Model structure on projective systems of -algebras and bivariant homology theories
arXiv:1508.04283
Abstract
Using the machinery of weak fibration categories due to Schlank and the first author, we construct a convenient model structure on the pro-category of separable -algebras . The opposite of this model category models the -category of pointed noncommutative spaces defined by the third author. Our model structure on extends the well-known category of fibrant objects structure on . We show that the pro-category also contains, as a full coreflective subcategory, the category of pro--algebras that are cofiltered limits of separable -algebras. By stabilizing our model category we produce a general model categorical formalism for triangulated and bivariant homology theories of -algebras (or, more generally, that of pointed noncommutative spaces), whose stable -categorical counterparts were constructed earlier by the third author. Finally, we use our model structure to develop a bivariant -theory for all projective systems of separable -algebras generalizing the construction of Bonkat and show that our theory naturally agrees with that of Bonkat under some reasonable assumptions.
47 pages; v2 revised according to referee's comments (to appear in New York J. Math.)