(0,1)-category

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Definition

A completely distributive lattice is a complete lattice in which arbitrary joins and arbitrary meets distribute over each other.

Properties

Algebraic lattices

Proposition

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

This appears as remark 4.3 in (Caramello 2011).

Constructive complete distributivity

A complete lattice $A$ is called constructively completely distributive (CCD) if the join-assigning morphism $D A \to A$ has a left adjoint, with $D A$ the poset of downsets.

Constructive complete distributivity is equivalent to complete distributivity if and only if the axiom of choice holds (Wood&Fawcett (1990)).

Constructively completely distributive lattices are an example of continuous algebras for a lax-idempotent 2-monad.

Remark

Completely distributive lattices correspond to tight Galois connections (Raney 1953). This generalizes to a correspondence between totally distributive toposes and essential localizations (Lucyshyn-Wright 2011).

CCD lattices are precisely the nuclear objects in the category of complete lattices.

The (bi-) category $\mathfrak{CCD}$ with CCD lattices and sup-preserving maps is the idempotent splitting of the (bi-) category of relations $\mathfrak{Rel}$. This plays an important role in domain-theoretical approaches to the semantics of linear logic.

References

• B. Fawcett, R. J. Wood, Constructive complete distributivity I , Math. Proc. Camb. Phil. Soc. 107 (1990) pp.81-89.
• R. Guitart, J. Riguet, Enveloppe Karoubienne de Catégories de Kleisli , Cah. Top. Geom. Diff. Cat. XXXIII no.3 (1992) pp.261-266. (pdf)

• R. Lucyshyn-Wright, Totally Distributive Toposes , arXiv.1108.4032 (2011). (pdf)

• G. N. Raney, Tight Galois Connections and Complete Distributivity , Trans.Amer.Math.Soc 97 (1960) pp.418-426. (pdf)

• R. Rosebrugh, R. J. Wood, Constructive complete distributivity IV , App. Cat. Struc. 2 (1994) pp.119-144. (preprint)

• I. Stubbe, Towards “Dynamic Domains”: Totally Continuous Complete Q-Categories , Theo. Comp. Sci. 373 no.1-2 (2007) pp.142-160.

Revised on September 19, 2014 19:12:19 by Thomas Holder (89.204.138.125)