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. Say we have a groupoid C. A (possibly evil) property of objects in C is a map F: Ob(C) -> {F,T} where Ob(C) is the class of objects of C and {F,T} is the set of truth values. I say the property is non-evil if it extends to a functor F: C -> {F,T} where now we regard {F,T} as a discrete groupoid. For example, suppose C is the groupoid of vector spaces over your favorite field. A typical non-evil property is "being 5-dimensional". A typical evil property is "having the empty set as its origin". (In ZF set theory, we can take any vector space and ask whether its zero element happens to be the empty set.) I think most mathematicians would be happy to see a theorem that begins Theorem: If V is a vector space that is 5-dimensional... but somewhat surprised to see a theorem that begins: Theorem: If V is a vector space with the empty set as its origin... We instinctively feel that any theorem of the second sort could have been phrased better: how could it really matter that the origin is the empty set? Now, structures. Again, let C be a groupoid. A (possibly evil) structure on objects in C is a map F: Ob(C) -> Set The idea is that for any object c in C, F(c) is the set of structures that can be put on that object. I say the structure is non-evil if it extends to a functor F: C -> Set If C is the groupoid of vector spaces, a typical non-evil structure is a basis: here F(c) is the set of bases of the vector space c. A typical evil structure would be "a basis, if the underlying set of c is the real numbers, but two bases otherwise." Again, I think most mathematicians would feel happy to work with the non-evil structure, but somewhat uncomfortable working with the evil one. That's the feeling that this concept of "evil" is trying to formalize. Note that a non-evil structure on finite sets is what you call a "species". You wisely avoided studying the evil ones. Best, jb [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Hi John, thanks for trying to move the discussion away from terminology and back to actual mathematical matters. In order to focus on the math and not on the terminology, let me today use the word "XXXX" instead of "evil". I don't think the notion used in your examples is general enough. For example, fix some groupoid C, and consider the property of an object x: "x is isomorphic to exactly 3 objects of C". To me, this is clearly XXXX, 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-XXXX. For a property P of objects x of a category C, "being invariant under isomorphisms of objects in C" is strictly weaker than "being invariant under equivalences of C". Proof: Clearly, any isomorphism of C can be mapped to an identity of some category C' by some equivalence of categories. Therefore, any property that is invariant under equivalences of categories is invariant under isomorphisms of objects. The above example shows that the converse is not true. I think for XXXXness of structures, a similar refinement is needed. To me, the intuitive concept of XXXX for structures is "cannot be transported along equivalences such that the equivalence becomes structure preserving". To say it more explicitly: if category C has the structure, and category C' is equivalent to C (as a category), then C' can be equipped with a structure in such a way that the equivalence (both directions) is structure preserving. This is a fairly subtle concept, not least because it depends on the precise 2-category in question (to fix what "equivalence" and "structure preserving" means). For example, whether the structure of being "strictly monoidal" is XXXX or not depends on what one means by "structure preserving" (e.g., strict monoidal or strong monoidal functors). The natural transformations need to be specified too, so that one can define "structure preserving equivalence". I tried to give a more general and precise 2-categorical definition on the categories list on January 3, 2010, but I am not sure I got it quite right. I think it was Mark Weber who also pointed out, around the same time, that one person's XXXX concept is another person's non-XXXX concept - in a different 2-category. The fact that being XXXX depends on an ambient 2-category means that it is not a moral judgment, and people should not be offended by it. Some perfectly useful things can be XXXX sometimes, and some perfectly useless things can be non-XXXX. For example, even a not-very-natural property like "there are exactly 3 objects isomorphic to x" can be non-XXXX when viewed in the right 2-category. For example, this is the case in the 2-category of categories, functors, and identity natural transformations. Last comment. Thomas Streicher brought up the example of a fibration P: XX -> BB as a concept that was XXXX but very useful. But I don't think this concept is actually XXXX. Certainly if one thinks of the fibration as a *structure* on BB, then this transports very nicely along equivalences. Namely, given any equivalence BB <--> BB', one can find a fibration P' : XX' -> BB' which is equivalent, as a fibration, to P. Right? So I don't think it is correct to identify the concept of XXXX with "having to talk about equality". Rather, it should be defined in some 2-categorical way. See also Mike Shulman's post from January 4, which discussed this distinction in more depth. -- Peter John Baez wrote:
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.
... [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (2)
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John Baez -
selinger＠mathstat.dal.ca