December 17th, 2000

beartato phd


I wonder if the lemma from yesterday could apply to
creation of weak w-categories?
Take the category Fc2c of finitely complete (strong)
2-categories, and the endofunctor Int : Fc2c -> Fc2c
which takes a finitely complete 2-category C to the
(er, hopefully finitely complete) 2-category of
weak 2-category objects in C. Assuming everything
has gone okay so far, we can point out that we
have a morphism f : Cat -> Int(Cat),
(which just takes a category to the weak 2-category
discrete at the 2-cell level) cross our fingers and
hope that Int preserves colimits, (it seems pretty
obvious that it preserves coproducts, but I don't
know about coequalizers...) and maybe get a
fixed point finitely complete 2-category U such that
Int(U) =~ U.
I don't know, though, maybe this is nothing like
a weak w-category. Somehow I thought it would be.
Also not sure how to guarantee its nontriviality.
Maybe that's what the retractions are for in Scott's
construction? For then you get an epi arrow going -from-
the colimit -to- the objects of the base of the cocone,
so you know the colimit can't be trivial...