dagger not evil (2)

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

Dear Peter and all,

In my last message, I wrote that the forgetful functor DCat ---> Cat 
is wrong because it is not a right Quillen functor. 
The argument is not good enough.

I believe that a forgetful functor XStruc--->Cat should reflect 
weak equivalences in addition to preserving them.  

The forgetful functor DCat ---> Cat preserves weak equivalences
but it does not reflect them. 
Because two objects in a dagger category can be isomorphic
without been unitary isomorphic.

Best,
aj

-------- Message d'origine--------
De: Joyal, André
Date: mar. 05/01/2010 15:04
À: Peter Selinger; Categories List
Objet : dagger not evil
 
Dear Peter and all,

I cannot resist adding my grain of salt to the ongoing 
discussion on dagger categories.

I will take the point of view of a homotopy theorist.
Recall that the category of small categories Cat admits a
"natural" model structure (called the "folk" model structure
for the wrong reason by the folks). 

The category of small dagger categories DCat also admits 
a "natural" model structure. A dagger functor f:A-->B
is a weak equivalence iff it is fully faithful and 
unitary surjective (this last condition means that every object of B
is unitary isomorphic to an object in the image of the functor f).
The cofibrations and the trivial fibrations are as in Cat.
A fibrations is a unitary isofibration (a map having the lifting
property for unitary isomorphisms).


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