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Open Access Publications from the University of California

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.

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