Thanks - I've gotten both (now all three!) finally and now sent out. Sorry that the order has gotten a messed up - that sort of thing seems to be inevitable when things get busy, and I know that mailers sometimes deliver out of order due to length anyway... And by the way, all very interesting posts! very best, Bob On Thu, 31 Dec 2009, Peter Selinger wrote:

Hi Bob,

I just bounced the message to you, and here I am forwarding it again just in case. Strange that it got lost. Let me know if you get this.

This logically precedes my message to Dusko, in which I say that I "state again" something that I first stated in this here message.

-- Peter

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From selinger Wed Dec 30 10:52:47 2009 Subject: Re: categories: Re: Quantum computation and categories To: toby+categories@ugcs.caltech.edu Date: Wed, 30 Dec 2009 10:52:47 -0400 (AST) Cc: categories@mta.ca (categories)

Several people attempted to give a "non-evil" definition of a dagger category. Not much of this makes sense.

Consider the following two categories:

(a) the category of finite dimensional complex vector spaces and linear maps, and (b) the category of finite dimensional Hilbert spaces and linear maps.

Clearly, they are equivalent categories. They have the same morphisms! Yet everyone knows that Hilbert spaces and complex vector spaces are not the same. For example, one can define unitary morphisms w.r.t. Hilbert spaces, but not w.r.t. complex vector spaces. The concept of "unitary" is itself "evil", because it is not preserved under isomorphism of objects in the category (b)!

So whatever extra structure the category (b) has, which allows a definition of "unitary", must be evil. Transporting this along equivalences does not make any sense whatsoever.

Specifically, take Toby's proposal, and consider two different objects A,B of (b) such that both A and B are two-dimensional Hilbert spaces. Let u:A->B be some non-unitary isomorphism. Then you can easily find an equivalence of categories which identifies both A and B with the two-dimensional vector space C^2, and which identifies u with the identity morphism on C^2. At this point, you have not equipped the category (a) with anything useful, because it does not induce a notion of unitary map on C^2.

It is tempting to say that what is wrong with the category (b) is that the morphisms don't accurately reflect the structure of the spaces. Perhaps one would prefer to equip the category of finite dimensional Hilbert spaces with unitary maps. Or with self-adjoint maps. Or with isometries. Or with positive maps. The fact that there are so many possible choices, and neither is strong enough to express all the others internally, shows that this is not a good solution. One nice feature of the dagger structure is that it does allow all of the above to be expressed internally. So one gets lots of "evils" for the price of one!

A sensible thing to do is to consider the category (b) of finite dimensional Hilbert spaces with linear maps, to be also *equipped*, as extra structure, with a distinguished lluf subcategory of isomorphisms (the "unitary") ones. There is a natural notion of equivalence between such categories-with-distinguished-subcategory (in particular, where each component of the natural isomorphisms FG -> id and GF -> id is required to lie in the subcategory). One can define a version of the dagger structure for such categories-with-distinguished-subcategory (in addition to the usual dagger axioms, one also must require that the notion of "unitary" induced by the dagger structure coincides with the distinguished subcategory that is a priori given).

Observe that the dagger structure can be transported along such equivalence of categories-with-distinguished-subcategory. So the dagger definition is non-evil on categories-with-distinguished-subcategory.

Unfortunately, the concept of a "distinguished subcategory" is itself evil, if one does not require the subcategory to contain all isomorphisms of the original category.

So it seems that, to define the extra structure of Hilbert spaces (on top of vector spaces), one needs at least one "evil" concept, be it that of unitary maps or the dagger structure.

-- Peter

Toby Bartels wrote:

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John Baez wrote in part:

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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.

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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.

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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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