in defense of evil

[selinger@mathstat.dal.ca (Peter Selinger)]

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.



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