# nLab model structure on operator algebras

Contents

### Context

#### Model category theory

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

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

Model structures

for $\infty$-groupoids

for ∞-groupoids

for equivariant $\infty$-groupoids

for rational $\infty$-groupoids

for rational equivariant $\infty$-groupoids

for $n$-groupoids

for $\infty$-groups

for $\infty$-algebras

general $\infty$-algebras

specific $\infty$-algebras

for stable/spectrum objects

for $(\infty,1)$-categories

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

for $(\infty,1)$-operads

for $(n,r)$-categories

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

#### Noncommutative geometry

noncommutative geometry

(geometry $\leftarrow$ Isbell duality $\to$ algebra)

# 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

• Michael Joachim, Mark W. Johnson, Realizing Kasparov’s KK-theory groups as the homotopy classes of maps of a Quillen model category, An alpine anthology of homotopy theory, Contemp. Math. 399, Amer. Math. Soc. 2006, pp. 163–197, MR2007c:46070

• Paul Arne Østvær, Homotopy theory of $C^*$-algebras, Frontiers in Mathematics, Springer Basel, 2010, (arxiv/0812.0154, pdf)

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 $C^\ast$-algebras but on a wider class of similar categories:

• Kasper K. S. Andersen, Jesper Grodal, A Baues fibration category structure on Banach and $C^\ast$-algebras, pdf

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

• Ralf Meyer, Ryszard Nest, Homological algebra in bivariant K-theory and other triangulated categories, math.KT/0702146

• Ralf Meyer, Ryszard Nest, The Baum-Connes conjecture via localisation of categories), Topology 45 (2006), no. 2, 209–259, math.KT/0312292, MR2006k:19013

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

There is also a related survey

• Ralf Meyer, Categorical aspects of bivariant K-theory, K-theory and noncommutative geometry, European Math. Soc. Ser. Congr. Rep., math.KT/0702145, MR2010h:19008

Last revised on January 10, 2014 at 07:43:22. See the history of this page for a list of all contributions to it.