nLab smooth locus

Idea

A smooth locus is the formal dual of a finitely generated smooth algebra (or $C^\infty$-ring):

a space that behaves as if its algebra of functions is a finitely generated smooth algebra.

Given that a smooth algebra is a smooth refinement of an ordinary ring with a morphism from $\mathbb{R}$, a smooth locus is the analog in well-adapted models for synthetic differential geometry for what in algebraic geometry is an affine variety over $\mathbb{R}$.

Definition

A finitely generated smooth algebra is one of the form $C^\infty(\mathbb{R}^n)/J$, for $J$ an ideal of the ordinary underlying algebra.

Write $C^\infty Ring^{fin}$ for the category of finitely generated smooth algebras.

Then the opposite category $\mathbb{L} := (C^\infty Ring^{fin})^{op}$ is the category of smooth loci.

Notation

For $A \in C^\infty Ring^{fin}$ one write $\ell A$ for the corresponding object in $\mathbb{L}$.

Often one also write

$$R := \ell C^\infty(\mathbb{R})$$

for the real line regarded as an object of $\mathbb{L}$.

Properties

The category $\mathbb{L}$ has the following properties:

Proposition

The canonical inclusion functor

$$SmthMfd \hookrightarrow \mathbb{L}$$

$$X \mapsto \ell C^\infty(X)$$

from the category SmthMfd of smooth manifolds is a full subcategory embedding (i.e. a full and faithful functor. Moreover, it preserves pullbacks along transversal maps.

Proposition

The Tietze extension theorem holds in $\mathbb{L}$: $R$-valued functions on closed subobjects in $\mathbb{L}$ have an extension.

Applications

There are various Grothendieck topologies on $\mathbb{L}$ and various of its subcategories, such that categories of sheaves on these are smooth toposes that are well-adapted models for synthetic differential geometry.

For more on this see

• Models for Smooth Infinitesimal Analysis

 Related concepts

• smooth algebra

References

See the references at C-infinity-ring.