# nLab condensed set

Contents

### Context

#### Algebra

higher algebra

universal 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

$ProfiniteSet^op \to Set$

such that the natural map

$T(S\sqcup S')\to T(S)\times T(S')$

is a bijection for any profinite sets $S$ and $S'$, whereas the natural fork

$T(S)\to T(S')\rightrightarrows T(S'\times_S S')$

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

Scholze, p.7 modifies this definition to deal with size issues. For any uncountable strong limit cardinal $\kappa$, the category of $\kappa$-condensed sets is the category of sheaves on the site of profinite sets of cardinality less than $\kappa$, with finite jointly surjective families of maps as covers.

The category of condensed sets is the (large) colimit of the category of $\kappa$-condensed sets along the filtered poset of all uncountable strong limit cardinals $\kappa$.

## References

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