Peter wrote:
In order to focus on the math and not on the terminology, let me today use the word "XXXX" instead of "evil".
Good. I think you're secretly on my side, though, because you're using four X's. :-)
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".
What a fiendishly clever example!
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.
True. By the way, in case anyone out there forgets: "Extending to a functor C -> {F,T}" was merely a pedantic way of saying "being a property of objects of C that is invariant under isomorphisms" - a pedantic way that lets us easily generalize this notion to "being a structure on objects of C that is covariant under isomorphisms". To generalize, we just replace {F,T} by Set. It may not be clear to everybody why I like this pedantic approach, so I should probably explain why. A (-1)-category is a truth value, a 0-category is a set, and a 1-category is a category. So, when we replace {F,T} by Set, we are replacing the 0-category of (-1)-categories by the 1-category of 0-categories. Since we are just increasing a certain parameter by 1, it becomes easy to see how to continue this game indefinitely. For more details, try this: http://arxiv.org/PS_cache/math/pdf/0608/0608420v2.pdf#page=12 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".
Yes. Here's my rejoinder: I had been fixing a groupoid C and asking whether a property of objects of that category was invariant under isomorphisms. When you say "x is isomorphic to exactly 3 objects of C", you are actually treating C not as fixed but as variable. The more things we let vary, the more invariance properties we can demand! In particular, there's a 2-groupoid Cat_* where an object is a "pointed category" (C,x), that is, a category C with chosen object x. I can treat "being an object x that is isomorphic to exactly 3 objects in C" as a property of pointed categories. And, I would call this property evil... whoops, I mean XXXX... because it determines a function Ob(Cat_*) -> {F,T} that does not extend to a 2-functor Cat_* -> {F,T} Again, this is just a pedantic way of saying what you're saying. I'm just trying to point out that I can fit it into my philosophy.
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 remember enjoying that post, but I'll need to reread it to remember what you said.
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.
Very much so! And you've also noticed here that sometimes a property of objects in a fixed category arises from a property of pointed categories.... so that we can take either a 1-categorical or a 2-categorical approach to the XXXXness of this property.
So I don't think it is correct to identify the concept of XXXX with "having to talk about equality".
I agree. I take it as a *rule of thumb* that when somebody writes down a property of categories that involves equations between objects, they're running the risk that this property is not invariant under equivalence of categories. But I don't know the general theorems that make this rule of thumb precise. Can every property of categories that is invariant under equivalence be expressed in some language that doesn't include equations between objects? Or conversely? Or what precise conditions are needed to get theorems along these lines? Best, jb [For admin and other information see: http://www.mta.ca/~cat-dist/ ]