nLab light condensed set

Idea

Proposed by Peter Scholze in the Analytic Stacks lectures in 2023 (Clausen-Scholze 2023) as a more well-behaved replacement for condensed sets and pyknotic sets in condensed mathematics that eliminate the size issues in the other two approaches. More specifically,

• unlike condensed sets, light condensed sets form a Grothendieck topos, rather than merely a pretopos

• unlike pyknotic sets, light condensed sets do not require universes

Definition

Recall that a light profinite set is a countable limit of finite sets. A light condensed set or $\aleph_1$-condensed set is a sheaf on the site of light profinite sets with finite jointly surjective families of maps as covers.

Category of light condensed sets

The category of light condensed sets $\mathrm{LightCondSet}$ is a Grothendieck topos.

The category of sequential topological spaces embed fully faithfully into the category of light condensed sets, and there is a functor that reflects the category of light condensed sets back onto the category of sequential topological spaces.

Internal logic

In the internal logic of the topos of light condensed sets, the weak limited principle of omniscience (WLPO) for the natural numbers is false (and thus excluded middle also fails).

On the other hand, the lesser limited principle of omniscience (LLPO) does hold in the topos of light condensed sets. Markov's principle also holds in the topos of light condensed sets, as does the fan theorem.

The full intermediate value theorem and the Brouwer's fixed point theorem can be proven in the topos of light condensed sets.

Related concepts

• Stone duality

• condensed mathematics

• condensed set, pyknotic set

• sequential topological space

• Johnstone's topological topos

References

• Dustin Clausen, Peter Scholze, Analytic Stacks, YouTube playlist, website

• David Wärn, On internally projective sheaves of groups (arXiv:2409.12835)

• Felix Cherubini, Thierry Coquand, Freek Geerligs, Hugo Moeneclaey, A Foundation for Synthetic Stone Duality (arXiv:2412.03203)

• What does the topos of (light) condensed sets classify? (MathOverflow)

• Clausen–Scholze’s Theorem 9.1 of Analytic.pdf, in view of light condensed sets, AKA is the Liquid Tensor Experiment easier now? (MathOverflow)

• Condensed vs pyknotic vs consequential (MathOverflow)

• David Roberts, answer to Notions of convergence not corresponding to topologies (MathOverflow)