model structure on operator algebras


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



Paths and cylinders

Homotopy groups

Basic facts


Model category theory

model category



Universal constructions


Producing new model structures

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

Model structures

for \infty-groupoids

for ∞-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras



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



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.


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 *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 *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 *C^\ast-algebras, pdf

See at homotopical structure on C*-algebras.

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

Revised on January 10, 2014 07:43:22 by Zoran Škoda (