condensed set




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

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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.



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 *


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?.


Condensed sets form a pretopos.

See \cite[Proposition 1.7]{ScholzeLCM} for the following 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(*).


Last revised on April 10, 2021 at 14:48:46. See the history of this page for a list of all contributions to it.