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

Contents

Context

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

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

Recall the notation

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

for the structured homotopy ∞-groupoid?.

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

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

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

the localizaiton monoid

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

is internal rationalization or Lie differentiation .

Definition

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

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

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

ch A=Π()R:H(X,A)H(Π(X),Π(A)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 RR contractible this is

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

Ordinary Chern character for spectra

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

Id Π(A)Func(Π(A),Π(A))H(A,Π(A))=H(A,LConstΠ(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 EE a spectrum, the Hurewicz isomorphism for spectra yields a canonical cocycle

Idhom(π *E,π *E)H 0(E,π *E) 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=C = CartSp and mathfbH=Sh (,1)(C)\mathfb{H} = Sh_{(\infty,1)}(C), a locally contractible (∞,1)-topos.

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

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

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

is the topological classifying space of GG. Its rationalization G\mathcal{B}G \otimes \mathbb{R} is the rational space whose rational cohomology ring is [P 1,,P k]\mathbb{Q}[P_1, \cdots , P_k], with P iP_i the generatong invariant polynomials on 𝔤\mathfrak{g}.

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

H(LConst(Π(BG)),B k)H(G,Γ(B k))H k(G,). 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)