nLab synthetic tangent bundle



Synthetic differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

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



smooth space


The magic algebraic facts




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 }


Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)



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)} \,.


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 TXpXT X \overset{p}{\to} 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


For Microlinear spaces

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


For lecture notes see

Last revised on October 29, 2021 at 03:24:47. See the history of this page for a list of all contributions to it.