evil (fwd) Re: Quantum computation and categories

[Dusko Pavlovic <Dusko.Pavlovic@comlab.ox.ac.uk>]

[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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