Model structure on projective systems of C∗C^*-algebras and bivariant homology theories

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 $\mathrm{Pro}(\mathtt{SC^*})$. The opposite of this model category models the $\infty$-category of pointed noncommutative spaces $\mathtt{N}\mathcal{S_*}$ defined by the third author. Our model structure on $\mathrm{Pro}(\mathtt{SC^*})$ extends the well-known category of fibrant objects structure on $\mathtt{SC^*}$. We show that the pro-category $\mathrm{Pro}(\mathtt{SC^*})$ 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 $\infty$-categorical counterparts were constructed earlier by the third author. Finally, we use our model structure to develop a bivariant $\mathrm{K}$-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.Comment: 47 pages; v2 revised according to referee's comments (to appear in New York J. Math.

Triangulated surfaces in triangulated categories

For a triangulated category A with a 2-periodic dg-enhancement and a triangulated oriented marked surface S we introduce a dg-category F(S,A) parametrizing systems of exact triangles in A labelled by triangles of S. Our main result is that F(S,A) is independent on the choice of a triangulation of S up to essentially unique Morita equivalence. In particular, it admits a canonical action of the mapping class group. The proof is based on general properties of cyclic 2-Segal spaces. In the simplest case, where A is the category of 2-periodic complexes of vector spaces, F(S,A) turns out to be a purely topological model for the Fukaya category of the surface S. Therefore, our construction can be seen as implementing a 2-dimensional instance of Kontsevich's program on localizing the Fukaya category along a singular Lagrangian spine.Comment: 55 pages, v2: references added and typos corrected, v3: expanded version, comments welcom