Re: Intuitionism's Limits

[categories <cat-dist@mta.ca>]

Date: Sun, 2 Mar 1997 15:27:28 -0500 (EST)
From: John Baez <baez@phys.psu.edu>

William James <wjames@arts.adelaide.edu.au> writes:

> I suppose the answer is that monicity is relative to a category,
> but what supports this as a claim? 

It seems to me that category theory takes the sensible viewpoint that
mathematical entities (e.g. objects and morphisms) only become
interesting through their relationship with other entities.  Every
arrow looks just like every other arrow if we consider it in
isolation.  Every arrow in a category C is an image of the what James
Dolan calls the "walking arrow" --- the nonidentity morphism in the
free category C0 on a single morphism --- under some functor F: C0 ->
C.  Studying an arrow in isolation is just like studying the walking
arrow, which is completely dull.  The fun begins only when we have a
bunch of arrows and start composing them.

This is one reason why I think n-category theory should be useful in
physics problems like quantum gravity, where it only makes sense to
speak of where or when an event occurs relative to other events, not
with respect to some spacetime manifold of fixed geometry.  For
some of the technical apsects of how this might go, see:

John Baez and James Dolan, Higher-dimensional algebra and topological
quantum field theory, Jour. Math. Phys. 36 (1995), 6073-6105.

Louis Crane, Clock and category: is quantum gravity algebraic?,
J. Math. Phys. 36 (1995), 6180-6195.  

These both appeared in a special issue on diffeomorphism-invariant 
physics.