Dear all, I apologize for sending out a third (long) email about dagger categories today. There have been some proposals on this list for defining "non-evil" analogues of dagger structure. None of them make sense, and I do feel an obligation to rebut them one by one. It's easy to give the subject two minutes of thought and produce some half-baked ideas, but unfortunately it takes much longer to explain why the ideas don't work. I am afraid that if I don't nib this in the bud, I will have to read it in referee's reports for the next 10 years ("why isn't Selinger using the non-strict dagger structure proposed by so-and-so...?"). I imagine most readers of this list are tired of this subject, and I invite you to ignore the rest of this email. Before I dissect Dusko's proposed definition, let me repeat the main reason why *every* such definition is doomed to failure. It is a fact that the notion of dagger, however defined, intrinsically does not transport along equivalences. This is a hard fact, in the sense 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. For this argument, it is irrelevant how dagger structure is 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 the latter coincides with the "usual" notion in the category of Hilbert spaces. Here is the specific rebuttal of Dusko's proposal.
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 monoidal category is symmetric monoidal, this is precisely the definition of a compact closed structure. The first two conditions are redundant. The whole 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, Dusko now assumes that each object A is equipped with a chosen isomorphism A* -> A. (He actually assumes 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 many things wrong with this: 1) The main example, which is the category of finite dimensional Hilbert spaces, does not have the structure Dusko requires. There are no such chosen isomorphisms or Frobenius structures or chosen bases. However, to continue the argument, let's assume that we have chosen such additional structure in some arbitrary way. 2) In the main example, the category of finite dimensional Hilbert spaces, Dusko's definition does not coincide with reality, i.e., the *actual* definition of the adjoint of a linear map. No matter how the bases are chosen, the resulting map B -> B* -> A* -> A is *not* the adjoint of f for most f. Indeed, when written as a matrix in the chosen basis, this is still the transpose, and not the adjoint, of the matrix of f. 3) In any case, the structure of "having a chosen Frobenius structure on each object" is itself an evil structure on categories. Quite clearly, there will be some isomorphisms of the category that don't preserve the Frobenius structure. There will be equivalences of categories sending those isomorphisms to identities. Identities always preserve Frobenius structure, so therefore Frobenius structure does not transport along such equivalences. The same argument shows that the structure of "having a chosen isomorphism A* -> A on each object" is evil too. In summary, Dusko's definition does not coincide with the intended example, and is evil anyway, so doesn't solve the problem. -- 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.
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