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