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

Context

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

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

Recall the notation

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

for the structured homotopy ∞-groupoid?.

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

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

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

the localizaiton monoid

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

$${\mathrm{ch}}_A:A\to \Pi (A)=\mathrm{LConst}\Pi (A)\to \mathrm{LConst}\Pi (A)\otimes ℝ.$$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 $\Pi (A)\otimes ℝ\simeq \Gamma (\mathrm{LConst}\Pi (A)\otimes R)$ and by adjointness it follows that the Chern character acts on $A$-cohomology as

$${\mathrm{ch}}_A=\Pi (-)\otimes R:H(X,A)\to H(\Pi (X),\Pi (A)\otimes R).$$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

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

Ordinary Chern character for spectra

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

$${\mathrm{Id}}_{\Pi (A)}\in \mathrm{Func}(\Pi (A),\Pi (A))\stackrel{\simeq }{\to }H(A,\Pi (A))=H(A,\mathrm{LConst}\Pi (A)).$$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

$$\mathrm{Id}\otimes ℝ\in \mathrm{hom}({\pi }_{*}E,{\pi }_{*}E\otimes ℝ)\stackrel{\simeq }{\to }{H}^0(E,{\pi }_{*}E\otimes ℝ)$$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 $mathfbH={\mathrm{Sh}}_{(\mathrm{\infty },1)}(C)$, 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

$$\mid \Pi (BG)\mid \simeq ℬG$$|\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 $\mathbb{Q}[P_1,\cdots ,P_k]$, with $P_i$ the generatong invariant polynomials on $𝔤$.

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

$$H(\mathrm{LConst}(\Pi (BG)\otimes ℝ),B^kℝ)\simeq H(ℬG\otimes ℝ,\Gamma (B^kℝ))\simeq H^k(ℬG,\mathbb{Q}).$$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}) \,.