Weakly enriched higher categories

Abstract

The goal of this thesis is to begin to lay the foundations for a theory of enriched [infinity]categories. We introduce a definition of such objects, based on a non-symmetric version of Lurie's theory of [infinity]-operads. Our first main result is a construction of the correct homotopy theory of enriched [infinity]-categories as a localization of an "algebraic" homotopy theory defined using [infinity]-operads; this is joint work with David Gepner. We then prove some comparison results: When a monoidal [infinity]-category arises from a nice monoidal model category we show that the associated homotopy theory of enriched [infinity]-categories is equivalent to the homotopy theory induced by the model category of enriched categories; when the monoidal structure is the Cartesian product we also show that this is equivalent to the homotopy theory of enriched Segal categories. Moreover, we prove that the homotopy theory of ([infinity], n)-categories enriched in spaces, obtained by iterating our enrichment procedure, is equivalent to that of n-fold complete Segal spaces. We also introduce notions of natural transformations and correspondences in the setting of enriched [infinity]-categories, and use these to construct (co,2)-categories of enriched cocategories, functors, and natural transformations, and double co-categories of enriched [infinity]-categories, functors, and correspondences. Finally, we briefly discuss a non-iterative definition of enriched ([infinity], n)-categories, based on a version of [infinity]-operads over Joyal's categories On, and define what should be the correct [infinity]-category of these.