Seminar

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An application of homotopy theoretic methods in non-commutative topology requires the development of the non-standard methods in homotopical algebra, since the category of $C^*$-algebras is very far from the familiar homotopy theories. The homotopy theory of homotopy theories is very useful for the study of the homotopy theory of noncommutative CW-complexes (see, e.g., the construction of the noncommmutative spectra by Arone, Barnea and Schlank). In order to extend these techniques to more general non-commutative spaces we suggest an analog for model categories of the results of Dwyer and Kan (87'), stating (in the modern terms) that a map of relative categories (A,U) -> (B,V) induces a Quillen equivalence of the categories of homotopy functors into simplicial sets iff the induced map between the respective simplicial localisations is an r-equivalence. Motivating example: the Quillen equivalence between the simplicial categories sSet and Top is well studied, but what about the categories of homotopy functors from these model categories to simplicial sets? We will present a framework that will make sense of this question and will provide an affirmative answer.

Joint work with David White.