Doctor of Philosophy
In homotopical algebra, 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.
Summary for Lay Audience
Modern homotopy theory has its origins in geometry, in particular in the study of homotopical properties of geometric objects. For familiar shapes, these are properties that are unaffected by continuous deformations, such as stretching or shrinking. For example, the number of connected components of a shape is a homotopical property, while properties like length, curvature and dimension are not.
A number of technical difficulties arise in studying the homotopy theory of shapes, and a large amount of abstract machinery has been developed to overcome these difficulties. This abstraction has led to widespread applications for homotopy theoretic methods throughout modern mathematics.
Homotopy colimits are important homotopy theoretic constructions. Classically, these provide a means of gluing shapes together to make new shapes with prescribed properties. In practice, working with homotopy colimits is technical, and requires that we keep track of large amounts information. Derivators provide an elegant framework for packaging this information, allowing us to carry out computations which might be intractable or impossible with other methods.
In a number of settings, homotopical information is available intrinsically in the form of an enrichment. In this context, the analogues of homotopy colimits are called weighted homotopy colimits. In this thesis, we define and study enriched derivators, which provide a framework for computing with weighted homotopy colimits and other constructions from enriched homotopy theory.
Richardson, James, "Enriched Derivators" (2019). Electronic Thesis and Dissertation Repository. 6748.