nLab
synthetic differential topology

Contents

Context

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 }

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          Models

          Lie theory, ∞-Lie theory

          differential equations, variational calculus

          Chern-Weil theory, ∞-Chern-Weil theory

          Cartan geometry (super, higher)

          Contents

          Idea

          Synthetic differential topology (SDT) is a synthetic axiomatization of differential topology in analogy to how synthetic differential geometry (SDG) is synthetic axiomatization of differential geometry.

          Where in SDG the concept of infinitesimal neighbourhoods is encoded by the axioms (the Kock-Lawvere axiom) in SDT it is germs of spaces that are being encoded by the axioms.

          Notice that the germ of a manifold around a point is in general “larger” than the formal neighbourhood of that point, reflecting, dually, the fact that there are smooth functions which are non-vanishing in every open neighbourhood of that point but all whose partial derivatives vanish at that point (see also at bump function).

          SDT uses the representability of germs by an object defined intrinsically as the points not well-separated from 0R n0 \in R^n, ¬¬{0}\neg \neg \{0\}.

          Examples of sequences of local structures

          geometrypointfirst order infinitesimal\subsetformal = arbitrary order infinitesimal\subsetlocal = stalkwise\subsetfinite
          \leftarrow differentiationintegration \to
          smooth functionsderivativeTaylor seriesgermsmooth function
          curve (path)tangent vectorjetgerm of curvecurve
          smooth spaceinfinitesimal neighbourhoodformal neighbourhoodgerm of a spaceopen neighbourhood
          function algebrasquare-0 ring extensionnilpotent ring extension/formal completionring extension
          arithmetic geometry𝔽 p\mathbb{F}_p finite field p\mathbb{Z}_p p-adic integers (p)\mathbb{Z}_{(p)} localization at (p)\mathbb{Z} integers
          Lie theoryLie algebraformal grouplocal Lie groupLie group
          symplectic geometryPoisson manifoldformal deformation quantizationlocal strict deformation quantizationstrict deformation quantization

          References

          Last revised on August 28, 2018 at 05:47:30. See the history of this page for a list of all contributions to it.