Re: dagger not evil

[selinger@mathstat.dal.ca (Peter Selinger)]

Toby Bartels wrote:
>
> Andr=E9 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?

There is, because Cat -> Grpd is right invertible, but that is
probably not the functor you had in mind.

> 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

As far as I know, you can't do this with a dagger structure alone;
however, if you regard Hilb+ as a dagger category with dagger
biproducts, then the short linear maps are definable as those maps
f: A -> B satisfying

(1)  there exists some g: A -> C such that
               (f^dagger o f) + (g^dagger o g) = id,

or equivalently,

(2)  there exists some h: D -> B such that
               (f o f^dagger) + (h o h^dagger) = id.

In Hilb+, conditions (1) and (2) are equivalent for a given f.  In a
general dagger category with dagger biproducts, they are not
equivalent (as the example Rel shows), and one should in general
define a "short linear" morphism to be an f that satisfies both (1)
and (2). The always form a dagger subcategory.

In general, invertible + short linear, in this sense, does not imply
unitary. A counterexample is chosen-basis vector spaces over Z_2 (with
dot product as inner product), where all maps are short linear, but
not all invertible maps are unitary.

However, if one further assumes that the category satisfies the
following strictness condition:

(3)      1 + (g o g^dagger)  = 1     implies    g = 0,

then invertible + short linear implies unitary. So something like your
"right" forgetful functor DCat' -> Cat can be defined, where DCat' is
DCat equipped with these additional hypotheses. I don't know whether
there is a more organic definition that works in greater generality.

-- Peter



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