Re: evil

[Toby Bartels <toby+categories@ugcs.caltech.edu>]

[Note from moderator: Several messages to categories apparently hung in a
mail system for several days. With apologies to posters, I am about to
post four from late Decemeber in what should have been their posting
order. Sorry about the delay,
Bob]

A dagger structure on a category should not really be considered evil at all.

If you have a functor F: C^op -> C and ask whether it is a dagger structure,
then this is (taken literally) an evil question; the answer is yes
iff F^2 = 1 and F is the identity on objects, both evil conditions.
More precisely, two isomorphic functors may have different answers.
(A non-evil version is to ask whether F is isomorphic to a dagger structure.)

However, it's not necessary to define a dagger-category as a category C
equipped with a functor F: C^op -> C such that F satisfies these conditions.
In lower-level language, we ask instead that C be equipped with an operation
that takes each morphism f: x -> y to a morphism f^\dag: y -> x
such that id^\dag = id, (f g)^\dag = g^\dag f^\dag, and (f^\dag)^\dag = f.
Nothing here refers to equality of objects; it can be formulated in a language
that (like FOLDS) does not have this concept.

Given a dagger structure on C, defined in this elementary way,
we can construct a functor \dag: C \to C^op that satisfies the evil property.
(Of course, it also satisfies the non-evil version of that property.)
But that is neither here nor there as to whether dagger structures are evil.

There is some new discussion on the nLab:
http://ncatlab.org/nlab/show/evil#daggers
In particular, Mike Shulman shows how to translate dagger structures
along equivalences of categories, proving that they are not evil.

My previous post on this subject should probably be ignored.
While any concept ~can~ be de-evilled in the way shown there,
this does not necessarily give you the concept that you want,
and indeed it need not even preserve already non-evil concepts.
(And in this case specifically, it does not seem to be correct,
as others have already argued here.)


--Toby


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