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 and definition

Condensed sets aim to provide a convenient setting in the framework of homotopical algebra for working with algebraic objects that have some sort of a topology on them.

Definition

A condensed set is a sheaf of sets on the pro-étale site of a point. That is, a condensed set is a functor

ProfiniteSet opSetProfiniteSet^op \to Set

such that the natural maps

T()*T(\emptyset)\to *

and

T(SS)T(S)×T(S)T(S\sqcup S')\to 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 the (large) colimit of the category of κ\kappa-condensed sets along the filtered poset of all uncountable strong limit cardinals κ\kappa.

References

Last revised on July 24, 2019 at 18:47:42. See the history of this page for a list of all contributions to it.