nLab
smooth locus

Context

Synthetic differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive Unknown characterUnknown character discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \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 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 ( n)/JC^\infty(\mathbb{R}^n)/J, for JJ an ideal of the ordinary underlying algebra.

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

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

Notation

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

Often one also write

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

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

Properties

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

Proposition

The canonical inclusion functor

SmthMfd𝕃 SmthMfd \hookrightarrow \mathbb{L}
X𝓁C (X) 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.

Proposition

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

Applications

There are various Grothendieck topologies on 𝕃\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

References

See the references at C-infinity-ring.

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