Schreiber
nonabelian Chern character

under construction

Idea

The familiar Chern character is a canonical cocycle in real cohomology defined on any spectrum $A$. Abelian differential cohomology is essentially defined in terms of the homotopy fibers of the induced morphism in cohomology.

This definition generalizes to the theory of differential nonabelian cohomology if a notion of Chern character is available for $E$ any smooth ∞-stack, i.e. any ∞-Lie groupoid.

Definition

Let $H=(\mathrm{SPSh}(C{)}_{\mathrm{proj}}^{\mathrm{loc}}{)}^{\circ }$ be a smooth (∞,1)-topos with line object $R$ over the ring object $Z$.

For $A\in \mathrm{SPSh}(C)$ cofibrant suppose there is a universal cocycle

$$c:A\to \prod _i{B}^{n_i}R//Z$$c : A \to \prod_i \mathbf{B}^{n_i} R//Z

that encodes the integral cohomology of $A$.

this may be resolved to a morphism in $[I,\mathrm{SPSh}(C{)}_{\mathrm{proj}}^{\mathrm{loc}}]$

$$\left[\begin{array}{c}A\\ \downarrow \\ \mathrm{cone}(A)\end{array}\right]\to \prod _i\left[\begin{array}{c}{B}^{n_i}\\ \downarrow \\ \mathrm{cone}(B^{n_i})\end{array}\right]\to \prod _i\left[\begin{array}{c}{B}^{n_i}R//Z\\ \downarrow \\ E{B}^{n_i}R//Z\end{array}\right]\to \prod _i\left[\begin{array}{c}*\\ \downarrow \\ B^{n_i+1}R//Z\end{array}\right].$$\left[ \array{ A \\ \downarrow \\ cone(A) } \right] \to \prod_i \left[ \array{ \mathbf{B}^{n_i} \\ \downarrow \\ cone(\mathbf{B}^{n_i}) } \right] \to \prod_i \left[ \array{ \mathbf{B}^{n_i} R//Z \\ \downarrow \\ \mathbf{E}\mathbf{B}^{n_i} R//Z } \right] \to \prod_i \left[ \array{ {*} \\ \downarrow \\ \mathbf{B}^{n_i+1} R//Z } \right] \,.

Using the path ∞-groupoid $\Pi :\mathrm{SPSh}(C)\to \mathrm{SPSh}(C)$ this induces a morphism in cohomology

$$[I,\mathrm{SPSh}(c)](\left[\begin{array}{c}X\\ \downarrow \\ \Pi (X)\end{array}\right],,\left[\begin{array}{c}A\\ \downarrow \\ \mathrm{cone}(A)\end{array}\right])\to \prod _i[I,\mathrm{SPSh}(c)](\left[\begin{array}{c}X\\ \downarrow \\ \Pi (X)\end{array}\right],,\left[\begin{array}{c}*\\ \downarrow \\ B^{n_i+1}R//Z\end{array}\right]).$$[I,SPSh(c)] \left( \left[ \array{ X \\ \downarrow \\ \Pi(X) } \right] ,\,, \left[ \array{ A \\ \downarrow \\ cone(A) } \right] \right) \;\;\; \to \;\;\; \prod_{i} [I,SPSh(c)] \left( \left[ \array{ X \\ \downarrow \\ \Pi(X) } \right] ,\,, \left[ \array{ {*} \\ \downarrow \\ \mathbf{B}^{n_i + 1} R//Z } \right] \right) \,.

The term on the left is just $A$-cohomology $H(X,A)$. The term on the right is in (abelian) nonabelian deRham cohomology. Since the $Z$-part of the cocycle only contributes on constant paths, and since there the cocycle is forced to be trivial, this is equivalent to the same expression with $R//Z$ replaced by just $R$. This way in summary we obtain a morphism

$$\mathrm{ch}:H(X,A)\to \prod _i{H}_{\mathrm{dR}}(X,B^{n_i+1}R)$$ch : \mathbf{H}(X,A) \to \prod_i \mathbf{H}_{dR}(X,\mathbf{B}^{n_i+1} R)

from $A$-cohomology to real cohomology. This is the nonabelian Chern character for us.