∞-categories of animae and ∞-categories

Axiom L. There exist \(\infty \)-categories \(\An \) and \(\Cat _{\infty }\), together with a fully faithful functor \(\An \hookrightarrow \Cat _{\infty }\). They satisfy the following properties:

• There is a functor \(\pi _{\univ }\colon (\Cat _{\infty })_{\bullet } \to \Cat _{\infty }\) called the universal cocartesian fibration.¹ Every object \(c \in \Cat _{\infty }\) defines an \(\infty \)-category \(C\) via the following pullback square:

  

  An \(\infty \)-category \(C\) arising this way is called a small \(\infty \)-category. If \(c\) is contained in the full subcategory \(\An \), then \(C\) is an anima; animae arising this way are called small animae.
• For objects \(c,d \in \Cat _{\infty }\) corresponding to \(\infty \)-categories \(C\) and \(D\), there is an equivalence of animae \[ \Hom _{\Cat _{\infty }}(c,d) \iso \Map (C,D), \] and in particular morphisms in \(\Cat _{\infty }\) correspond to functors of \(\infty \)-categories. Composition in \(\Cat _{\infty }\) corresponds to composition of functors.
• The \(\infty \)-categories \(\An \) and \(\Cat _{\infty }\) admit all small limits and colimits; here the adjective ‘small’ means that the indexing \(\infty \)-category is required to be small. The inclusion \(\An \hookrightarrow \Cat _{\infty }\) preserves small limits and colimits.
• The initial and terminal \(\infty \)-categories are small. Small \(\infty \)-categories are closed under products, coproducts, pullbacks, pushouts, functor categories, subcategories, and localizations.
• There is a fully faithful functor \[ \simp \hookrightarrow \Cat _{\infty } \] given on objects by sending a linearly ordered set \(P\) to its associated \(\infty \)-category. In particular, the \(\infty \)-categories \([n]\) are small for all \(n \geq 0\).
• There is a fully faithful functor \[ \Set \hookrightarrow \An \] given on objects by sending a set to its associated anima.