Re: Not invariant but good

[John Baez <baez@math.ucr.edu>]

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/ ]