nLab
synthetic tangent bundle

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

          In synthetic differential geometry, the tangent bundle of an object XX is the internal hom X 𝔻 1X^{\mathbb{D}^1} out of the 1d first order infinitesimal disk 𝔻 1\mathbb{D}^1, equipped with the projection to XX induced from the unique point *𝔻 1\ast \to \mathbb{D}^1:

          X (*𝔻 1):X (𝔻 1)X. X^{(\ast \to \mathbb{D}^1)} \;\colon\; X^{(\mathbb{D}^1)} \longrightarrow X \,.

          (Here we are using that the internal hom-functor () ()(-)^{(-)} is a contravariant functor in its “exponent” variable.)

          This makes manifest and precise the intuitive idea that a tangent vector on XX is an “infinitesimal curve” in XX, see also the Examples below.

          In this formulation the operation of differentiation is simply the internal hom-functor:

          Given a function:

          XfY. X \overset{f}{\longrightarrow} Y \,.

          its differential is its image under the internal hom () (𝔻 1)(-)^{(\mathbb{D}^1)}:

          X (𝔻 1)AAf (𝔻 1)AAY (𝔻 1). X^{(\mathbb{D}^1)} \overset{ \phantom{AA} f^{(\mathbb{D}^1)} \phantom{AA} }{\longrightarrow} Y^{(\mathbb{D}^1)} \,.

          Examples

          In a standard model for synthetic differential geometry/differential cohesion such as the Cahiers topos H\mathbf{H}, for XSmthMfdyHX \in SmthMfd \overset{y}{\hookrightarrow} \mathbf{H} an ordinary smooth manifold, its synthetic tangent bundle coincides with the traditional tangent bundle TXpTXT X \overset{p}{\to} T X:

          X (𝔻 1) TX X (*𝔻 1) p X X * X \array{ X^{(\mathbb{D}^1)} & \simeq & T X \\ {}^{\mathllap{ X^{ (\ast \to \mathbb{D}^1) } }}\big\downarrow && \big\downarrow{}^{ \mathrlap{p_X} } \\ X^{\ast} & \simeq & X }

          Moreover, if YSmthMfdHY \in SmthMfd \hookrightarrow \mathbf{H} is another smooth manifold, then a morphism

          XAAfAAY X \overset{\phantom{AA} f \phantom{AA} }{\longrightarrow} Y

          is equivalently a smooth function in the traditional sense (i.e. the external Yoneda embedding-functor SmthMfdyHSmthMfd \overset{y}{\hookrightarrow} \mathbf{H} is fully faithful ) and its image under the internal hom is its traditional differentiation dfd f:

          X (𝔻 1) AAf (𝔻 1)AA Y (𝔻 1) TX AAdfAA TY \array{ X^{(\mathbb{D}^1)} &\overset{ \phantom{AA} f^{(\mathbb{D}^1)} \phantom{AA} }{\longrightarrow}& Y^{(\mathbb{D}^1)} \\ {}^\simeq\big\downarrow && \big\downarrow{}^\simeq \\ T X &\overset{ \phantom{AA} d f \phantom{AA} }{\longrightarrow}& T Y }

          This way the evident functoriality of the internal hom () (𝔻 1)(-)^{(\mathbb{D}^1)} is identified with the chain rule of traditional differentiation.

          For more on this see

          Properties

          For Microlinear spaces

          For XX a microlinear space the synthetic tangent bundle shares many of the expected properties of a tangent bundle.

          References

          For lecture notes see

          Last revised on September 14, 2018 at 07:29:39. See the history of this page for a list of all contributions to it.