Remark 4.7.0.5. Throughout this book, we will make reference to a dichotomy between “small” and “large” mathematical objects. We will generally take a somewhat informal view of this dichotomy, taking care only to avoid maneuvers which are obviously illegitimate (see Example 4.7.0.1). However, the reader who wishes to adopt a more scrupulous approach could proceed (within the framework of Zermelo-Fraenkel set theory) as follows:
Assume the existence of an uncountable strongly inaccessible cardinal $\kappa $ (see Definition 4.7.3.20).
Declare that an $\infty $-category $\operatorname{\mathcal{C}}$ is small (essentially small, locally small) if it is $\kappa $-small (essentially $\kappa $-small, locally $\kappa $-small), and apply similar conventions to other mathematical objects of interest (such as sets and categories).