Re: terminology in definitions of limits

[John Baez <john.c.baez@gmail.com>]

Dear Categorists -

On Tue, Jan 20, 2009 at 11:34 PM, Vaughan Pratt <pratt@cs.stanford.edu>wrote:


> Colin McLarty wrote:
>
>> I often call them "test objects" in talking with students (by analogy with
>> "test particles" in General Relativity).  I don't think I have ever done it
>> in print.
>
>

> From a game-theoretic standpoint one can be either taking the test or
> administering it.   [..]  What you're calling a "test" object there is for
> me merely the variable being universally quantified over in the definition
> of "all."


 When I teach limits I call Colin's "test object" a "competitor" to the true
limit, or "pretender to the throne", and describe the universal property as
saying "whatever you can do, I can do better".

This game-theoretic approach to universal properties becomes more
interesting when dealing with n-categorical weak limits: the two players
take turns making moves.  First the proponent picks a cone, then the
challenger picks a cone, then the proponent picks a map between cones, then
the challenger picks a map between cones, then the proponent picks a map
between maps between cones, etc..

This idea is important in opetopic n-categories, and there's also an
omega-categorical version - a nice discussion appears starting at the bottom
of page 32 of this paper by Makkai:

http://www.math.mcgill.ca/makkai/equivalence/equivinpdf/equivalence.pdf

"The Hero has to answer each move of the Challenger [...] If Hero can keep
it up forever, he wins; otherwise he loses."

Best,
jb