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Details of talk

TitleGenerator notions in $\infty$-cosmology
PresenterDominic Verity (Macquarie University)
Author(s)Prof Dominic Verity, Assoc Prof Emily Riehl
SessionCategory Theory, Algebraic Topology, K-Theory
Time16:00:00 2017-12-15

   $\infty$-Cosmoi provide a framework in which to develop the abstract category
theory of various kinds of $(\infty,1)$-categorical structures. In essence, an
$\infty$-cosmos is simply a finitely complete $(\infty,2)$-category, although
for expository reasons they are often taken to be categories of fibrant objects
enriched in the Joyal model structure. This notion is general enough to
immediately encompass most of the common models of $(\infty,1)$-categories;
quasi-categories, complete Segal-spaces, $\Theta_1$-spaces and such like. At the
same time, it is powerful enough to develop a theory of (co)cartesian
fibrations, a calculus of two-sided modules (pro-functors), Yoneda's lemma,
theories of adjunction and Kan extension and so forth. Indeed, much of this
theory can be developed in the setting of the (strict, classical) \emph{homotopy
2-category\/} obtained from the $\infty$-cosmos by applying the homotopy
category construction to its hom-spaces.

    In this talk we briefly recap the cosmological approach to the category
theory of $\infty$-categorical structures and discuss how it encompasses fibred
categorical notions. This leads us naturally to the study of certain generating
sets of ``compact'' objects in an $\infty$-cosmos, a mechanism which allows us
to adapt certain fibrewise arguments into the $\infty$-cosmos framework.