Full faithfulness of the smooth-function functor

Statement

For finite-dimensional Hausdorff second-countable smooth manifolds $M,N$ , pullback gives a natural bijection

\operatorname{Hom}_{\mathbf{Man}}(M,N) \;\cong\; \operatorname{Hom}_{C^\infty\mathbf{Ring}} \bigl(C^\infty(N),C^\infty(M)\bigr), \qquad F\longmapsto F^*.

Thus the assignment

C^\infty(-): \mathbf{Man}^{\mathrm{op}} \longrightarrow C^\infty\mathbf{Ring}

is a covariant fully faithful functor from the opposite category of the smooth-manifold category. Equivalently, $M\mapsto C^\infty(M)$ is a fully faithful contravariant functor on $\mathbf{Man}$ . One should not call the displayed functor on $\mathbf{Man}^{\mathrm{op}}$ itself contravariant.

Why a homomorphism determines a map

For each $x\in M$ , composing a $C^\infty$ -ring morphism $\varphi:C^\infty(N)\to C^\infty(M)$ with evaluation at $x$ gives a character $C^\infty(N)\to\mathbb R$ . Such characters are evaluations at unique points of $N$ . Writing that point as $F(x)$ defines a set map $F:M\to N$ . In local coordinates, the coordinate functions pull back to smooth functions, which proves that $F$ is smooth and $\varphi=F^*$ .

Essential image and limitations

The theorem is full faithfulness, not an equivalence with all $C^\infty$ -rings. Its essential image consists precisely of those $C^\infty$ -rings isomorphic to $C^\infty(M)$ for manifolds $M$ . Quotients with singular or infinitesimal behavior generally lie outside that image.

References

Jet Nestruev, Smooth Manifolds and Observables, Springer, 2003. DOI record. Relevant: “Algebras and Points,” “Smooth Manifolds,” and “Smooth Maps.”
Ieke Moerdijk and Gonzalo E. Reyes, Models for Smooth Infinitesimal Analysis, Springer, 1991. DOI record. Relevant: Chapter I, $C^\infty$ -rings and smooth loci.