Local C-infinity-ringed space

Definition

A local $C^\infty$ -ringed space is a pair $(X,\mathcal O_X)$ in which $X$ is a topological space and $\mathcal O_X$ is a sheaf of $C^\infty$ -rings whose stalks are local rings after forgetting the extra smooth operations. A morphism

(f,f^\#):(X,\mathcal O_X)\longrightarrow(Y,\mathcal O_Y)

consists of a continuous map $f:X\to Y$ and a morphism

f^\#:\mathcal O_Y\longrightarrow f_*\mathcal O_X

of sheaves of $C^\infty$ -rings that induces local homomorphisms on stalks.

Forgetting the operations $\Phi_g$ but retaining addition and multiplication produces an ordinary locally ringed space. The converse is false: a sheaf of real algebras need not carry, or determine, operations for every smooth $g:\mathbb R^n\to\mathbb R$ .

Smooth manifolds as structured spaces

Every smooth manifold $M$ determines

(M,C^\infty_M).

Here $C^\infty_M$ is its sheaf of smooth functions. Each point has a neighborhood for which this pair is isomorphic to $(U,C^\infty_U)$ for an open subset $U\subseteq\mathbb R^n$ . Conversely, a Hausdorff second-countable locally $C^\infty$ -ringed space locally modeled in this way recovers a smooth manifold. Morphisms between these manifold models correspond exactly to smooth maps.

Broader smooth spaces

General locally $C^\infty$ -ringed spaces need not be manifolds. Allowing quotient $C^\infty$ -rings and more general local models admits singular and infinitesimal smooth spaces. Accordingly, “locally $C^\infty$ -ringed” is the ambient structured-space notion; “locally modeled on open subsets of Euclidean space” is the additional manifold condition.

References

Dominic Joyce, “Algebraic Geometry over $C^\infty$ -rings,” Memoirs of the AMS 260 (2019). arXiv version. Relevant: §§2–3, local $C^\infty$ -rings and $C^\infty$ -ringed spaces.
Ieke Moerdijk and Gonzalo E. Reyes, Models for Smooth Infinitesimal Analysis, Springer, 1991. DOI record. Relevant: Chapter I, smooth algebras and loci.