Set theory for category theory

TL;DR

This paper compares various set-theoretic foundations used in category theory, highlighting how different formalizations influence the scope of categorical constructions and their practical implications.

Contribution

It provides an expository overview and comparison of multiple set-theoretic frameworks for category theory, clarifying their effects on categorical practice.

Findings

Different set-theoretic formalizations impact categorical constructions.

The choice of foundations influences the scope of permissible category-theoretic concepts.

The paper clarifies implications of set-theoretic choices for everyday category theory use.

Abstract

Questions of set-theoretic size play an essential role in category theory, especially the distinction between sets and proper classes (or small sets and large sets). There are many different ways to formalize this, and which choice is made can have noticeable effects on what categorical constructions are permissible. In this expository paper we summarize and compare a number of such "set-theoretic foundations for category theory," and describe their implications for the everyday use of category theory. We assume the reader has some basic knowledge of category theory, but little or no prior experience with formal logic or set theory.