nLab smooth locus

Contents

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

infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \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}{}& \esh &\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}
XC (X) 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 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.

Last revised on July 7, 2023 at 19:54:31. See the history of this page for a list of all contributions to it.