Quillen’s theorem A and categories as model for homotopy types

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Date(s) - 12/11/2015
11 h 00 min - 12 h 00 min

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Quillen proved his Theorem A in the ’70s in order to have a powerful tool to investigate weak equivalences between nerves of small categories. This in turn allowed him to lie the foundations of a higher K-theory. Furthermore, in the ’80s Grothendieck was studying the homotopy theory of small categories and he realized that a relative version of Quillen’s Theorem A was a fundamental ingredient for an axiomatic description of any class of functor which qualify to be a class of “weak equivalences” for the category of small categories; Grothendieck calls these classes of weak equivalences, satisfying Theorem A and some more axioms, fundamental localisers.

Assuming a basic result about model categories, which we are going to recall with some detail, we give a quick proof of Quillen’s Theorem A in its relative version. Then, using very similar techniques, we give a proof of the equivalence of categories between the homotopy category of small categories and the homotopy category of simplicial sets, which in turn is equivalent to the category of CW-complexes and continuous maps modulo homotopy.

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Olivier CHABROL
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