Localization theory for derivators

Abstract

We outline the theory of reflections for prederivators, derivators and stable derivators. In order to parallel the classical theory valid for categories, we outline how reflections can be equivalently described as categories of fractions, reflective factorization systems, and categories of algebras for idempotent monads. This is a further development of the theory of monads and factorization systems for derivators.

Accessibility and presentability in 2-categories

“…We end the paper with a collection of relevant examples; in §5 and §6 we show how different choices of K and よ generate the different notions of accessibility and presentability: this yields a unified treatment of different shapes of Gabriel-Ulmer representation and duality. Several of these examples carry an homotopical flavour; this fits the goals of the present work in a bigger framework: finding a uniform treatment of accessibility in those 2-categorical settings capturing the theory of higher categories, like ∞-cosmoi [RV15a,RV15b] and derivators [Gro13] (a statement of purpose in this direction is contained at the end of [Lor18]).…”

“…The case of derivators. The preprint [Lor18] ends with a short statement of purpose drawing a connection with the present work. As the main objective of [Lor18] was to lay down the theory of co/reflective localizations of derivators, it has been natural to surmise that there exists a notion of locally presentable and accessible derivator allowing to restate the representation theorem that we prove in Theorem 3.8; and this is easy to believe especially because at least two definitions echoing the content of [AR94, 1.C] have already been given in the context of derivators in [MR16] and [Ren09].…”