Title: Three models for the homotopy theory of homotopy theories
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Author: Julia E. Bergner \vskip .2 in
Author's e-mail address: bergnerj@member.ams.org \vskip .2 in
AMS classification number: Primary: 55U35; Secondary 18G30, 18E35
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arXiv submission number: math.AT/0504334 \vskip .2 in
Abstract: Given any model category, or more generally any category
with weak equivalences, its simplicial localization is a
simplicial category which can rightfully be called the ``homotopy
theory" of the model category. There is a model category structure
on the category of simplicial categories, so taking its simplicial
localization yields a ``homotopy theory of homotopy theories." In
this paper we show that there are two different categories of
diagrams of simplicial sets, each equipped with an appropriate
definition of weak equivalence, such that the resulting homotopy
theories are each equivalent to the homotopy theory arising from
the model category structure on simplicial categories. Thus, any
of these three categories with their respective weak equivalences
could be considered a model for the homotopy theory of homotopy
theories. One of them in particular, Rezk's complete Segal space
model category structure on the category of simplicial spaces, is
much more convenient from the perspective of making calculations
and therefore obtaining information about a given homotopy theory.