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 C*-algebras Pro(SC*). The opposite of this model category models the ∞-category of pointed noncommutative spaces NS_∗ defined by the third author. Our model structure on Pro(SC*) extends the well-known category of fibrant objects structure on SC*. We show that the pro-category Pro(SC*) also contains, as a full coreflective subcategory, the category of pro-C*-algebras that are cofiltered limits of separable C*-algebras. By stabilizing our model category we produce a general model categorical formalism for triangulated and bivariant homology theories of C*-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 K-theory for all projective systems of separable C*-algebras generalizing the construction of Bonkat and show that our theory naturally agrees with that of Bonkat under some reasonable assumptions.

The first author was supported by the Alexander von Humboldt Foundation (Humboldt Professorship of Michael Weiss).
The third author was supported by the Deutsche Forschungsgemeinschaft (SFB 878 and SFB 1085), ERC through AdG 267079, and the Alexander von Humboldt Foundation (Humboldt Professorship of Michael Weiss).