differential cohomology - abelian case

This is a subentry of the entry differential cohomology.

We discuss here how the general theory of differential nonabelian cohomology developed there relates to the standard theory of abelian differential cohomology.


raw material for the moment

The crucial input for translating the general setup to the familar abelian differential cohomology setup is

  • the “naturality of the differential Quillen adjunction” as discussed at path ∞-groupoid which gives commutative diagram

    Y A flat Π(Y) A \array{ Y &\to & A_{flat} \\ \downarrow && \downarrow \\ \Pi(Y) &\to& A }

    and their version for the infinitesimal path ∞-groupoid?

    Y A flat inf Π inf(Y) A \array{ Y &\to & A_{flat}^{inf} \\ \downarrow && \downarrow \\ \Pi^{inf}(Y) &\to& A }
  • the deRham theorem for ∞-Lie groupoids? which for A=B nA = \mathbf{B}^n \mathbb{R} identifies

    • YA flat infY \to A_{flat}^{inf} with a degree nn real cococyle;

    • Π inf(Y)A\Pi^{inf}(Y) \to A with the corresponding deRham cocycle.

Ordinary abelian differential cohomology starts (following Hopkins-Singer) with specifying real cohomology classes

ι:AB n c \iota : A \to \mathbf{B}^n\mathbb{R}_{c}

on an abelian AA, then pulling these back along a given AA-cocycle g:YAg : Y \to A to the composite

YgAιB n c Y \stackrel{g}{\to} A \stackrel{\iota}{\to} \mathbf{B}^n\mathbb{R}_{c}

and then identifying them there by a homotopy hh with a differential form

Y g A ι B n c Π inf(Y) ω B n \array{ Y &\stackrel{g}{\to}& A &\stackrel{\iota}{\to}& \mathbf{B}^n\mathbb{R}_c \\ \downarrow &&&& \downarrow \\ \Pi^{inf}(Y) &\stackrel{\omega}{\to}&&\to& \mathbf{B}^n\mathbb{R} }
Created on September 29, 2009 12:57:41 by Urs Schreiber (