Abhijeet Vats (HU Berlin)
It has been known for some time that one can do homotopy theory on categories of C*-algebras or Banach algebras in an analogous way to how one does it on spaces or chain complexes. Our goal is to explain this basic theory by highlighting the key constructions that will be abstracted in the follow-up talk. This includes a reasonable notion of fibration which, using the appropriate formal constructions of category theory, gives rise to Kasparov's KK-theory when the latter is viewed from a homotopical lens. Our eventual goal is to address the question of whether or not the Calkin projection, on an infinite-dimensional separable complex Hilbert space, is a fibration. This question, due to Claude Schochet, is old and seems to have neat connections back to Brown-Douglas-Fillmore theory. We will share some recent results in this direction.
