Let 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 gives the constant path inclusion .
If the -topos has rational structure
the localizaiton monoid
is internal rationalization or Lie differentiation .
For the Chern character is the characteristic class induced by the rationalization of the constant path inclusion
If has a well-adapted rational structure we have and by adjointness it follows that the Chern character acts on -cohomology as
With the internal line object contractible this is
Ordinary Chern character for spectra
We may think of as the characteristic class map induced from the canonical -cocycle on 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 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 CartSp and , a locally contractible (∞,1)-topos.
For a compact Lie group, regarded as an object of , write for its delooping.
From the discussion at homotopy ∞-groupoid? we have that
is the topological classifying space of . Its rationalization is the rational space whose rational cohomology ring is , with the generatong invariant polynomials on .
We find that the cohomology of the Chevalley-Eilenberg algebra of in degree is