smooth locus

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 $\mathbb{R}$.

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**.

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 $\mathbb{L}$.

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

The canonical inclusion functor

$SmthMfd \hookrightarrow \mathbb{L}$

$X \mapsto \mathcal{l}C^\infty(X)$

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.

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

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

See the references at C-infinity-ring.

Revised on August 29, 2016 04:05:43
by Urs Schreiber
(89.15.236.152)