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.

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?.

References

Last revised on April 13, 2020 at 04:02:22. See the history of this page for a list of all contributions to it.