Pointwise extensions and sketches in bicategories

TL;DR

This paper explores pointwise extensions in bicategories, including enriched categories, showing they can be replaced by extensions along special maps and introducing the concept of limit sketches internal to bicategories.

Contribution

It introduces a method to replace extensions with those along fully faithful maps and proposes the concept of limit sketches internal to bicategories.

Findings

Extensions can be replaced by those along fully faithful maps

Introduction of limit sketches internal to bicategories

Applicable to bicategories of enriched categories

Abstract

We make a few remarks concerning pointwise extensions in a bicategory which include the case of bicategories of enriched categories. We show that extensions, pointwise or not, can be replaced by extensions along very special fully faithful maps. This leads us to suggest a concept of limit sketch internal to the bicategory.