🚀 go-pugleaf

Vaughan Pratt writes:

>The only arbitrariness in my definition is the choice of neighborhood
>of the manifold and its retraction onto the manifold.  No chopping, no
>charts.
>
>Furthermore my definition is a *lot* simpler, I'd say, even without
>going into the notion of smooth coordination required to define an
>atlas.
>
>There has to be some other reason why the complicated notion of atlas
>is essential.

Well, yes, and I just explained a couple.  I should note that usually
one takes a maximal atlas, which eliminates any arbitrariness in the
atlas.  But to be general about it: it's just bad taste to define a
certain kind of mathematical object as being "a subobject of X with
properties..." when it frequently, indeed typically, arises in contexts
where it is NOT given as a subobject of X.  

For example: "a group is a subset of the set of permutations of a set
that is closed under taking inverses and products".  Yuck!

These things should be theorems, not definitions.