Bruno Lindan (University of Manchester) will give a talk on
Enrichment in bicategories and Cauchy completion
Abstract:
The notion of Cauchy completeness for enriched categories (i.e. cocompleteness with respect to absolute colimits) turns out to encompass a wide range of idempotent completions in mathematics. The name itself derives from an example due to Lawvere: there is a certain monoidal category R such that R-categories are (generalised) metric spaces, and R-Cauchy-completion is precisely Cauchy completion in the usual sense. In this talk, we will investigate the corresponding notion for categories enriched in a bicategory.
The main conceptual difference between enrichment in a monoidal category V and in a bicategory W is that each object in a W-category lies over an object in the base W (such data trivialising when V is viewed as a one-object bicategory BV); this makes the theory considerably more flexible. Regarding the objects of W as ‘generalised arities’ suggests the viewpoint that absolute tensors in a W-category may be used to encode general algebraic operations. We consider in detail an example of a bicategory W such that Cauchy complete W-categories are precisely categories with finite products. The base of enrichment in this case is not biclosed – this means W-modules may fail to compose, forcing the consideration of the virtual equipment of W-modules and introducing various subtleties. We discuss generalisations of this perspective with motivations from categorical logic.