Re: dagger not evil

[Toby Bartels <toby+categories@ugcs.caltech.edu>]

André Joyal wrote in small part:

>In other words the forgetful functor DCat ---> Cat is wrong.

This reminds me of the fact that the obvious forgetful functor
Ban -> Set or Hilb -> Set (where now Hilb is just a category,
with short linear maps as morphisms, not a dagger category) is wrong.
That is, the wrong forgetful functor takes the set of points,
whereas the right one takes the set of points in the closed unit ball.
(Of course, both functors exist, but the right one is representable.)

While the obvious forgetful functor DCat -> Cat is wrong,
is there a right one?  In particular, we have a functor
Cat -> Grpd that takes the lluf subcategory (LS) of invertible morphisms
and the functor DCat -> Grpd that takes the LS of unitary morphisms;
is there a functor DCat -> Cat that completes a commutative triangle?

Less rigorously but more concretely, can we start with Hilb+
(the dagger category with all bounded linear maps as morphisms)
and systematically derive the class of short linear maps,
much as we can systematically derive the class of unitary maps?
Offhand, I don't see how to do this.

(Note: "short" = "Lipschitz with Lipschitz constant at most 1",
so "short linear" = "bounded linear with norm at most 1".)


--Toby


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]