nLab synthetic tangent bundle

Idea

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

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

(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 $X$ is an “infinitesimal curve” in $X$, see also the Examples below.

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

Given a function:

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

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

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

Examples

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

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

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

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

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

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

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

For more on this see

• at Cahiers topos – Synthetic tangent spaces.

• at geometry of physics – supergeometry the section Super mapping spaces

Properties

For Microlinear spaces

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

Related concepts

• odd tangent bundle (analog in supergeometry)

• kinematic tangent bundle, operational tangent bundle

• jet bundle via jet comonad

References

For lecture notes see

• geometry of physics – supergeometry the section Super mapping spaces