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. ellis.cooper@vsea.com