Abhijeet Vats (HU Berlin)
It is a fact of life that higher categories are very pervasive in modern homotopy theory and algebraic K-theory. Nevertheless, the model-dependent theory, unsurprisingly, still relies on the model categories of Daniel Quillen and indeed, these structures were introduced as a unified way of dealing with several, seemingly disparate examples of situations where there is some reasonable notion of ``homotopy''. Since the latter does hold for categories of operator algebras, it might be reasonable to hope that model categories also give a unified treatment of homotopy theory on such categories. Indeed, this was an implicit question asked by Claude Schochet in an old paper.
In this talk, we will recall the basics of model category theory and also recall an impossibility theorem in this direction. Nevertheless, we will show that there exist structures, known as categories of fibrant objects, which do capture homotopy theory on interesting subcategories of the category of C*-algebras. We will recall a result, due to Kenneth Brown, that this set-up is sufficient to have reasonable control over their 1-categorical localizations. We will also relate these categories to those introduced by Friedhelm Waldhausen as input for the Segal S-construction. Finally, we will discuss a general construction, the so-called Spanier-Whitehead construction, which underlies common constructions of bivariant invariants of operator algebras in the literature. We will see that this construction is best done by passing to the Dwyer-Kan localization, thus demonstrating that higher categories form the cleanest approach to this construction in this generality.
