Abstract
?  $\infty$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;
quasicategories, complete Segalspaces, $\Theta_1$spaces and such like. At the
same time, it is powerful enough to develop a theory of (co)cartesian
fibrations, a calculus of twosided modules (profunctors), 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
2category\/} obtained from the $\infty$cosmos by applying the homotopy
category construction to its homspaces.
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.

