Re: evil

[David Yetter <dyetter@math.ksu.edu>]

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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