condensed set




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

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


Basic concepts

Universal constructions

Extra stuff, structure, properties


Basic statements


Analysis Theorems

topological homotopy theory



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.


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 map

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

is a bijection 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.


Last revised on June 12, 2019 at 01:55:57. See the history of this page for a list of all contributions to it.