Schreiber nonabelian Chern character


under construction



The familiar Chern character is a canonical cocycle in real cohomology defined on any spectrum AA. 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 EE any smooth ∞-stack, i.e. any ∞-Lie groupoid.


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

For ASPSh(C)A\in SPSh(C) cofibrant suppose there is a universal cocycle

c:A iB n iR//Z c : A \to \prod_i \mathbf{B}^{n_i} R//Z

that encodes the integral cohomology of AA.

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

[A cone(A)] i[B n i cone(B n i)] i[B n iR//Z EB n iR//Z] i[* B n i+1R//Z]. \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 Π:SPSh(C)SPSh(C)\Pi : SPSh(C) \to SPSh(C) this induces a morphism in cohomology

[I,SPSh(c)]([X Π(X)],,[A cone(A)]) i[I,SPSh(c)]([X Π(X)],,[* B n i+1R//Z]). [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 AA-cohomology H(X,A)\mathbf{H}(X,A). The term on the right is in (abelian) nonabelian deRham cohomology. Since the ZZ-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//ZR//Z replaced by just RR. This way in summary we obtain a morphism

ch:H(X,A) iH dR(X,B n i+1R) ch : \mathbf{H}(X,A) \to \prod_i \mathbf{H}_{dR}(X,\mathbf{B}^{n_i+1} R)

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

Last revised on January 22, 2010 at 13:26:45. See the history of this page for a list of all contributions to it.