nLab
model structure on operator algebras

Context

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Model category theory

model category, model \infty -category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for equivariant \infty-groupoids

for rational \infty-groupoids

for rational equivariant \infty-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general

specific

for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\infty,1)-sheaves / \infty-stacks

Noncommutative geometry

Contents

Overview

So far there are only few works on homotopy theory for operator algebras. One of the basic checks for good homotopy theory of operator algebras is that the Kasparov KK-groups should be obtained from Hom-s in the appropriate stable category of operator algebras. This subject is important in order to introduce more systematic homotopic methods in noncommutative geometry à la Alain Connes.

References

Quillen model category structures on various categories of operator algebras have been introduced and studied in

and a category of fibrant objects approach in

In fact, Baues fibration category structure has been constructed in 1997, not only on the category of C *C^\ast-algebras but on a wider class of similar categories:

See at homotopical structure on C*-algebras.

The derived category/triangulated category approach to operator algebras is introduced in

where a noncommutative analogue of the stable category of spectra is introduced.

There is also a related survey