The Joyal model structure is cofibrantly generated. The generating cofibrations are the same as Kan--Quillen: $I = \{\partial\Delta[n] \hookrightarrow \Delta[n]\}$. The generating trivial cofibrations include the inner horn inclusions $\Lambda^n_k \hookrightarrow \Delta[n]$ for $0 < k < n$, together with additional maps encoding the contractibility of the space of composites.

Fibrations between quasi-categories are the isofibrations: inner fibrations $p: \mathcal{C} \to \mathcal{D}$ such that for every equivalence $e: p(x) \xrightarrow{\sim} d$ in $\mathcal{D}$, there exists an equivalence $\bar{e}: x \xrightarrow{\sim} \bar{d}$ in $\mathcal{C}$ with $p(\bar{e}) = e$.

The Joyal weak equivalences (categorical equivalences) contain the Kan--Quillen weak equivalences but are strictly larger. The identity functor $\mathrm{id}: \mathbf{sSet}_{\mathrm{KQ}} \to \mathbf{sSet}_{\mathrm{Joyal}}$ is a left Quillen functor but NOT a Quillen equivalence.

Key difference: the nerve of the category $[1] = (0 \to 1)$, i.e., $\Delta[1]$, is categorically equivalent to $\Delta[0]$ (since $[1]$ has a terminal object, it is equivalent to the terminal category). But $|\Delta[1]| = [0,1] \not\simeq \{*\}$ as spaces. Wait -- actually $[0,1] \simeq \{*\}$ topologically. So let us use $N(B\mathbb{Z}) \to *$: this is NOT a Joyal weak equivalence (since $B\mathbb{Z}$ is not equivalent to the terminal category) but it IS a Kan--Quillen weak equivalence... no, $|N(B\mathbb{Z})| = S^1$, which is not contractible. Better example: the map $\Delta[0] \sqcup \Delta[0] \to J$ (two points mapping to the two objects of the walking isomorphism) is a Kan--Quillen weak equivalence (since $|J| \simeq *$ is contractible) but NOT a categorical equivalence ($J$ has one connected component while $\Delta[0] \sqcup \Delta[0]$ has two). Actually $J$ is connected with $|J| \simeq [0,1] \simeq *$. A proper example: consider $\Delta[0] \to J$. This IS both a categorical equivalence and a KQ equivalence. The distinction is more subtle.

In fact, the Joyal weak equivalences are detected by the homotopy category: $f: X \to Y$ is a Joyal equivalence iff $\operatorname{h}(f): \operatorname{h}X \to \operatorname{h}Y$ is an equivalence of categories and $f$ is a weak equivalence on mapping spaces.

Kan--Quillen fibrant = Kan complexes (all horns fill). Joyal fibrant = quasi-categories (inner horns fill).

Every Kan complex is a quasi-category, but not conversely. The nerve $N(\mathcal{C})$ of a non-groupoid category is a quasi-category but NOT a Kan complex.

The Joyal fibrant replacement of a simplicial set $X$ produces a quasi-category; the Kan--Quillen fibrant replacement produces a Kan complex (which "forgets" the direction of morphisms by making them all invertible).

For ordinary categories, a functor $p: \mathcal{E} \to \mathcal{B}$ is a Grothendieck fibration if and only if $N(p): N(\mathcal{E}) \to N(\mathcal{B})$ is an inner fibration satisfying an additional lifting condition for outer horns. The $\infty$-categorical analogue of Grothendieck fibrations are Cartesian fibrations (inner fibrations with a Cartesian lifting property).

A left fibration $p: X \to S$ has the RLP against $\Lambda^n_k \hookrightarrow \Delta[n]$ for $0 \leq k < n$ (all horns except the last). A right fibration uses $0 < k \leq n$ (all except the first).

Left fibrations over $S$ model covariant functors $S \to \mathcal{S}$ (from $S$ to the $\infty$-category of spaces). Right fibrations model contravariant functors. This is the $\infty$-categorical Grothendieck construction, formalized by Lurie's straightening/unstraightening equivalence.

The Joyal model structure is left proper (pushouts along cofibrations preserve weak equivalences) but NOT right proper. This is a technical difference from the Kan--Quillen model structure (which is both left and right proper).

The failure of right properness means one must be careful with pullbacks in the Joyal model structure. Lurie addresses this using categorical fibrations (isofibrations) instead of general fibrations.

The Joyal model structure is a monoidal model category with respect to the join operation $X \star Y$, not the Cartesian product. The Cartesian product does not satisfy the pushout-product axiom for the Joyal model structure.

However, the Cartesian product preserves categorical equivalences between cofibrant objects (which is all objects, since cofibrations are monomorphisms), so the product is well-behaved on the homotopy category.

The Joyal model structure is enriched over itself in the following sense: for quasi-categories $\mathcal{C}$ and $\mathcal{D}$, the functor category $\operatorname{Fun}(\mathcal{C}, \mathcal{D})$ is a quasi-category, and this construction is homotopy invariant.

The "mapping space" in the Joyal sense is $\operatorname{Map}^J(X, Y) = \operatorname{Fun}(X, Y)^\simeq$ (the core of the functor category), which is a Kan complex.

Given a simplicial set $X$, its Joyal fibrant replacement is a quasi-category $\mathcal{C}$ with a categorical equivalence $X \xrightarrow{\sim} \mathcal{C}$. This can be constructed via:

1. Small object argument: Iteratively fill all inner horns.
2. Coherent nerve: If $X$ comes from a simplicial category, use the coherent nerve $N_\Delta$.
3. Localization: If $X = N(\mathcal{M})$ for a model category, the Dwyer--Kan localization $N(\mathcal{M})[W^{-1}]$ is a quasi-category.

The Joyal fibrant replacement of a Kan complex $K$ is $K$ itself (Kan complexes are quasi-categories).

For a quasi-category $\mathcal{C}$:

• Joyal fibrant replacement: $\mathcal{C}$ itself (already fibrant).
• Kan fibrant replacement: The underlying $\infty$-groupoid $\mathcal{C}^\simeq$ (the core) -- but this loses the non-invertible morphisms.

For the nerve $N([1]) = \Delta[1]$:

• Joyal fibrant: $\Delta[1]$ (already a quasi-category).
• Kan fibrant: $\operatorname{Ex}^\infty(\Delta[1])$, which is a contractible Kan complex (since $[0,1]$ is contractible).

The Kan fibrant replacement "groupoid-ifies" by making all morphisms invertible.