Dear Bob, if you managed to intercept my previous message (subject: in defense of evil), could you please substitute the following body. I am afraid to submit it directly to categories, in case you missed my request to stop the first one, and end up forwarding them both. Sorry for the trouble. I should know better than to press "send" on stuff like that. Thanks, -- Peter ---------------------------------------------------------------------- Dear Dusko, your proposed definition of a non-evil dagger structure is a nice try, but it doesn't work at all. Before I dissect your definition, let me state again the reason why *every* such definition will fail. It is a hard fact that the notion of dagger, however defined, intrinsically does not transport along equivalences. By "hard fact", I mean that it is a consequence of examples, rather than definitions. There is no way to fudge the definitions to make this fact go away, unless one changes the definitions so radically that they no longer fit the fundamental example. The fundamental example is the category of finite dimensional complex vector spaces vs. the category of finite dimensional Hilbert spaces. They are equivalent, the latter has a dagger structure, and the former does not. This shows that the notion is evil. For this argument, it is irrelevant how dagger structure is actually defined (strict, weak, abstract, concrete...): all that is required is that the definition is strong enough to yield a notion of unitary map, and that this unitary notion coincides with the "usual" one in the category of Hilbert spaces. Now let me comment on your definitions in detail.
DEF. Let CC be a monoidal category. a *dagger* on CC is a functor P: CC^op ---> CC which is
* self-adjoint * equivalence * given together with the dinaturals ** e_X : X (x) PX ---> I ** h_X : I--->PX (x) X which make PX -| X.
Assuming here that the category is symmetric monoidal, this is precisely the definition of a compact closed structure. The first two conditions are redundant. The point of Abramsky and Coecke's work on dagger categories was to explain, in categorical terms, that the adjoint (i.e., dagger) of a linear function f:A->B is *not* the same as the transpose. The adjoint goes B -> A, whereas the transpose goes B* -> A*. This is something people used to be confused about. Abramsky and Coecke cleared up the confusion; the above definition reintroduces it. To remove the distinction between a morphism B* -> A* and a morphism B -> A, you next assume that each object A is equipped with a chosen isomorphism A* -> A. (You actually assume chosen Frobenius algebra structures, which is a stonger assumption, but only the isomorphisms are needed for the present purpose). With this assumption, given a map f : A -> B, we can take the transpose f* : B* -> A*, and then compose it with the given isomorphisms to get a map B -> B* -> A* -> A. This is of the type required for a (strict) dagger structure. There are a number of things wrong with this: 1) The main example, which is the category of finite dimensional Hilbert spaces, does not have the structure you require. It isn't equipped with chosen Frobenius structures (equivalently chosen bases), nor with chosen isomorphisms A* -> A. However, to continue the argument, let's assume that we have arbitrarily chosen such additional structure. 2) In the main example, the category of finite dimensional Hilbert spaces, your definition does not coincide with reality. Namely, the *actual* definition of the adjoint of a linear map does not coincide with what one gets as the result of your definition. To see this, assume chosen bases, and note that the matrix of the map B -> B* -> A* -> A is exactly the transpose of the matrix of f : A -> B, in the given bases of A and B. On the other hand, the matrix of f+ : B -> A is the adjoint. 3) Moreover, contrary to what you wrote, the structure of "having a chosen Frobenius structure on each object" is itself an evil structure on categories. Namely, there will be some isomorphisms of the category that don't preserve the Frobenius structure (think linear map that does not preserve the chosen bases). Choose one such isomorphism s : U -> V, and construct an equivalence F to some other category such that the chosen isomorphism is sent to an identity. By transporting the Frobenius structure along F, you end up with two different Frobenius structures on F(U)=F(V). So there is no coherent way to transport. The same argument shows that the structure of "having a chosen isomorphism A* -> A on each object" is also evil. In summary, your definition does not coincide with the intended example (linear adjoints in Hilbert spaces), and ends up being evil anyway, so doesn't solve the problem. It also assumes too much structure (for example, the usual definition of a dagger category neither requires a monoidal structure nor a closed one. It works on any category). I hope we can agree that, to capture something essential about Hilbert spaces, some amount of evil structure is required, and we should embrace the evil definition and live with it. In fact, I am surprised that there are not more well-known examples of evil structures in category theory. In principle, any structure that allows the definition of a distinguished subcategory of isomorphisms (in this case, the unitary ones) should be evil. Does anybody know further examples? -- Peter Dusko Pavlovic wrote:
[yesterday john baez sent his message only to me, and i replied only to him. he actually meant to send it to the list, and encouraged me to resend the reply. i apologize for posting so much these days. -- dusko (in bed with a flu and a computer)]
hi john,
thanks for your note. the notion of evil is an interesting challenge in any context.
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!
let me try.
DEF. Let CC be a monoidal category. a *dagger* on CC is a functor P: CC^op ---> CC which is
* self-adjoint * equivalence * given together with the dinaturals ** e_X : X (x) PX ---> I ** h_X : I--->PX (x) X which make PX -| X.
LEMMA. Suppose that every object in CC comes with a Frobenius algebra structure. Then there are coherent natural isomorphisms PX-->X.
PROOF. The Frobenius algebra structure induces
** ee_X : X (x) X ---> I ** hh_X : I---> X (x) X
which make X-|X. the natural isomorphisms PX-->X are composed from the adjunction equipment (along the proof that an adjoint is unique up to a coherent iso). QED
DEF. A strict dagger is a functor D:CC^op ---> CC obtained by transferring a dagger along the canonical isomorphisms from the lemma.
COROLLARY. strict daggers are not evil: they are preserved under the equivalences.
PROOF. daggers are obviously preserved. frobenius algebras are preserved. hence the canonical isomorphisms are preserved.
FACT 1. a frobenius algebra structure on a hilbert space is just a choice of a basis. (hence we can a non-evil adjoint by first defining a preadjoint to be the conjugate of the dual operator, and then transferring along the isomorphism X^* ---> X induced by the chosen basis.)
FACT 2. a frobenius algebra structure in nCob is the underwear structure.
-- dusko
PS the hope is that this provides a nonevil view of the daggers in FHilb and nCob. i guess the general suggestion might be to define dagger compact structure by a self-adjoint equivalence, plus a requirement that every object admits a frobenius algebra structure. that structure is not evil, and it is carried by all examples considered so far.
i don't think that there is a general solution for the problem of evil in categories: we can only pin down a particular object, as an element of an isomorphism class, in the lucky cases when there is some additional structure that characterizes it. but in general, evil exists. every functor can be factored as an identity-on-the-objects-functor (ioof), followed by an embedding. the embedding is good, but ioofs are evil, and i think that they deserve their name. lord knows how much we use them.
in a sense, category theory can be distinguished from set theory by the presence of evil.