Re: dagger not evil

["Joyal, André" <joyal.andre@uqam.ca>]

Dear Toby

You wrote:

>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?

I will try answer your question, but my answer is wonkish.

First, a category can be regarded as a (simplicial) diagram of groupoids.
More precisely, every category C has a "Rezk nerve" RN(C)
which is a simplicial object in the category of groupoids.
By definition, we have

RN(C)_n=IsoNat([n],C)

for every non-negative integer n,
where IsoNat([n],C) denotes the groupoid of natural isomorphisms
in the category of functors [n]-->C.
The nerve RN(C) was first introduced by Charles Rezk in
http://arxiv.org/abs/math/9811037
The functor RN has very nice properties. 
It embeds the category Cat in the category
of simplicial groupoids Simp(Grpd). 
The embedding respects (ie preserves and reflects)
the equivalences defined on both sides,
where a map of simplicial groupoids f:X-->Y  
is defined to be an equivalence if it is an equivalence levelwise.
It can be proved that RN is a right Quillen functor
with respect to the natural model structure on Cat
and with respect to the Reedy model structure on Simp(Grpd).

A dagger category can also be regarded as a (dagger simplicial) diagram of groupoids.
More precisely, every dagger category C has a "unitary nerve" UN(C)
which is a dagger simplicial object in the category of groupoids.
By definition, we have

UN(C)_n=UIsoNat([n],C)

for every non-negative integer n,
where UIsoNat([n],C) denotes the groupoid of unitary natural isomorphisms
in the category of functors [n]-->C.
The functor UN embeds the category DCat in the category
of dagger simplicial groupoids DSimp(Grpd). 
The embedding respects the equivalences defined on both sides,
where a map of dagger simplicial groupoids f:X-->Y  
is defined to be an equivalence if it is an equivalence levelwise.

You wrote:

>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.


I am afraid I dont have an answer to this question. 
But I will think about the problem.


Best,
André




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