RE : categories: Re: Not invariant but good

["Joyal, André" <joyal.andre@uqam.ca>]

Dear John,

I agree with you that some of the examples 
in my list can be regarded as covariant structures.
But not all of them. Especially the example of
pullback squares in a model category. In fact,
the notion of fibration in a model category is
also not invariant under weak equivalences, since
every map is, up to a weak equivalence, a fibration. 
The notion of Grothendieck fibration is also not invariant 
under equivalences of categories, since the composite of a 
Grothendieck fibration with an equivalence is not a 
Grothendieck fibration in general. One could introduce 
a weaker notion of Grothendieck fibration which repairs this absence
of invariance but the usual notion of a Grothendieck 
fibration will remain important. I am reluctant to
call the notion of Grothendieck fibrations "evil".

I feel that the whole controversy about the "evil" terminology
is preventing us from discussing rationally and fruithfully
important foundational issues. The word is very negative
and polarising. Nobody likes to be told that he has
done something "evil" when he has done nothing so.

I guess you have introduced the "evil" terminology
because you wanted peoples to pay attention to
the fact that certain constructions in category theory
and higher category theory are not invariant under
equivalences. If this is so, you have succeeded in your goal.
But please, could you agree to change the terminology?

Best,
André


-------- Message d'origine--------
De: John Baez [mailto:baez@math.ucr.edu]
Date: sam. 25/09/2010 23:29
À: categories
Objet : categories: Re: Not invariant but good
 
Dear Andre -

> Many good things in mathematics are depending on the choice
> of a representation which is not invariant under equivalences,
> or under isomorphisms. Modern geometry would not exists
> without coordinate systems.

I agree.  I think you're arguing against a position that nobody
here has espoused.

A coordinate system is a structure, not a property.  In my
earlier email I said a *property* is evil if it's not invariant under
equivalences.  But I'd say a *structure* is evil if it's not *covariant*
under equivalences.  Coordinate systems are covariant under
equivalences, so they're not evil.

Let me expand on this a bit, first for properties and then for
structures.

...


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