nLab flat distributive lattice

Idea

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The category of cubes in triangulated type theory is given by the category of finite flat distributive lattices.

Properties

Theorem

Given a flat distributive lattice $L$, if $a \vee b = 1$, then $a = 1$ or $b = 1$.

Proof

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 Examples

• Every finitely generated distributive lattice is flat.

Related concepts

• distributive lattice

• flat module, flat functor

• Phoa's principle

• category of cubes

• triangulated type theory

References

• Bas Spitters, Cubical sets and the topological topos (arXiv:1610.05270)

• Daniel Gratzer, Jonathan Weinberger, Ulrik Buchholtz, Directed univalence in simplicial homotopy type theory (arXiv:2407.09146)