(0,1)-category

(0,1)-topos

# Contents

## Definition

A completely distributive lattice is a

## Properties

### Algebraic lattices

###### Proposition

The category of Alexandroff locales is equivalent to that of completely distributive algebraic lattices.

This appears as (Caramello, remark 4.3).

### Constructive complete distributivity

A complete lattice is called constructive completely distributive if the join-assigning morphism $D A \to A$, with $D A$ the poset of downsets. This is equivalent to complete distributivity if and only if the axiom of choice holds; see (WoodFawcett). Constructive completely distributive lattices are an example of continuous algebras for a lax-idempotent 2-monad.

## References

• Richard Wood? and Barry Fawcett, “Constructive complete distributivity. I”. Math. Proc. Camb. Phil. Soc. (1990), 107, 81

Revised on November 18, 2013 10:33:03 by Mike Shulman (192.195.154.58)