Contents

# Contents

## Chains

Let $U$ denote an open subset of $\mathbb{R}^n$ and

$C(U) = \bigoplus_{r \ge 0} C_r(U)$

be the space of chains on $U$.

Let $X$ be a smooth space. For any plot $\phi:U\to X$, and any chain $c\in C(U)$, the push forward

$\phi_* c$

is a chain in $X$. The collection of all plots allow us to push forward all such spaces of chains to obtain the space of chains on $X$, i.e.

$C(X) = \bigoplus_{r\ge 0} C_r(X).$

## Cochains

Using plots $\phi:U\to X$ on chains on the domains of plots $C(U)$ gives chains $C(X)$ on $X$. Once we have chains on $X$, we can define cochains on $X$ to be the formal duals of chains, i.e.

$C^*(X) = \bigoplus_{r\ge 0} C^*_r(X).$

For example, if $\alpha\in C^*(X)$ we can define its value on $\phi_* c\in C(X)$ via

$\langle \alpha,\phi_* c\rangle = \langle \phi^*\alpha, c\rangle.$

## Differential Forms

A differential form $\alpha\in\Omega^p(X)$ on $X$ is a cochain such that its pull back via any plot is a differential form on the domain of that plot, i.e.

$\phi^*\alpha\in\Omega^r(U)\forall\phi:U\to X\implies\alpha\in\Omega^r(X).$

## Integration

Let $\phi_i:U_i\to X$ denumerate a collection of plots such that any chain $c\in C(X)$ can be expressed as

$c = \sum_i (\phi_i)_* c_i$

for some chain $c_i\in C(U_i)$. Then we can define integration on $X$ via

$\int_c \alpha = \sum_i \int_{(\phi_i)_* c_i} \alpha = \sum_i \int_{c_i} \phi_i^* \alpha.$

## Boundary

$\partial\circ \phi_* = \phi_*\circ\partial$

## Coboundary

$d\circ \phi^* = \phi^*\circ d$

## Stokes Theorem

$\int_c d\alpha = \sum_i \int_{c_i} (\phi_i)^* (d\alpha) = \sum_i \int_{c_i} d(\phi_i^*\alpha) = \sum_i \int_{(\phi_i)_*(\partial c_i)} \alpha = \sum_i \int_{\partial (\phi_i)_* c_i} \alpha = \int_{\partial c}\alpha.$

## Literature

Eventually the following will be a commented list – promised.

• John Baez and Alexander Hoffnung, Convenient Categories of Smooth Spaces (arXiv, blog)

• Patrick Iglesias-Zemmour, Diffeology (pdf)

• Matthias Kreck, Stratifolds and differential algebraic topology (pdf)

• William Lawvere, Taking categories seriously (pdf)

• David Spivak, Quasi-smooth derived manifolds (pdf)

• Andrew Stacey, Comparative Smootheology (arXiv)

• Martin Laubinger, Differential Geometry in Cartesian Closed Categories of Smooth Spaces (pdf)

• Alexander Hoffnung, Smooth spaces: convenient categories for differential geometry, (pdf slides)

• Alexander Hoffnung, From Smooth Spaces to Smooth Categories, (pdf slides)

There are also Hofer’s polyfolds.

Concerning smooth ∞-stacks there is useful material in

• Daniel Dugger, Sheaves and homotopy theory (web, pdf)

## Remarks

Dual to generalized smooth spaces are generalized smooth algebras of functions on them, according to the general lore of space and quantity.

## Further discussion

We had extensive discussion of generalized smooth spaces at the $n$-Café:

Revised on May 2, 2010 at 09:56:02 by Eric Forgy