# nLab vertical differential form

Contents

### Context

#### 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

cohesion

tangent cohesion

differential cohesion

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

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## Definition

### In differential geometry

Let $\pi : P \to X$ be a bundle in the category Diff of smooth manifolds.

The dg-algebra $\Omega^\bullet_{vert}(P)$ of vertical differential forms on $P$ is the quotient of the de Rham complex dg-algebra $\Omega^\bullet(P)$ of all forms on $P$, by the dg-ideal of horizontal differential forms, hence of all those forms that vanish when any one vector in their arguments is a vertical vector field in that it is in the kernel of the differential $d \pi : T P \to T X$.

For a trivial bundle $P = X \times F$ the underlying complex of $\Omega^\bullet_{vert}(P)$ is $\wedge^\bullet_{C^\infty(X \times F)} \Gamma(T^* F)$.

## References

Last revised on July 23, 2018 at 14:50:13. See the history of this page for a list of all contributions to it.