nLab
condensed set

Contents

Context

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

Algebra

Contents

Idea

Condensed sets are basic objects in condensed mathematics, whose aim is to provide a convenient setting in the framework for working with algebraic objects that are equipped with sort of a topology. A related alternative is provided by pyknotic sets.

Topological spaces formalize the idea of spaces with a notion of “nearness” of points. However, they fail to handle the idea of “points that are infinitely near, but distinct” in a useful way. Condensed sets handle this idea in a useful way. (Scholze 21)

For instance, /\mathbb{R}/\mathbb{Q} and 2()/ 1()\ell^2(\mathbb{N})/\ell^1(\mathbb{N}) are indiscrete as topological spaces but retain structure as condensed sets.

Condensed sets contain information about limits of images of any set, AA, along the ultrafilters of AA.

Definition

Definition

A condensed set is a sheaf of sets on the pro-étale site of a point — in other words, on the category of profinite spaces with finite jointly surjective families of maps as covers — which is the colimit of a small diagram of representables (a small sheaf?).

That is, a condensed set is a functor

ProfiniteSet opSet ProfiniteSet^op \longrightarrow Set

such that the natural maps

T()* T(\emptyset) \longrightarrow *

and

T(SS)T(S)×T(S) T(S\sqcup S') \longrightarrow T(S) \times T(S')

are bijections for any profinite sets SS and SS', whereas the natural fork

T(S)T(S)T(S× SS) T(S)\to T(S') \rightrightarrows T(S'\times_S S')

is an equalizer for any surjection of profinite sets SSS'\to S.

Scholze, p.7 modifies this definition to deal with size issues:

For any uncountable strong limit cardinal κ\kappa, the category of κ\kappa-condensed sets is the category of sheaves on the site of profinite sets of cardinality less than κ\kappa, with finite jointly surjective families of maps as covers.

The category of condensed sets is then the (large) colimit of the category of κ\kappa-condensed sets along the filtered poset of all uncountable strong limit cardinals κ\kappa, hence is the category of small sheaves?.

Properties

Condensed sets form a locally small, well-powered, locally cartesian closed infinitary-pretopos, that is neither a Grothendieck topos nor an elementary topos – since it lacks both a small separator (indeed, it is not even total) and a subobject classifier. It has a large separator of finitely presentable projectives, and hence is algebraically exact. (Campbell 20)

See \cite[Proposition 1.7]{ScholzeLCM} for the following proposition.

Proposition

The forgetful functor from the category of topological spaces to condensed sets is a faithful functor. It becomes fully faithful when restricted to compactly generated spaces. (In the case of κ-condensed sets, one must take κ-compactly generated spaces instead.)

This functor admits a left adjoint, which sends a condensed set TT to the topological space given by the underlying set T(*)T(*) of TT equipped with the quotient topology induced by the map

STST(*),\coprod_{S\to T}S\to T(*),

where SS runs over all (κ-small) profinite sets mapping into TT. The counit of this adjunction coincides with the counit X cgXX^{cg}\to X of the adjunction between (κ-small) compactly generated spaces and topological spaces.

The left adjoint exists also for topological groups and other algebraic structures, but in this case, the underlying set is not T(*)T(*).

References

Last revised on October 27, 2021 at 03:06:32. See the history of this page for a list of all contributions to it.