Thanks. I have added pointer also to Shanks’s announcement.

Also, I adjusted the wording of the attribution paragraph (here).

]]>Yes. There is also this announcement by Pursell’s PhD advisor M. E. Shanks: https://www.ams.org/journals/bull/1951-57-04/S0002-9904-1951-09521-X/S0002-9904-1951-09521-X.pdf, see page 295.

They never published any of this, except that they have a similar paper about Lie algebras: https://www.ams.org/journals/bull/1951-57-04/S0002-9904-1951-09521-X/S0002-9904-1951-09521-X.pdf.

]]>Interesting that such an early reference exists.

In trying to check it out on my phone, I only get to see the first 19 pages. Do you mean to say Pursell’s proof covers only diffeos/ring-isos, but not non-invertible maps?

]]>Added:

The case of the category of smooth manifolds and diffeomorphisms is proved in

- Lyle Eugene Pursell,
*Algebraic structures associated with smooth manifolds*, PhD dissertation, Purdue University, 1952. 93 pp. ISBN: 978-1392-88143-9. PDF.

Added redirect: Milnor duality. To satisfy a link at duality between geometry and algebra.

]]>added pointer to Milnor’s original statement

- John Milnor, James Stasheff, Problem 1-C (p. 11) in:
*Characteristic Classes*, Annals of Mathematics Studies, Princeton University Press 1974 (ISBN:9780691081229)

and to these proofs for the case of isomorphisms:

Janusz Grabowski,

*Isomorphisms and ideals of the Lie algebras of vector fields*, Inventiones mathematicae volume 50, pages 13–33 (1978) (doi:10.1007/BF01406466)Jerrold Marsden, Ratiu, Abraham, Theorem 4.2.36 in:

*Manifolds, tensor analysis, and applications*, Springer 2003 (ISBN:978-1-4612-1029-0)Janusz Grabowski,

*Isomorphisms of algebras of smooth functions revisited*, Arch. Math. 85 (2005), 190-196 (arXiv:math/0310295)

added doi to

- Ivan Kolar, Jan Slovák, Peter Michor,
*Natural operations in differential geometry*, Springer (1993) (book webpage, doi:10.1007/978-3-662-02950-3)

and

- Jet Nestruev,
*Smooth manifolds and Observables*, Graduate Texts in Mathematics 218, Springer 2003 (doi:10.1007/b98871)

Ah, I was just wondering what sort of things break for uncountable disjoint unions of second countable manifolds.

EDIT: I was thinking continuum-many summands, which is still better behaved than for general uncountable coproducts.

]]>Thanks. I have made the standard regularity assumptions explicit in the entry now and added pointer to this MO discussioon.

]]>Such a statement is true for manifolds with finitely many or countably many connected components.

As Theo Johnson-Freyd once pointed out to me, and later expanded in this answer on MO: http://mathoverflow.net/a/91445 it is false for arbitrary paracompact Hausdorff manifolds, in particular, given two uncountable (discrete) sets S and T one can find a morphism of real algebras C^∞(T)→C^∞(S) that is not induced by a function S→T.

However, the construction is a very subtle set-theoretical argument that uses measurable cardinals.

]]>I finally gave this statement its own entry, in order to be able to conveniently point to it:

*embedding of smooth manifolds into formal duals of R-algebras*