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