# Schreiber Chern character in an (∞,1)-topos

differential cohomology in an (∞,1)-topos -- survey

structures in an (∞,1)-topos

(…)

# Contents

## Context

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

$\left(\Pi ⊣\mathrm{LConst}⊣\Gamma \right):H\stackrel{\stackrel{\Pi }{\to }}{\stackrel{\stackrel{\mathrm{LConst}}{←}}{\underset{\Gamma }{\to }}}\infty \mathrm{Grpd}\phantom{\rule{thinmathspace}{0ex}}.$(\Pi \dashv LConst \dashv \Gamma) : \mathbf{H} \stackrel{\overset{\Pi}{\to}}{\stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}}} \infty Grpd \,.

Recall the notation

$\left(\Pi ⊣♭\right):=\left(\mathrm{LConst}\circ \Pi ⊣\mathrm{LConst}\circ \Gamma \right)$(\mathbf{\Pi} \dashv \mathbf{\flat}) := (LConst \circ \Pi \dashv LConst \circ \Gamma)

for the structured homotopy ∞-groupoid?.

The unit of the adjunction $\left(\Pi ⊣\mathrm{LConst}\right)$ gives the constant path inclusion $A\to \Pi \left(A\right)$.

If the $\left(\infty ,1\right)$-topos $H$ has rational structure

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

the localizaiton monoid

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

is internal rationalization or Lie differentiation .

## Definition

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

${\mathrm{ch}}_{A}:A\to \Pi \left(A\right)=\mathrm{LConst}\Pi \left(A\right)\to \mathrm{LConst}\Pi \left(A\right)\otimes ℝ\phantom{\rule{thinmathspace}{0ex}}.$ch_A : A \to \mathbf{\Pi}(A) = LConst \Pi(A) \to LConst \Pi(A)\otimes \mathbb{R} \,.

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

${\mathrm{ch}}_{A}=\Pi \left(-\right)\otimes R:H\left(X,A\right)\to H\left(\Pi \left(X\right),\Pi \left(A\right)\otimes R\right)\phantom{\rule{thinmathspace}{0ex}}.$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

$\simeq {H}_{\mathrm{dR}}\left(X,\Pi \left(A\right)\otimes R\right)\phantom{\rule{thinmathspace}{0ex}}.$\simeq \mathbf{H}_{dR}(X, \mathbf{\Pi}(A)\otimes R) \,.

## Ordinary Chern character for spectra

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

${\mathrm{Id}}_{\Pi \left(A\right)}\in \mathrm{Func}\left(\Pi \left(A\right),\Pi \left(A\right)\right)\stackrel{\simeq }{\to }H\left(A,\Pi \left(A\right)\right)=H\left(A,\mathrm{LConst}\Pi \left(A\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$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

$\mathrm{Id}\otimes ℝ\in \mathrm{hom}\left({\pi }_{*}E,{\pi }_{*}E\otimes ℝ\right)\stackrel{\simeq }{\to }{H}^{0}\left(E,{\pi }_{*}E\otimes ℝ\right)$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.

## Examples

Let $C=$ CartSp and $mathfbH={\mathrm{Sh}}_{\left(\infty ,1\right)}\left(C\right)$, a locally contractible (∞,1)-topos.

For $G$ a compact Lie group, regarded as an object of $H$, write $BG$ for its delooping.

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

$\mid \Pi \left(BG\right)\mid \simeq ℬG$|\Pi(\mathbf{B}G)| \simeq \mathcal{B}G

is the topological classifying space of $G$. Its rationalization $ℬG\otimes ℝ$ is the rational space whose rational cohomology ring is $ℚ\left[{P}_{1},\cdots ,{P}_{k}\right]$, with ${P}_{i}$ the generatong invariant polynomials on $𝔤$.

We find that the cohomology of the Chevalley-Eilenberg algebra of $\mathrm{LConst}\left(\Pi \left(BG\right)\otimes ℝ\right)$ in degree $k$ is

$H\left(\mathrm{LConst}\left(\Pi \left(BG\right)\otimes ℝ\right),{B}^{k}ℝ\right)\simeq H\left(ℬG\otimes ℝ,\Gamma \left({B}^{k}ℝ\right)\right)\simeq {H}^{k}\left(ℬG,ℚ\right)\phantom{\rule{thinmathspace}{0ex}}.$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)