# nLab smooth locus

Contents

### Context

#### Synthetic differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## 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

See the references at C-infinity-ring.