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, model $\infty$-category
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
(geometry $\leftarrow$ Isbell duality $\to$ algebra)
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 $C^*$-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:
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