evil

evil

[John Baez]

Sorry, if it's not too late please post this one instead:

From: John Baez <baez@math.ucr.edu>
Date: Tue, Sep 14, 2010 at 3:33 PM
Subject: Evil
To: categories <categories@mta.ca>


David wrote:

Jean Benabou wrote:
>> Maybe my english isn't so "beautiful", but in all cases where "evil" has
>> been used, what is wrong with "wrong" instead?
>

I'm not so enamoured with the use of the word 'evil', but it seems to
> be more entrenched than perhaps it was intended, namely as a joke.
>

It's supposed to be funny, but I'm glad to see it become entrenched.

Why?

First, it has a very specific meaning.  A property of objects of some
category C is said to be "evil" if it holds for some object x of C but not
some isomorphic object y.  More generally: a property of objects of some
n-category is "evil" if it holds for some object x but not some equivalent
object y.  For details, see:

http://ncatlab.org/nlab/show/evil

Second, it captures the interesting state of affairs in category theory
where some definitions can be well-formed yet somehow "suboptimal" because
equations were used when isomorphisms should have been specified.

"Wrong" doesn't work here, since mathematicians use it in other important
ways: for example, "false", "incorrect" or "inappropriate".  "Evil" is, to
the best of my knowledge, never used in mathematics except in this one
technical sense.

If anybody finds the term "evil" upsettingly strong, I suggest "naughty" as
an alternative.

Best,
jb


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Re: Small is beautiful

[Colin McLarty]

I'm not sure I understand this

> In particular, syntax is NOT the adjoint of semantics. Cratylus,
> Chomsky, and their 21st century followers can be refuted by
> looking soberly at the actual practice of mathematics (wherein
> the construction of sequences of words and of diagrams
> is pursued with great care for the purpose of communication.
> That syntax is only remotely dependent on the structure of the
> content that is to be communicated).
>
> Both of the functors
>
> ?--------------> theories -------------->Large categories
>         Syntax                                      Semantics
>
> are needed.  The domain category of the first can be chosen
> in various useful ways: sketches or diagrams of signatures et cetera.

Do you mean that if we choose some kind of sketches for the domain
category then theories are a reflective subcategory, more or less the
'definitionally closed' sketches?   Then a presentation of a theory T
would be (up to isomorphism) any unit arrow of the adjunction with T
as codomain?

best, Colin


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Re: evil

[claudio pisani]

Isn't there something "evil" in the definition of dual category itself?

Claudio





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Re: evil

[Toby Bartels]

[Note from moderator: Several messages to categories apparently hung in a
mail system for several days. With apologies to posters, I am about to
post four from late Decemeber in what should have been their posting
order. Sorry about the delay,
Bob]

A dagger structure on a category should not really be considered evil at all.

If you have a functor F: C^op -> C and ask whether it is a dagger structure,
then this is (taken literally) an evil question; the answer is yes
iff F^2 = 1 and F is the identity on objects, both evil conditions.
More precisely, two isomorphic functors may have different answers.
(A non-evil version is to ask whether F is isomorphic to a dagger structure.)

However, it's not necessary to define a dagger-category as a category C
equipped with a functor F: C^op -> C such that F satisfies these conditions.
In lower-level language, we ask instead that C be equipped with an operation
that takes each morphism f: x -> y to a morphism f^\dag: y -> x
such that id^\dag = id, (f g)^\dag = g^\dag f^\dag, and (f^\dag)^\dag = f.
Nothing here refers to equality of objects; it can be formulated in a language
that (like FOLDS) does not have this concept.

Given a dagger structure on C, defined in this elementary way,
we can construct a functor \dag: C \to C^op that satisfies the evil property.
(Of course, it also satisfies the non-evil version of that property.)
But that is neither here nor there as to whether dagger structures are evil.

There is some new discussion on the nLab:
http://ncatlab.org/nlab/show/evil#daggers
In particular, Mike Shulman shows how to translate dagger structures
along equivalences of categories, proving that they are not evil.

My previous post on this subject should probably be ignored.
While any concept ~can~ be de-evilled in the way shown there,
this does not necessarily give you the concept that you want,
and indeed it need not even preserve already non-evil concepts.
(And in this case specifically, it does not seem to be correct,
as others have already argued here.)


--Toby


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Re: evil

[David Yetter]

I have been following with some amusement the discussion of 'evil'.

I am mostly amused because long ago, I think in my grad student days
standing in Peter Freyd's office, I had made the suggestion that to
avoid overloading adjectives (normal and regular being particularly
abominable examples of the phenomenon) mathematicians should resort to
using adjectives that usually have a moral denotation.

Once one realizes that 'evil' exists not just for objects in categories,
but for 0- and 1-arrows in bicategories, 0-, 1- and 2-arrows in
tricategories, . . . it seems to me the proper attitude to take toward
'evil' (or strictness) is given by Saunders' dictum about generality:
"good general theory does not search for the maximum generality, but for
the right generality".  So a good (higher) categorical structure should
not search for the maximum weakness, but for the right weakness. (Or, if
you want, not search for the minimum 'evil', but the the right amount of
'evil'.)

For instance, it seems to me that structure of the category of framed
tangles in which arrows are ambient isotopy classes (rel boundary) of
framed tangles, which Shum's beautiful coherence theorem tells us is
monoidally equivalent to the free ribbon (née tortile) category one one
object generator is marred and made less useful (certainly for
application to knot theory and 3- and 4-manifold topology) by deciding
one should work instead with a (2,infinity)-category with one object,
framed point sets as 1-arrows, framed tangles as 2-arrows, isotopies as
3-arrows, isotopies of isotopies as 4-arrows, and so ad infinitum.

Or, maybe not, but only if one has an application for which the
(2,infinity)-category is the right level of weakness.

Best Thoughts,
David Yetter




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