nLab condensed infinity-groupoid

Contents

Context

Higher geometry

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

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

Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

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 pp-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 ‘pp-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 𝒰\mathcal{U}.

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

T:ProfiniteSpaces opGrpd 𝒰T: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 𝒰\mathcal{U}, such that the natural maps

T(𝟘)𝟙T(\mathbb{0}) \to \mathbb{1}

and

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

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

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

is a homotopy equivalence for any hypercover of profinite spaces S SS_\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 opGrpd 𝒰T:ExtrDisc^\op \longrightarrow \infty Grpd_\mathcal{U}, see (Barwick-Haine, corollary 2.4.4).

Properties

The (infinity,1)-category CondGrpd\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 CondGrpd\mathrm{Cond}\infty\mathrm{Grpd} seem to form a predicative elementary (infinity,1)-topos. If true, then CondGrpd\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

See condensed mathematics and infinity-groupoid#Anima.

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