
To: David Rusin <rusin@mp.cs.niu.edu>
Subject: Re: What's a manifold?
Date: 05 Nov 92 11:08:15 PST (Thu)
From: pratt@cs.stanford.edu

	Good point, but there is a way 'round this too: Define a manifold to
	be a pair (M,F) where F is a MAXIMAL atlas

So why can't I do the same?  Take the minimal n for which M embeds in
R^n, delete all points p of R^n for which there is no unique nearest
point of M to p, and retract each point of the remaining set,
necessarily open, to the nearest point of M.  (One could even perform
this retraction homotopically, but this is an unnecessary complication
here.)  Why won't that work?

	(PS: I don't find this to be the most helpful way to view a manifold in
	some contexts, but it does have that air of elegance that tells you
	REAL MATHEMATICIANS probably work this way :-) )

Was it more or less elegant than defining a manifold as a smooth
retract of an open subset of R^n?  That's the *complete* definition,
not a hand-waving approximation.  Moreover it has the advantage that
you can immediately define a TD-bundle for any category D in which
idempotents split, starting with D = Set or Vct_K, but many common
categories have that property.  You define it as *any* functor T:C->D
where C is the category of open subsets of R^n for any n and their
smooth maps.  You then use that idempotents split in D to extend T to a
functor from the category of manifolds to D.  Now *that's* what I call
elegant.

-v

