If $p: X \to S$ is an inner fibration and $s \in S_0$ is a vertex, the fiber $X_s = p^{-1}(s)$ is a quasi-category. This follows because inner horn fillers in $X$ lying over degenerate simplices in $S$ (at the vertex $s$) provide inner horn fillers in $X_s$.

This shows that inner fibrations are indeed "families of quasi-categories."

The lifting theorem for equivalences is stronger than just lifting inner horns. It states that in an inner fibration $p: \mathcal{C} \to \mathcal{D}$, equivalences in $\mathcal{D}$ can always be lifted to equivalences in $\mathcal{C}$ (given a lift of the source).

This is the $\infty$-categorical analogue of the path-lifting property for covering spaces and fibrations. In an isofibration (Joyal fibration), the lift can be chosen to start at any specified point.

A map $f: \mathcal{C} \to \mathcal{D}$ between quasi-categories is a Joyal weak equivalence (categorical equivalence) if and only if:

1. The induced functor $\operatorname{h}f: \operatorname{h}\mathcal{C} \to \operatorname{h}\mathcal{D}$ on homotopy categories is an equivalence of ordinary categories.
2. For all objects $x, y \in \mathcal{C}$, the induced map $\operatorname{Map}_{\mathcal{C}}(x, y) \to \operatorname{Map}_{\mathcal{D}}(f(x), f(y))$ is a weak homotopy equivalence.

Equivalently, $f$ is a categorical equivalence if it is essentially surjective (every object of $\mathcal{D}$ is equivalent to one in the image of $f$) and fully faithful (the mapping space comparison is a weak equivalence).

A crucial consequence: if $\mathcal{D}$ is a quasi-category and $K$ is any simplicial set, then $\operatorname{Fun}(K, \mathcal{D}) = \mathcal{D}^K$ is a quasi-category. The proof uses the Joyal model structure: the exponential $\mathcal{D}^K$ is fibrant (a quasi-category) whenever $\mathcal{D}$ is fibrant.

This justifies the use of functor categories throughout $\infty$-category theory without needing to verify the quasi-category axiom each time.

For a quasi-category $\mathcal{C}$ and an object $c \in \mathcal{C}$, the overcategory (or slice) $\mathcal{C}_{/c}$ and undercategory $\mathcal{C}_{c/}$ are quasi-categories. These are defined as simplicial sets with:

$(\mathcal{C}_{/c})_n = \operatorname{Hom}_c(\Delta[n] \star \Delta[0], \mathcal{C})$

The Joyal lifting theorem ensures that the forgetful functor $\mathcal{C}_{/c} \to \mathcal{C}$ is a right fibration, and the forgetful functor $\mathcal{C}_{c/} \to \mathcal{C}$ is a left fibration.

The Joyal model structure and lifting theorem provide the foundation for Lurie's straightening/unstraightening equivalence:

$\operatorname{St}: \mathbf{sSet}_{/S}^{\text{left fib}} \xrightarrow{\;\simeq\;} \operatorname{Fun}(S, \mathcal{S}): \operatorname{Un}$

This equivalence identifies left fibrations $X \to S$ with functors $S \to \mathcal{S}$ (from $S$ to the $\infty$-category of spaces). It is the $\infty$-categorical Grothendieck construction and is fundamental to the entire theory.

The Joyal lifting theorem ensures that the fibers of a left fibration are Kan complexes (spaces), making this identification possible.

The Joyal model structure on $\mathbf{sSet}$ is Quillen equivalent to:

• The Bergner model structure on simplicial categories (via the coherent nerve $N_\Delta \dashv \mathfrak{C}$).
• The Rezk model structure on bisimplicial sets (complete Segal spaces).
• The Segal category model structure.

These Quillen equivalences show that quasi-categories, simplicial categories, complete Segal spaces, and Segal categories all provide equivalent models for $(\infty, 1)$-categories.

The Joyal model structure provides the technical foundation for proving that quasi-categories have limits and colimits. Specifically:

• Representability: A functor $F: K \to \mathcal{C}$ has a limit if and only if the canonical map $\mathcal{C} \to \operatorname{Fun}(K, \mathcal{C})$ has a right adjoint.
• Construction: Limits in $\mathcal{C}$ are computed as terminal objects of the appropriate slice category $\mathcal{C}_{/F}$.

The lifting properties of isofibrations are essential for establishing the universal properties of these constructions.