Re: Evil in bicategories

[Toby Bartels <toby+categories@ugcs.caltech.edu>]

David Roberts wrote in part:

>Jean Bénabou 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.
>Regardless of my personal convictions, I like to remain a mathematical
>agnostic, so 'wrong' seems to me to be too strong.

I feel the same way, which is why I *prefer* to say "evil" instead of "wrong".
The word "evil" is so over-the-top that someone who uses it *must* be kidding.
However, the word "wrong" sounds like it should be taken seriously,
but the mathematics of strict categories is valid, not wrong at all.
(It's just not the mathematics that I'm doing when I do category theory.)

Incidentally, this usage of "evil" fits in with a usage of "morally"
examined by Eugenia Cheng: http://www.cheng.staff.shef.ac.uk/morality/.
"Morally", one cannot compare objects of a given category for equality
(because the results are not preserved by an equivalence of categories);
even if it is possible, it is "evil".

>Toby Bartels calls categories where one is not allowed to test for
>equality between arbitrary objects 'weak' and those where one can do
>so 'strict'

Right: http://ncatlab.org/nlab/show/strict+category

In many cases, we can use "strict" instead of "evil".
For example, here is David Leduc's original post:

>>>In a bicategory, composition of 1-cells is associative up to
>>>isomorphism. Because it would be evil to insist that h o (g o f) is
>>>equal to (h o g) o f. However the source and target objects of those
>>>compositions must be equal. Isn't it evil? Why not weaken this
>>>requirement by saying that the sources (respectively, targets) of h o
>>>(g o f) and (h o g) o f must only be isomorphic?

Let us replace each usage of "evil" by "too strict":

>>>In a bicategory, composition of 1-cells is associative up to
>>>isomorphism. Because it would be too strict to insist that h o (g o f)  is
>>>equal to (h o g) o f. However the source and target objects of those
>>>compositions must be equal. Isn't it too strict? Why not weaken this
>>>requirement by saying that the sources (respectively, targets) of h o
>>>(g o f) and (h o g) o f must only be isomorphic?

This even makes David's use of the verb "weaken" look very nice.

So while I like the noun "evil" to encapsulate the whole idea
(with the understanding that is not too be taken too seriously),
one can probably avoid it in serious mathematical questions.
(I don't mean to criticise David for using it, however.)


--Toby


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