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

Context

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 .

Definition

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

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

Ordinary Chern character for spectra

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.

Examples

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

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