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

  • (shape modality \dashv flat modality \dashv sharp modality)

    (ʃ)(ʃ \dashv \flat \dashv \sharp )

  • dR-shape modality\dashv dR-flat modality

    ʃ dR dRʃ_{dR} \dashv \flat_{dR}

  • tangent cohesion

    • differential cohomology diagram
    • differential cohesion

      • (reduction modality \dashv infinitesimal shape modality \dashv infinitesimal flat modality)

        (&)(\Re \dashv \Im \dashv \&)

      • graded differential cohesion

        • fermionic modality\dashv bosonic modality \dashv rheonomy modality

          (Rh)(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)

        • 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& 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 }

          </semantics></math></div>

          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.

          Last revised on August 29, 2016 at 04:05:43. See the history of this page for a list of all contributions to it.