2-categorical Brown representability and the relation between derivators and infinity-categories
- Author(s): Arlin, Kevin
- Advisor(s): Balmer, Paul
- et al.
In this thesis, we study the 2-category of infinity-categories, largely
with attention to its relationships with the 2-category of prederivators.
We prove that the 2-category of infinity-categories admits a small set of objects detecting equivalences and
satisfies a Brown representability theorem, which we formulate using a new notion of compactly generated 2-category. We show that the canonical
2-functor from the 2-category of infinity-categories into the 2-category of prederivators detects equivalences and, under appropriate
size conditions, induces an equivalence on hom-categories.
We explain how to extend prederivators defined on the 2-category of
ordinary categories to the domain of all infinity-categories using the delocalization theorem. We use the
Brown representability theorem to give conditions under which a prederivator is representable by an infinity-category. We also show how to extend derivators defined on categories
and satisfying a mild size condition to derivators on infinity-categories, using an extension
of Cisinski's theorem on the universality of derivators of spaces. This extension allows us to give conditions under which the small sub-prederivators of quite general derivators are all representable by infinity-categories.