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