Hi - You wrote: thanks for the examples. But I don't think the notion used in your
examples is general enough. Fox some groupoid C, and consider the property of an object x: "x is isomorphic to exactly 3 objects of C". Clearly it is evil, because it is not invariant under equivalences of C. Yet, according to the definition you used in this email, it extends to a functor C -> {F,T}, and therefore is non-evil.
Right. Your example is fiendishly clever! I think this property is "non-evil in C" but "evil in the 2-category of categories equipped with an object". If we can take C for granted, it's okay to talk about this property, but if we think of C as no better than any equivalent category, we can't. I've always thought that the question of whether properties are evil concerns objects of an n-category, and whether they are or are not invariant under equivalence. But here you are taking the object of the 1-category C and obtaining from it an object in the 2-category of categories equipped with an object. We can keep playing this trick! Now we've got an object in a 2-category. Given an object in a 2-category we can get an object in the 3-category of 2-categories equipped with an object. And so on. When are we allowed to stop? We could be seeing something Makkai has emphasized all along, namely that the first "fixed point" for a mathematician trying to systematically avoid evil in all its forms is omega-categorical mathematics. Thanks for your email. Maybe you didn't want to post it on the list because there's a war going on and you didn't want to be seen as attacking "my side". But it's a lot more interesting than the dispute over whether the word "evil" is acceptable terminology. It'd be sad if that sort of stuff drives actual math out of the category theory mailing list... so please feel free to post about this. Best, jb