Enriched Derivators

Abstract

The theory of derivators provides a convenient abstract setting for computing with homotopy limits and colimits. In enriched homotopy theory, the analogues of homotopy (co)limits are weighted homotopy (co)limits. In this thesis, we develop a theory of derivators and, more generally, prederivators enriched over a monoidal derivator E. In parallel to the unenriched case, these E-prederivators provide a framework for studying the constructions of enriched homotopy theory, in particular weighted homotopy (co)limits. As a precursor to E-(pre)derivators, we study E-categories, which are categories enriched over a bicategory Prof(E) associated to E. We prove a number of fundamental results about E-categories, which parallel classical results for enriched categories. In particular, we prove an E-category Yoneda lemma, and study representable maps of E-categories. In any E-category, we define notions of weighted homotopy limits and colimits. We define E-derivators to be E-categories with a number of further properties; in particular, they admit all weighted homotopy (co)limits. We show that the closed E-modules studied by Groth, Ponto and Shulman give rise to associated E-derivators, so that the theory of E-(pre)derivators captures these examples. However, by working in the more general context of E-prederivators, we can study weighted homotopy (co)limits in other settings, in particular in settings where not all weighted homotopy (co)limits exist. Using the E-category Yoneda lemma, we prove a representability theorem for E-prederivators. We show that we can use this result to deduce representability theorems for closed E-modules from representability results for their underlying categories.