Re: Quantum computation and categories

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

John Baez wrote in part:

>A dagger-category is a category C with a functor
>F: C -> C^{op}
>which is the identity on objects and has F^2 = 1.

>Category theorists will note that the above definition is "evil", in the
>technical sense of that term:
>http://ncatlab.org/nlab/show/evil
>Namely, it imposes equations between objects, so we cannot transport a
>dagger-category structure along an equivalence of categories.

>Often evil concepts (like the concept of "strict monoidal category") have
>non-evil counterparts (like the concept of "monoidal category").  But in
>this particular case I know no way to express the idea without equations
>between objects.  Both Hilb and nCob are dagger-categories.  This fact is
>important.  Try saying it in a non-evil way!

By default, there is a non-evil way to say it:

Given a category C, a _non-evil dagger-category structure_ on C
consists of a dagger-category C' and an equivalence F: C -> C' of categories.

So one question is whether there is a less long-winded way to say that.
Another question (which logically comes before the first question)
is what is the right notion of equivalence of such structures.


--Toby


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