nLab comparison map between algebraic and topological K-theory

Idea

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)

Statement

For complex K-theory

There is a canonical homomorphism of spectra

$$K \mathbb{C} \longrightarrow KU$$

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.

Formulation in cohesive homotopy-type theory

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

$$\mathcal{K}(V)\in Stab(\mathbf{H})\hookrightarrow T \mathbf{H}$$

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

$$\flat (\mathcal{K}(V)) \longrightarrow \Pi (\mathcal{K}(V)) \,.$$

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.

References

• Henri Gillet, Comparing algebraic and topological K-theory, in Higher Algebraic K-Theory: an overview Lecture Notes in Mathematics Volume 1491, 1992, pp 55-99

Discussion relating algebraic K-theory of varieties to complex topological K-theory is in

• Michael Paluch, Algebraic $K$-theory and topological spaces, K-theory 471 (2001) (web)

and for sheaves of spectra of twisted K-theory in

• Benjamin Antieau, Ben Williams, section 2.3 of The period-index problem for twisted topological K-theory, Geometry & Topology (arXiv:1104.4654)

Discussion in terms of Banach algebras is in

• Jonathan Rosenberg, Comparison between algebraic and topological K-theory for Banach algebras and $C^\ast$-algebras (pdf)