### Question on Smooth Functions

#### Posted by Urs Schreiber

There must be some standard textbook reference for the following statement:

The functor from smooth manifolds to algebras which sends each smooth manifold to the algebra of smooth functions on it, and which sends morphisms of smooth manifolds to the algebra homomorphisms obtained by the corresponding pullbacks of functions $(X \stackrel{\phi}{\to} Y) \mapsto ( C^\infty(X) \stackrel{\phi^*}{\leftarrow}C^\infty(Y))$ is

full: every algebra homomorphism between algebras of smooth functions comes from a pullback along a smooth map of the underlying smooth manifolds.

This is of course closely akin to Gel’fand’s equivalence (for instace recalled as theorem 1 in Spaceoids), although a little different.

A proof should consist of two steps:

a) every homomorphism of function algebras comes from pullback along a map of the underlying sets.

b) only pullback along smooth maps will take all smooth functions to smooth functions.

What’s a good reference for this?

Posted at February 11, 2008 8:53 AM UTC
## Re: Question on Smooth Functions

Natural Operations in Differential Geometryhas a simple proof of this statement. It is Corollary 35.10. The preceding material in Section 35 is overkill for this result; 35.9 has an elementary proof in the “short story” in the introduction to Chapter VIII.