# nLab synthetic tangent bundle

Contents

### Context

#### Synthetic differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

• ( $\dashv$ $\dashv$ )

$(ʃ \dashv \flat \dashv \sharp )$

• , ,

• $\dashv$

$ʃ_{dR} \dashv \flat_{dR}$

• ( $\dashv$ $\dashv$ )

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

• $\dashv$ $\dashv$

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

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

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 $X$ 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 $T X \overset{p}{\to} T X$:

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

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

## Properties

### For Microlinear spaces

For $X$ 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.