dagger not evil

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

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).

The forgetful functor DCat ---> Cat is a right adjoint
but it is not a right Quillen functor with respect to the
natural model structures on these categories.
In other words the forgetful functor DCat ---> Cat is wrong.
This may explains why a dagger category cannot be
regarded as a category equipped a homotopy invariant structure.
But I claim that the notion of dagger category is perfectly reasonable
from an homotopy theoretic point of view.
This is because the model category DCat is combinatorial.
It follows, by a general result, that the notion of
of dagger category is homotopy essentially algebraic
There a homotopy limit sketch whose category of models (in spaces)
is Quillen equivalent to the model category DCat.
This is true also for the model category Cat.

There should be a notion of dagger quasi-category.
A dagger simplicial set can be defined to be a simplicial set X
equipped with an involutive isomorphism dag:X-->X^o  
which is the identity on 0-cells.
The category of dagger simplicial sets (and dagger preserving maps)
is the category of presheaves on the category whose objects are the ordinals [n]
but where the maps [m]-->[n] are order reversing or preserving.

Finally, the (homotopy) trace of a category (resp. quasi-category) has
the structure of a cyclic set in the sense of Connes.
I conjecture that the (homotopy) trace of a dagger category (resp. dagger quasi-category)
has the structure of a dihedral set in the sense
of Fiedorowicz and Loday.

Happy New Year to all!
andré

PS: I will be quiet during the next few weeks.


-------- Message d'origine--------
De: categories@mta.ca de la part de Peter Selinger
Date: dim. 03/01/2010 02:23
À: Categories List
Objet : categories: the definition of "evil"
 
Dear all,

sorry for sending yet another message on the topic of "evil"
structures on categories. After some interesting private replies, as
well as Dusko's latest message (which should have appeared on the list
by the time you read this), I noticed that not everyone is agreeing on
the technical meaning of the term "evil". I will therefore attempt to
state a more precise technical definition of the term as I have used
it. Perhaps 2-category theorists already have another name for this.

The information definition I had used is that a structure is "evil" if
it does not "transport along equivalences of categories". I thought it
was reasonably obvious what was meant by "transport along", but there
is actually a lot of variation in what people understand this phrase
to mean.

John Baez gave a pointer to a website containing a technical
definition of "evil": http://ncatlab.org/nlab/show/evil.
Unfortunately, this site only speaks of properties, not structures. It
is easy to state what it means for a property of categories to be
transported along equivalences: namely, if C has the property, and C
and C' are equivalent, then C' has the property. Structures are more
tricky.

....

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