🚀 go-pugleaf

In article <1992Nov7.022034.26120@CSD-NewsHost.Stanford.EDU> pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt) writes:
>In article <1992Nov5.174751.2086@galois.mit.edu> tycchow@riesz.mit.edu (Timothy Y. Chow) writes:
>>In fact, I think that despite appearances your retract definition of
>>manifold actually contains topological axioms implicitly.  Don't you
>>really want to say that a manifold is something *homeomorphic* to a
>>smooth retract of an open subset of R^n?
>
>(Homeomorphic is for topological spaces, did you mean diffeomorphic?)
>What does "something diffeomorphic to" mean?  Some *what* thing?  How
>is diffeomorphism defined for objects not yet already covered under my
>definition of manifold?
>
>I'm still contemplating the issue of noncanonicality of the retract
>(tubular neighborhood) definition.  Opinions are running strongly
>against.  None of the arguments have been "knock-down" arguments for
>me, but all the same I'm starting to get the picture.

I think that in the long run one of the better ways to treat this issue is
via ringed spaces.  This is not for the person just starting to learn about
manifolds, but for the person who knows about topological manifolds, smooth
manifolds, piecewise linear manifolds, complex manifolds, almost complex
manifolds, algebraic varieties, and so on.  All these kinds of spaces can
be regarded as "spaces with structure" in the following way: they are
all equipped with a sheaf of rings, namely the sheaf of "nice" functions 
appropriate to the sort of space at hand: continuous, smooth, piecewise linear,
holomorphic, almost holomorphic [that's what they *should* call but I don't
think they do], and algebraic functions, respectively.  

So we may define a topological manifold as a (locally compact Hausdorff)
space such that each point has a neighborhood such that the sheaf of
continuous real-valued functions on that neighborhood is isomorphic to
the sheaf of continuous real-valued functions on R^n.  We may define a
smooth manifold as a topological manifold with a dense subsheaf of its
sheaf of continuous functions -- which we call the sheaf of "smooth" 
functions -- such that each point has a neighborhood such that the sheaf
of smooth functions on that neighborhood is isomorphic to the sheaf
of smooth functions on R^n.  And so on for the other types of spaces.

This is, of course, how algebraic geometers do it these days.  You can
learn about ringed spaces and much more in the chapter on "schemes" in 
Hartshorne's Algebraic Geometry.  (Algebraic geometers have lots of sneaky
schemes.  A "scheme" is a specially nice sort of ringed space.)

Note that this avoids the use of atlases, although it is equivalent to
the atlas approach, or even that irritating tubular neighborhood approach
in some cases!  (Now try to to show that every algebraic variety is
the retract of a tubular neighborhood in C^n!)