Icons

TL;DR

The paper introduces icons, a new type of 2-cell in a 2-category of bicategories, providing a novel perspective and applications in various areas of higher category theory.

Contribution

It presents icons as a new 2-cell concept in bicategory 2-categories, offering both elementary and technical insights with multiple applications.

Findings

Icons serve as oplax natural transformations with identity components.

Icons enable new constructions in monoidal categories and 2-nerves.

Applications include 2-dimensional Lawvere theories and bicategory bundles.

Abstract

Categorical orthodoxy has it that collections of ordinary mathematical structures such as groups, rings, or spaces, form categories (such as the category of groups); collections of 1-dimensional categorical structures, such as categories, monoidal categories, or categories with finite limits, form 2-categories; and collections of 2-dimensional categorical structures, such as 2-categories or bicategories, form 3-categories. We describe a useful way in which to regard bicategories as objects of a 2-category. This is a bit surprising both for technical and for conceptual reasons. The 2-cells of this 2-category are the crucial new ingredient; they are the icons of the title. These can be thought of as ``the oplax natural transformations whose components are identities'', but we shall also give a more elementary description. We describe some properties of these icons, and give applications to monoidal categories, to 2-nerves of bicategories, to 2-dimensional Lawvere theories, and to bundles of bicategories.