Eric Forgy
Smooth Spaces

Chains

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

$$C(U)=\underset{r\ge 0}{⨁}{C}_r(U)$$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 $\varphi :U\to X$, and any chain $c\in C(U)$, the push forward

$${\varphi }_{*}c$$\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.

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

Cochains

Using plots $\varphi :U\to X$ 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.

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

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

$$⟨\alpha ,{\varphi }_{*}c⟩=⟨{\varphi }^{*}\alpha ,c⟩.$$\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.

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

Integration

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

$$c=\sum _i({\varphi }_i{)}_{*}{c}_i$$c = \sum_i (\phi_i)_* c_i

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

$${\int }_c\alpha =\sum _i{\int }_{({\varphi }_i{)}_{*}{c}_i}\alpha =\sum _i{\int }_{c_i}{\varphi }_i^{*}\alpha .$$\int_c \alpha = \sum_i \int_{(\phi_i)_* c_i} \alpha = \sum_i \int_{c_i} \phi_i^* \alpha.

Boundary

$$\partial \circ {\varphi }_{*}={\varphi }_{*}\circ \partial$$\partial\circ \phi_* = \phi_*\circ\partial

Coboundary

$$d\circ {\varphi }^{*}={\varphi }^{*}\circ d$$d\circ \phi^* = \phi^*\circ d

Stokes Theorem

$${\int }_{c}d\alpha =\sum _i{\int }_{c_i}({\varphi }_i{)}^{*}(d\alpha )=\sum _i{\int }_{c_i}d({\varphi }_i^{*}\alpha )=\sum _i{\int }_{({\varphi }_i{)}_{*}(\partial c_i)}\alpha =\sum _i{\int }_{\partial ({\varphi }_i{)}_{*}{c}_i}\alpha ={\int }_{\partial c}\alpha .$$\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:

• Comparative Smootheology (I, II, III)

• Convenient categories of smooth spaces

• Spivak on derived manifolds

• Bär on fiber integration in differential cohomology