evil (fwd) Re: Quantum computation and categories

evil (fwd) Re: Quantum computation and categories

[Dusko Pavlovic]

[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.







[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

in defense of evil

[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.



[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: in defense of evil

[Dusko Pavlovic]

hi peter,

happy new year! may the daggers over hilbert spaces be the worst evil cast
upon us :)

but i think that this "defense" view of evil suggests that you may be
missing the point.


1) THE PROBLEM OF EVIL

if someone gives you two big complicated hilbert spaces, or two big
complicated groups --- how do you decide whether they are the same? what
does it mean that to be the same group?

well, one thing you could try is to map the elements of one to the
elements of the other one. this may give you an isomorphism.

OR you may look at the actual presentations, say of the group structures
that they gave you, and see whether these are completely equal. if you are
extremely lucky, the two groups may be given by the same equational
presentation. otherwise, if you live in a set theoretic universe, each of
the groups would be given to you as a set, and the two sets can be equal,
or not. in a computer system, each of the groups would be given as a
software module, and they may, or may not be identical as executable
binaries.

the main problem with this is that the various implementation conventions
need to be consistent and interoperable. i think it was peter freyd who
once asked something like: "How can you tell that the set corresponding to
the Monster Group is not the same as the set corresponding to an initial
segment of the number pi?" the same question applies to the binaries, that
may represent an executable group operation, or a piece of data in another
encoding.

note that the problem is NOT a mere philosophic disagreement between a
categorical/structuralist and a set-theoretical/positivist view of math.
it is an eminently PRACTICAL problem of computation.

every crypto system consists of some groups. all these groups are
implemented. some attacks depend on the implementations. other attacks
only depend on the structure of a group, and apply to all implementations.
the latter attacks are more dangerous, but easier to eliminate --- by
avoiding all evil definitions. the former attacks cannot be completely
eradicated: at the end of the day, every new implementation opens some new
avenues of attack.

((the distinction between the evil and the non-evil definitions is
becoming a big deal in cryptography, since mixing them led to some awkward
attacks. eg, one of the most standardized key agreement protocols, ECMQV,
is all in terms picard groups --- and then at one point it suddenly
truncates a key from 160 to 80 bits. which is of course evil. this led to
an attack. and this protocol is "critical for national security" and
brings about $50M annyally just to one company...))

no one is proposing to eradicate evil. like other things in science, evil
is neither good mor bad; it's just there. our goal should be to understand
it.


2) EVIL IN HILBERT SPACES

john baez asked if we can define the dagger structure of FHilb and nCob in
a non-evil way. i think that the structure that i described achieves that
goal.

[[i am sorry if the rest of this gets longer than desirable, but since
peter "dissected" my post, the code of flames requires a clarification.
pls skip or enjoy.]]

> *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.

the fact that an operation does not transport on the nose does not mean
that it is impossible to find an isomorphic copy that will be preserved.
that is what i proposed.

> 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.

i am sorry if i am being extremely stupid, but how does this example
differ from saying: "ordered group of integers is equivalent as a poset
with the ordered ring of integers; the latter has multiplication and the
former does not"? what do i learn from the fact that an algebraic
operation (inner product, multiplication...) not specified in a signature,
turns out to be derivable?

>> 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.

sorry: the word "dinaturals" should be omitted. i corrected this after the
original post. i am sorry about the confusion.

if e_X and h_X happen to be dinatural, then P is just the star functor. in
general, all we know about P is that it is a functor whose object part
maps each object to some dual. the duals are determined up to isomorphism;
any will do.

> 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.

except in the cases when it is, eg in Rel etc.

> 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.

oh this was nice: people were confused, the confusion got cleared up,
the period of enlightment ensued.

but the story never ends, as we can always refine the picture. the point
of evil is that your first sentence in the above paragraph does not refer
to anything that can be stated in categorical terms. if i give you B and
B*, how do you really know which is which? you look for a star? think
about it: what do you need to be able to really tell when they are equal
and when not?

> 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.

there is no need for the *chosen* ones. the entire subject of linear
algebra is based on the assumption that every space has *a*
basis. that means that every hilbert space has *a* frobenius
structure. non-canonical, and not preserved by the morphisms of the
category.

> 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.

this is not true. maybe you used the dinaturality of the initial
(overconstrained) definition P in your calculation.

> 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.
[snip]
> 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.

i don't think that i could have required a *chosen structure* at any
point, even under torture, or coherent transport, or canonical
isomorphisms. all of that is in the eye of the beholder. i was talking
about the actual practice of linear algebra: spaces have bases, the bases
are not unique, nor preserved under the morphisms. in the finite case, the
isomoprhisms B~B* are not natural, but the non-canonical ones are
available when needed. -- my only point was that this can be used to avoid
evil. (this was already used in our work on dagger monoidal
structuralism.)

i would be the last person to suggest that a simple evil structure
should be *replaced* by a less simple non-evil structure. i use evil
as much as everyone else, and more. but it may be useful (and an
interesting exercise) to observe that a particular evil structure may
be isomorphic to something non-evil, and that it can be derived from
it.

sorry again about clogging everyones holiday mailboxes. i hope that peter
and i won't seriously annoy each other. lets stop defending things, or
taking ouresleves too seriously. we do math, and the damage to the
environment is minimal.

and i think i am going to stop posting for a while.

with the very best wishes to all,
-- dusko





>
> -- 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.
>>

...


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Re: evil (fwd) Re: Quantum computation and categories

[David Roberts]

> 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.

An ioof is 'evil' in a subtly different way to what was discussed, in
my opinion, in that the property of being such a functor is not
invariant under natural isomorphism. This is then really a
2-categorical notion of evil. Are there many examples/other commonly
used properties of functors that are evil in this way?


Happy new year

David


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