University of Regensburg
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From SFB1085 - Higher Invariants
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I will review the construction of the higher homotopy categories associated to an infinity-category and discuss some of their properties, especially in connection with higher weak (co)limits. These objects define a natural sequence of refinements for the comparison between homotopy commutativity and homotopy coherence, but their study seems to have received less attention than the classical homotopy category. Moreover, I will discuss some ongoing work on adjoint functor theorems in this context and a version of the classical Brown representability theorem for higher homotopy categories. I will then introduce a definition of K-theory for these objects and present some results about the comparison with Waldhausen K-theory. Lastly, I will also briefly discuss generalizations of Grothendieck derivators and derivator K-theory to the context of (n,1)-categories, and present analogous results about the comparison with Waldhausen K-theory.

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