Subject: categories: Re: Does duality categorify?

From: Vaughan Pratt <[email-masked]>
Date: Sat, 29 Apr 2006 10:37:16 -0700

John Baez wrote:
> Vaughn Pratt asks:
>>For example distributive lattices categorify to distributive algebras.
>>The dual of a distributive lattice is a partially ordered Stone space.
>>What does categorification do here?
>
> Unfortunately I don't know what a distributive algebra is
> or in what sense it's a categorified distributive lattice.

Sorry, I meant to write distributive category, a la Chapter 3 of Bob
Walters' book "Categories and Computer Science".  (I just finished
writing an article on algebra, I seem to have algebras on the brain.)

> If I had to think about this, I'd probably start with something
> I understand ever so slightly better, like the duality between
> finite posets and finite distributive lattices.

Right, that was the example I had in mind.
Distributive lattices:posets :: distributive categories:?

As an initial guess: "categories", with dualizer Set, which is both a
category and a distributive category.  So is CAT(C,Set) a distributive
category?  And if so, is DCAT(CAT(C,Set),Set) equivalent to C?  And what
about the enriched case V-DCAT(V-CAT(C,V),V)?

Vaughan