Short question:


Let $M$ and $N$ be smooth manifold, with appropriate smooth function algebras
$C^\infty(M,\mathbb{R})$ and $C^\infty(N,\mathbb{R})$. 


Can we express the smooth function algebra of the cartesian product manifold
in terms of $C^\infty(M,\mathbb{R})$ and $C^\infty(N,\mathbb{R})$? 


I know it is neither (equivalent to) 
$C^\infty(M,\mathbb{R}) \oplus C^\infty(N,\mathbb{R})$ nor
$C^\infty(M,\mathbb{R})\otimes\_{\mathbb{R}}C^\infty(N,\mathbb{R})$.


...


A more general question is,if there is a general rule to get from categorical
constructions on manifolds to constructions on the appropriate smooth function algebra.


Maybe this boils don to the question whether or not the functor $C^\infty(\cdot,\mathbb{R})$
from smooth manifolds to ass. comm. unitary $\mathbb{R}$-algebras preserves (co)limits.


That's indeed the deeper question.


...


P.S.: I tagged it in particular as algebraic-geometry related, do to the category theory
related part...



Smooth function algebra on cartesian product and beyond 
If $M$ and $N$ are compact, then $C^\infty(M\times N)=C^\infty(M)\bar\otimes\_{i}C^\infty(N)$, the completed injective tensor product which coincides with the completed projective tensor product, since the locally convex spaces involved are nuclear.


Edit: This also holds for for non-compact $M,N$; see [Treves: Topological Vector Spaces, Distributions, and Kernels, Page 530]. 


If the manifolds are not compact, the same holds for the space of smooth functions with compact support. 


