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/ ]