differential function complex in nLab

Context

Differential cohomology

• Idea
• Definitions
• Cocycles with values in graded vector spaces
• Differential functions
• Differential function complexes
• Differential $E$-cohomology
• Properties
• Homotopy groups
• Relation to differential cohomology in cohesive $(\infty,1)$-toposes
• Examples
• Line bundles with connection
• Differential K-cocycles
• Higher filtration degree
• References

Definition

For

• $E$ a topological space;

• $\iota \in Z^n(E,\mathbb{R})$ a cocycle on $E$ for real-valued singular cohomology on $E$,

a differential function on a smooth manifold $U$ with values in $(E,\iota)$ is a triple $(c,h,\omega)$ with

• $c : U \to E$ a continuous map;

• $\omega \in \Omega^n(S)$ a smooth differential form on $S$;

• $h \in C^{n-1}(U,\mathbb{R})$ a cochain in real cohomology on (the topological space underlying) $U$;

such that in the abelian group $Z^n(S,\mathbb{R})$ of singular cochains the equation

$$\omega = c^*\iota + \delta h$$

holds, where

• $\omega$ is here regarded as a singular cochain (that sends a chain to the integral of $\omega$ over it, as discussed at de Rham theorem),

• $\delta$ denotes the coboundary operator,(the Moore complex differential of the singular simplicial complex).

Definition

For

• $E$ be a topological space and

• $\iota \in Z^n(E,\mathbb{R})$ a cocycle on $E$ for real-valued singular cohomology on $X$,

• $U$ a smooth manifold,

the differential function complex

$$(E,\iota)^U$$

of all differential functions $S \to (X,\iota)$ is the simplicial set whose $k$-simplices are differential functions, def,

$$U \times \Delta^k_{Top} \to (E,\iota) \,.$$

Definition

For $s \in \mathbb{N}$ write

• $filt_s \Omega^\bullet(U \times \Delta^k)$

  for the sub-simplicial set of differential forms that vanish when evaluated on more than $s$ vector fields tangent to the simplex;

• $filt_s (X,\iota)^S \subset (X,\iota)^S$

  for the sub-simplicial set of those differential functions whose differential form component is in $filt_s \Omega^\bullet(U \times \Delta^k)$.

Proposition

The complex $filt_s (E,\iota)^U$ is (up to equivalence, of course) the homotopy pullback

$$\array{ filt_s (E,\iota)^U &\to& filt_s \Omega^n_{cl}(U \times \Delta^\bullet, \mathcal{V}) \\ \downarrow && \downarrow \\ Sing E^U &\to& Z^\bullet(U \times \Delta^\bullet, \mathcal{V}) }$$

in sSet (regarded as equipped with its standard model structure on simplicial sets).

Proposition

We have generally

$$\pi_k Z(S \times \Delta^\bullet_{Diff}, \mathcal{V}) = H^{n-m}(S; \mathcal{V})$$

(for instance by the Dold-Kan correspondence).

The simplicial homotopy groups of $filt_s \Omega^n_{cl}(S \times \Delta^\bullet_{Diff})$ are

$$\pi_k filt_s \Omega^n_{cl}(S \times \Delta^\bullet_{Diff}) = \left\{ \array{ H_{dR}^{n-k}(S, \mathcal{V}) & | k \lt s \\ \Omega_{cl}(S; \mathcal{V})^{n-s} & | k = s \\ 0 & | k \gt s } \right\} \,.$$

This implies isomorphisms

$$\pi_k filt_s(X; \iota)^S \stackrel{\simeq}{\to} \left\{ \array{ \pi_k X^S & | k \lt s \\ H^{n-k-1}(S; \mathcal{V})/ \pi_{k+1} X^S | k \gt s } \right. \,.$$

Proposition

For $S$ a smooth manifold, and $s \in \mathbb{N}$, the sequence of differential function complexes, def. ,

$$filt_{s + n}(E_n; \iota_n)^S \stackrel{\simeq}{\to} \Omega filt_{s + (n + 1)}(E_{n+1}; \iota_{n+1})^S$$

forms an Omega-spectrum.

This is the differential function spectrum for $E$, $S$, $s$.

Definition

The differential $E$-cohomology group of the smooth manifold $S$ in degree $n$ is

$$H_{diff}^n(S,E) := \pi_0 filt_0(E_n \iota_n)^S$$

Proposition

For $E$ an Omega-spectrum, $S$ a smooth manifold, we have for all $s,n \in \mathbb{N}$, a weak homotopy equivalence

$$\Omega filt_{s+1}(E_{n}; \iota_{n})^S \stackrel{\simeq}{\to} filt_{s}(E_{n-1}; \iota_{n-1})^S \,,$$

identifying the loop space object (at the canonical base point) of the differential function complex of $E_{n}$ at filtration level $s+1$ with that differential function complex of $E_{n-1}$ at filtration level $s$.

Proposition

For $E_\bullet$ a spectrum as above,
we have an (∞,1)-pullback square

$$\array{ filt_0 (E_n; \iota_n)^{(-)} &\to& \prod_i \Omega^{n_i}_{cl}(-) \\ \downarrow && \downarrow \\ Disc E_n & \stackrel{}{\to} & \prod_i \mathbf{B}^{n_i} \mathbb{R}_{disc} } \,.$$

Proof

By prop. we have that

• $filt_\infty (E; \iota_n)^S \simeq Sing X^S$;

• $filt_0 \Omega_{cl}(S \times \Delta^\bullet) \simeq \Omega_{cl}(S)$.

The statement then follows with the pasting law for homotopy pullbacks

$$\array{ filt_0 (E_n; \iota_n)^S &\to& \Omega^n_{cl}(S; \mathcal{V}) \\ \downarrow && \downarrow \\ filt_\infty (E_n; \iota_n)^S &\to& filt_\infty \Omega_{cl}(S \times \Delta^\bullet; \mathcal{V}) \\ \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} \\ Sing X^S &\to& Z(S \times \Delta^\bullet; \mathcal{V}) } \,.$$

(…)