# nLab differential calculus

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

## Idea

Differential calculus is one of the two halves of the infinitesimal calculus, the other being integral calculus. (The two are linked by the fundamental theorem of calculus.) The differential calculus was developed in the 18th century by Isaac Newton and Gottfried Leibniz (acting independently).

In modern terms, the original differential calculus describes the behaviour of differentiation of functions on the real line. From here, we move to the study of differential equations and then to differential geometry (de Rham complex) and eventually generalize to a wide class of analogous situations (see for instance synthetic differential geometry and D-modules).

Differential calculus on non-finite dimensional spaces is also known as variational calculus.

In the presence of Lie algebra actions a variant of differential caclulus is Cartan calculus.

Since Hochschild homology and cyclic homology (see there) is closely related to differential forms and de Rham differentials, generalizations and abstractions of Hochschild (co)homology are also used as generalized differential calculi .

## In cohesive homotopy theory

We list how aspects of differential calculus are captured by the axioms of cohesion and differential cohesion.

With the axioms of cohesion one may characterize the line object $\mathbb{A}^1 = \mathbb{R}^1$ (by the fact that its A1-homotopy localization is the localization modality which is exhibited by the given shape modality) and then one may essentially characterize a function

$\mathbf{d} \;\colon\; \mathbb{R} \longrightarrow \mathbf{\Omega}^1_{cl}$

which is the “universal de Rham differential”. For any function $f \;\colon\; X \longrightarrow \mathbb{R}$, the composite

$\mathbf{d}f \;\colon\; X \stackrel{f}{\longrightarrow} \mathbb{R} \stackrel{\mathbf{d}}{\longrightarrow} \mathbf{\Omega}^1_{cl}$

is interpreted as the de Rham differential of $f$. This is discussed in some detail at geometry of physics in the section 4. Differentiation.

A slight variant of this construction, using pullbacks of flat modal types yields variational calculus. See there the section In terms of smooth spaces.

With the stronger axioms of differential cohesion one may similarly ask for the infinitesimal interval $D \hookrightarrow \mathbb{R}$. With this one can proceeed as in synthetic differential geometry and formulate for instance differential equations synthetically, as discussed there in the section In terms of synthetic differential geometry.

Moreover, differential cohesion induces de Rham spaces and hence the geometry in the slice over them, dependent types over the infinitesimal shape modality. This is D-geometry which is a general way of talking about differential equations.

Discussion of differential calculus in terms of coinduction is in