Subject: categories: The Category of all Smooth Manifolds a la Lawvere

From: Ellis Cooper <[email-masked]>
Date: Fri, 18 Feb 2000 17:09:51 -0500

Hello, 

In his paper "Qualitative distinctions between some toposes of generalized
graphs" (Contemporary Mathematics Volume 92, 1989, pp. 261-299) Lawvere
mentions a "powerful theorem" that "justifies bypassing the complicated
considerations" usually associated with defining a smooth manifold (charts,
atlases, etc.). This is in the context of something called "closed under
splitting of idempotents" and the kind of idempotent he is talking about, I
think, is what you get if you embed a manifold in a sufficiently
high-dimensional space, wrap it inside and out with foam, call that foamy
thing an open set, and then the idempotent is the projection of the foam
back onto the embedded manifold. What I would like is a carefully written,
fully spelled out statement and proof of his theorem. Please advise.

In fact, I would be even more delighted by a more standard (motivation,
definition, theorem, proof) version of his entire paper, but I suppose that
is too much to ask. 

Please reply directly to me, at

Ellis D. Cooper
Senior Software Engineer
Varian Semiconductor Equipment Associates, Inc.
[email-masked]