group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
For suitable setups, there is a canonical homomorphism from algebraic K-theory to topological K-theory with given coefficients.
This usually goes just by “the comparison map”. (e.g. Rosenberg, theorem 2.1)
There is a canonical homomorphism of spectra
from the algebraic K-theory spectrum of the complex numbers to KU (e.g. Paluch 01, lemma 2.6).
In fact in terms of cohesion of smooth spectra, this is a component of a natural transformation, this we discuss below.
Given a cohesive (infinity,1)-topos and a symmetric monoidal (∞,1)-category $V\in CMon_\infty(Cat_\infty(\mathbf{H}))$, internal to $\mathbf{H}$ write
for its K-theory of a symmetric monoidal (∞,1)-category (formed locally and then ∞-stackified). This is a stable homotopy type in the tangent cohesive (∞,1)-topos.
The flat modality part $\flat \mathcal{K}(V)$ is the algebraic K-theory of $\flat V \in CMon_\infty(Cat_\infty)$. The shape modality part on the other hand is a “topological” version of this
The points-to-pieces transform $\flat \to \Pi$ provides a natural comparison map
For instance for $\mathbf{H}=$ Smooth∞Grpd and $V = \mathbf{Vect}^{\oplus}$ the stack of smooth vector bundles with direct sum monoidal structure, then $\flat \mathcal{K}(V)\simeq K \mathbb{C}$ is the algebraic K-theory of the complex numbers and $\Pi \mathcal{K}(V)\simeq$ KU is complex topological K-theory.
See at differential cohomology diagram for more on this.
Discussion relating algebraic K-theory of varieties to complex topological K-theory is in
and for sheaves of spectra of twisted K-theory in
Discussion in terms of Banach algebras is in
Last revised on May 1, 2014 at 00:55:44. See the history of this page for a list of all contributions to it.