differential cohomology in an (∞,1)-topos – survey
internal homotopy ∞-groupoid?
(…)
Background fields in twisted differential nonabelian cohomology?
Let $\mathbf{H}$ be a locally contractible (∞,1)-topos with global section essential geometric morphism
Recall the notation
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
the localizaiton monoid
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
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
With the internal line object $R$ contractible this is
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
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
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
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
Last revised on July 27, 2010 at 08:45:50. See the history of this page for a list of all contributions to it.