Enhanced 2-categories and limits for lax morphisms

TL;DR

This paper investigates limits in 2-categories with structured objects and functors that preserve structure up to coherent comparison, using 2-monads and enriched category theory to characterize the existence of such limits.

Contribution

It introduces an enhanced framework for 2-categories of categories with extra structure, characterizing limits via F-enriched category theory.

Findings

Characterization of limits in 2-categories with weak and strict morphisms

Use of F-enriched category theory for describing limits

Complete description of when limits exist in these structured 2-categories

Abstract

We study limits in 2-categories whose objects are categories with extra structure and whose morphisms are functors preserving the structure only up to a coherent comparison map, which may or may not be required to be invertible. This is done using the framework of 2-monads. In order to characterize the limits which exist in this context, we need to consider also the functors which do strictly preserve the extra structure. We show how such a 2-category of weak morphisms which is "enhanced", by specifying which of these weak morphisms are actually strict, can be thought of as category enriched over a particular base cartesian closed category F. We give a complete characterization, in terms of F-enriched category theory, of the limits which exist in such 2-categories of categories with extra structure.