Large ∞-categories

Remark 1.8.1 (Large \(\infty \)-categories). The \(\infty \)-categories \(\An \) and \(\Cat _{\infty }\) are sometimes referred to as the universe of animae and the universe of \(\infty \)-categories. It will sometimes be convenient to assume there are two more such universes \(\widehat {\An }\) and \(\widehat {\Cat }_{\infty }\) that satisfy all the same conditions as \(\An \) and \(\Cat _{\infty }\), with the following two further conditions:

• There is a fully faithful inclusion \(\Cat _{\infty } \hookrightarrow \widehat {\Cat }_{\infty }\) which on small animae restricts to a functor \(\An \hookrightarrow \widehat {\An }\);
• The \(\infty \)-categories \(\An \) and \(\Cat _{\infty }\) are small with respect to \(\widehat {\Cat }_{\infty }\).

We will refer to \(\infty \)-categories that are small with respect to \(\widehat {\Cat }_{\infty }\) as large \(\infty \)-categories. The two points may thus be summarized by saying that every small \(\infty \)-category is also large, and that the \(\infty \)-categories \(\An \) and \(\Cat _{\infty }\) are large.

All categorical constructions are understood relative to fixed ambient universes, which we silently enlarge when necessary.