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.
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