Chern character in an (∞,1)-topos

differential cohomology in an (∞,1)-topos -- survey
**structures in an (∞,1)-topos**
* **shape**
* **cohomology**
* cocycle/characteristic class
* twisted cohomology
* principal ∞-bundle
* ∞-vector bundle
* **homotopy**
* covering ∞-bundles
* Postnikov system
* path ∞-groupoid
* geometric realization
* Galois theory
* internal homotopy ∞-groupoid?
* Whitehead system
* **rational homotopy**
* ∞-Lie algebroid
* ordinary rational homotopy
* internal rational homotopy
* Chern-character
* **differential cohomology**
* flat differential cohomology
* de Rham cohomology
* de Rham theorem
* **relative theory over a base**
* relative homotopy theory
* Lie theory
## Examples
(...)
## Applications
* Background fields in twisted differential nonabelian cohomology
* Differential twisted String and Fivebrane structures
* D'Auria-Fre formulation of supergravity

Let $\mathbf{H}$ be a locally contractible (∞,1)-topos with global section essential geometric morphism

$(\Pi \dashv LConst \dashv \Gamma)
:
\mathbf{H}
\stackrel{\overset{\Pi}{\to}}{\stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}}}
\infty Grpd
\,.$

Recall the notation

$(\mathbf{\Pi} \dashv \mathbf{\flat})
:=
(LConst \circ \Pi \dashv LConst \circ \Gamma)$

for the structured homotopy ∞-groupoid?.

The unit of the adjunction $(\Pi \dashv LConst)$ gives the constant path inclusion $A \to \mathbf{\Pi}(A)$.

If the $(\infty,1)$-topos $\mathbf{H}$ has rational structure

$\mathbf{L} \stackrel{\overset{Lie}{\leftarrow}}{\underset{i}{\hookrightarrow}}
\mathbf{H}$

the localizaiton monoid

$(-)\otimes R := \mathbf{H} \stackrel{Lie}{\to} \mathbf{L}
\hookrightarrow \mathbf{H}$

is *internal rationalization* or *Lie differentiation* .

For $A \in \mathbf{H}$ the **Chern character** is the characteristic class induced by the rationalization of the constant path inclusion

$ch_A : A \to \mathbf{\Pi}(A) = LConst \Pi(A) \to LConst \Pi(A)\otimes \mathbb{R}
\,.$

If $\mathbf{H}$ has a well-adapted rational structure we have $\Pi(A)\otimes \mathbb{R} \simeq \Gamma (LConst \Pi(A)\otimes R)$ and by adjointness it follows that the Chern character acts on $A$-cohomology as

$ch_A = \mathbf{\Pi}(-)\otimes R : \mathbf{H}(X,A) \to \mathbf{H}(\mathbf{\Pi}(X), \mathbf{\Pi}(A)\otimes R)
\,.$

With the internal line object $R$ contractible this is

$\simeq \mathbf{H}_{dR}(X, \mathbf{\Pi}(A)\otimes R)
\,.$

We may think of $ch : \mathbf{H}(-,A) \to \mathbf{H}(-,\mathbf{\Pi}(A))$ as the characteristic class map induced from the canonical $\mathbf{\Pi}(A)$-cocycle on $A$ itself under the equivalence

$Id_{\Pi(A)} \in Func(\Pi(A), \Pi(A)) \stackrel{\simeq}{\to}
\mathbf{H}(A, \mathbf{\Pi}(A))
=
\mathbf{H}(A, LConst \Pi(A))
\,.$

Notice that (up to rationalization) this is indeed the way the Chern character is usually defined on spectra, see HoSi, def 4.56.

For $E$ a spectrum, the Hurewicz isomorphism for spectra yields a canonical cocycle

$Id\otimes \mathbb{R} \in hom(\pi_* E , \pi_* E \otimes \mathbb{R}) \stackrel{\simeq}{\to}
H^0(E, \pi_* E\otimes \mathbb{R})$

And the Chern character map on generalized (Eilenberg-Steenrod) cohomology is postcomposition with this cocycle, as in our definition above.

Let $C =$ CartSp and $\mathfb{H} = Sh_{(\infty,1)}(C)$, a locally contractible (∞,1)-topos.

For $G$ a compact Lie group, regarded as an object of $\mathbf{H}$, write $\mathbf{B}G$ for its delooping.

From the discussion at homotopy ∞-groupoid? we have that

$|\Pi(\mathbf{B}G)| \simeq \mathcal{B}G$

is the topological classifying space of $G$. Its rationalization $\mathcal{B}G \otimes \mathbb{R}$ is the rational space whose rational cohomology ring is $\mathbb{Q}[P_1, \cdots , P_k]$, with $P_i$ the generatong invariant polynomials on $\mathfrak{g}$.

We find that the cohomology of the Chevalley-Eilenberg algebra of $LConst (\Pi(\mathbf{B}G)\otimes \mathbb{R})$ in degree $k$ is

$H(LConst (\Pi(\mathbf{B}G)\otimes \mathbb{R}), \mathbf{B}^k \mathbb{R})
\simeq
H(\mathcal{B}G\otimes \mathbb{R}, \Gamma(\mathbf{B}^k \mathbb{R}))
\simeq
H^k(\mathcal{B} G, \mathbb{Q})
\,.$

Revised on July 27, 2010 08:45:50
by Urs Schreiber
(134.100.32.207)