$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Remark 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 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

  • 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).