Indexed profunctors over 2-categories

TL;DR

This paper introduces indexed profunctors over 2-categories to develop an abstract, unified theory of various limits and their preservation properties, generalizing classical concepts in category theory.

Contribution

It defines indexed profunctors over 2-categories and uses them to unify and generalize the theory of limits, including Kan extensions and adjoint functor theorems.

Findings

Abstract version of right adjoint functors preserving limits

General account of naturality of comparison arrows in limit preservation

Unified framework for various types of limits in 2-category theory

Abstract

We define the notion of an indexed profunctor over a 2-category, and use it to develop an abstract theory of limits. The theory subsumes (conical) limits, weighted limits, ends and Kan extensions. Results include an abstract version of the theorem that right adjoint functors preserve limits, and an abstract account of the phenomenon that the comparison arrow implicated in the preservation of a limit by a functor is natural in any functorial variable the functor happens to depend on. These results make an extensive use of the data and axioms of an indexed profunctor over a 2-category.