

Models

#### Higher category theory

higher category theory

# Contents

## Idea

The original motivation behind condensed infinity-groupoids is to create well-behaved categories of mathematical structures such as condensed HZ-module spectra in which one could do derived analytic geometry using category-theoretic methods without resorting to not-so-well behaved categories of topological spaces.

However, David Corfield offers another motivation for condensed infinity-groupoids in the blog post Pyknoticity vs Cohesiveness on the nCafé, as a way to extend cohesion to the $p$-adic case (see at condensed cohesion):

Mathematics rather than physics is the subject of chapter 5 of my book, where I’m presenting cohesive HoTT as a means to gain some kind of conceptual traction over the vast terrain that is modern geometry. However I’m aware that there are some apparent limitations, problems with ‘$p$-adic’ forms of cohesion, cohesion in algebraic geometry, and so on. In the briefest note (p. 158) I mention the closely related pyknotic and condensed approaches of, respectively, (Barwick and Haine) and (Clausen and Scholze).

## Definition

A condensed infinity-groupoid or condensed anima is a hypercomplete (infinity,1)-sheaf of infinity-groupoids on the pro-étale (infinity,1)-site of the point, small relative to a universe $\mathcal{U}$.

Equivalently, it is an (infinity,1)-functor

$T:ProfiniteSpaces^\op \longrightarrow \infty Grpd_\mathcal{U}$

from the opposite (infinity,1)-category of profinite spaces to the (infinity,1)-category Infinity-Grpd of infinity-groupoids which are small relative to a universe $\mathcal{U}$, such that the natural maps

$T(\mathbb{0}) \to \mathbb{1}$

and

$T(S + S^{'}) \to T(S) \times T(S^{'})$

are homotopy equivalences for any profinite space $S$ and $S^{'}$, and the natural map

$T(S) \to \operatorname{lim}_{\Delta} T(S_\bullet)$

is a homotopy equivalence for any hypercover of profinite spaces $S_\bullet \to S$.

When restricted to the site of extremally disconnected compact Hausdorff spaces, hyperdescent is automatic, so the simplest equivalent definition of a condensed infinity-groupoid is simply a finite-product-preserving (infinity,1)-functor $T:ExtrDisc^\op \longrightarrow \infty Grpd_\mathcal{U}$, see (Barwick-Haine, corollary 2.4.4).

## Properties

The (infinity,1)-category $\mathrm{Cond}\infty\mathrm{Grpd}$ of condensed infinity-groupoids is a locally small locally cartesian closed (infinity,1)-pretopos.

Peter Scholze speculates in this comment on the nCafé that $\mathrm{Cond}\infty\mathrm{Grpd}$ seem to form a predicative elementary (infinity,1)-topos. If true, then $\mathrm{Cond}\infty\mathrm{Grpd}$ is an example of an elementary (infinity,1)-topos which is not a Grothendieck (infinity,1)-topos.

## Terminology

In the model of infinity-groupoids as simplicial sets/Kan complexes, condensed infinity-groupoids are sometimes referred to as condensed anima.

## References

Last revised on July 29, 2022 at 17:02:27. See the history of this page for a list of all contributions to it.