Formulating Categorical Concepts using Classes

TL;DR

This paper explores the use of classes to formalize categorical concepts, introducing a structure for subobjects and a hierarchy of k-classes, applicable to both ordinary and higher categories.

Contribution

It proposes a new explicit structure for subobjects and a hierarchy of k-classes, advancing the formalization of categorical notions.

Findings

Introduces a structure for subobjects using classes

Develops a hierarchy of k-classes for categorical theory

Applies the hierarchy to higher categories

Abstract

We examine the use of classes to formulate several categorical notions. This leads to two proposals: an explicit structure for working with subobjects, and a hierarchy of $k$-classes. We apply the latter to both ordinary and higher categories.