Cartesian Double Categories with an Emphasis on Characterizing Spans

TL;DR

This thesis develops Cartesian double categories, focusing on their properties and characterizing the double category of Spans, offering a new framework for understanding profunctors within this context.

Contribution

It introduces Cartesian double categories, characterizes the double category of Spans as such, and proposes a framework for profunctors in this setting.

Findings

Characterization of the double category of Spans as a Cartesian double category

Development of properties of Cartesian and fibrant double categories

Proposed framework for profunctors as Cartesian double categories

Abstract

In this thesis, we introduce Cartesian double categories, motivated by the work of Carboni, Kelly, Walters, and Wood on Cartesian bicategories. Moving from bicategories to the slightly more generalized notion of double categories allows us to set the whole theory inside the welcoming 2-category of double categories, and to overcome technical problems that were caused by working with left adjoints inside a general bicategory. Cartesian double categories that are also fibrant are of particular interest to us. After describing some important properties of Cartesian and fibrant double categories, we give a characterization of the double category of Spans as a Cartesian double category. Lastly, we talk about profunctors and give a potential framework for their characterization as Cartesian double categories.