Since the initial object $\emptyset$ maps to any $X$ by the unique map $\emptyset \hookrightarrow X$ (which is a monomorphism = cofibration), every simplicial set is cofibrant. This is a major technical advantage of $\mathbf{sSet}$ over $\mathbf{Top}$ (where not every space is cofibrant).

Consequently, to compute $[X, Y]$ in the homotopy category, we only need a fibrant replacement of $Y$: $[X, Y] = \pi_0 \operatorname{Map}(X, RY)$ where $RY$ is a Kan complex weakly equivalent to $Y$.

An object $X$ is fibrant (i.e., $X \to \Delta[0]$ is a Kan fibration) if and only if $X$ is a Kan complex. Not every simplicial set is fibrant: for instance, $\Delta[2]$ is not a Kan complex (the outer horn $\Lambda^2_0$ with $d_1 = \mathrm{id}_{01}$ and $d_2 =$ identity at $0$ has a unique filler, but for a general $X$ with edges that are not composable in the nerve sense, outer horns may fail to fill).

Actually, $\Delta[n]$ is always a Kan complex since every horn in a representable has a filler (it is the nerve of a groupoid completion). The simplicial set $\Delta[1] \sqcup_{\Delta[0]} \Delta[1]$ (two edges glued at a vertex) is NOT a Kan complex.

A map $p: X \to Y$ is a trivial fibration if and only if it has the RLP against all boundary inclusions $\partial\Delta[n] \hookrightarrow \Delta[n]$. This is equivalent to: for every $n \geq 0$ and every compatible system of $(n-1)$-simplices in $X$ over an $n$-simplex in $Y$, there exists an $n$-simplex in $X$ mapping to the given simplex in $Y$ with the given faces.

In particular, trivial fibrations are surjective on all levels: $X_n \twoheadrightarrow Y_n$ for all $n \geq 0$.

The weak equivalences (maps whose geometric realization is a weak homotopy equivalence) satisfy two-out-of-three because weak homotopy equivalences in $\mathbf{Top}$ satisfy two-out-of-three, and geometric realization is a functor: if $|f|$ and $|g|$ are weak homotopy equivalences, so is $|g| \circ |f| = |g \circ f|$.

The lifting axiom (MC4) states:

• Trivial cofibrations (anodyne extensions = monomorphisms that are weak equivalences) have the LLP against Kan fibrations.
• Cofibrations (monomorphisms) have the LLP against trivial Kan fibrations.

The first is essentially the definition of Kan fibration composed with the characterization of anodyne extensions. The second follows from the characterization of trivial fibrations via boundary inclusions.

Both factorizations are constructed via the small object argument:

1. Cofibration + trivial fibration: Given $f: X \to Y$, iteratively solve all lifting problems for $\partial\Delta[n] \hookrightarrow \Delta[n]$ by attaching cells. After $\omega$ steps, obtain $X \hookrightarrow Z \xrightarrow{\sim} Y$ where $Z \to Y$ has the RLP against all boundary inclusions.

2. Trivial cofibration + fibration: Given $f: X \to Y$, iteratively solve all lifting problems for $\Lambda^n_k \hookrightarrow \Delta[n]$ by attaching cells. After $\omega$ steps, obtain $X \xrightarrow{\sim} Z \twoheadrightarrow Y$ where $Z \to Y$ has the RLP against all horns (is a Kan fibration).

The Kan--Quillen model structure makes $\mathbf{sSet}$ a simplicial model category: it is enriched, tensored, and cotensored over itself, and the pushout-product axiom holds.

The mapping space $\operatorname{Map}(X, Y)_n = \operatorname{Hom}_{\mathbf{sSet}}(X \times \Delta[n], Y)$ is a simplicial set. When $Y$ is a Kan complex, $\operatorname{Map}(X, Y)$ is also a Kan complex.

The pushout-product axiom states: if $i: A \hookrightarrow B$ and $j: C \hookrightarrow D$ are cofibrations, then the pushout-product $A \times D \sqcup_{A \times C} B \times C \hookrightarrow B \times D$ is a cofibration, which is trivial if either $i$ or $j$ is.

The Kan--Quillen model structure is both left proper and right proper:

• Left proper: Pushouts of weak equivalences along cofibrations are weak equivalences. Since every simplicial set is cofibrant, this says pushouts preserve weak equivalences along monomorphisms.

• Right proper: Pullbacks of weak equivalences along fibrations are weak equivalences. This says pulling back a weak equivalence along a Kan fibration gives a weak equivalence.

Right properness is used extensively in the theory of Bousfield localizations and in the construction of model structures on functor categories.

The category $\mathbf{sSet}$ also supports the Joyal model structure:

• Weak equivalences: Categorical equivalences (maps inducing equivalences on fundamental $\infty$-categories).
• Cofibrations: Monomorphisms (same as Kan--Quillen).
• Fibrations: Inner fibrations between quasi-categories (maps with RLP against inner horn inclusions $\Lambda^n_k \hookrightarrow \Delta[n]$ for $0 < k < n$).

Every Kan--Quillen weak equivalence is a Joyal weak equivalence, but not conversely. The Kan--Quillen fibrant objects (Kan complexes) form a proper subset of Joyal fibrant objects (quasi-categories).

The identity functor $\mathrm{id}: \mathbf{sSet}_{\mathrm{KQ}} \to \mathbf{sSet}_{\mathrm{Joyal}}$ is a left Quillen functor.

For simplicial objects in a model category $\mathcal{M}$ (functors $\Delta^{\mathrm{op}} \to \mathcal{M}$), there is a Reedy model structure where:

• Weak equivalences are levelwise.
• Cofibrations and fibrations use matching and latching maps.

This interacts with the Kan--Quillen model structure on $\mathbf{sSet} = \mathbf{Fun}(\Delta^{\mathrm{op}}, \mathbf{Set})$ via the discrete model structure on $\mathbf{Set}$ (every map is a fibration and cofibration, isomorphisms are weak equivalences).

The Kan--Quillen model structure makes $\mathbf{sSet}$ a monoidal model category with the Cartesian product $\times$. The unit is $\Delta[0]$, and the pushout-product axiom is satisfied. This means:

• The product $X \times Y$ of a cofibrant $X$ with any $Y$ preserves weak equivalences in $Y$ (homotopy invariance of products).
• The internal hom $\operatorname{Map}(X, Y)$ preserves fibrations and trivial fibrations in $Y$ when $X$ is cofibrant.

The monoidal structure is crucial for developing the theory of simplicial categories, simplicial algebras, and operads.

Several methods produce fibrant replacements (Kan complexes weakly equivalent to a given simplicial set):

1. Kan's $\operatorname{Ex}^\infty$: Iterate the subdivision right adjoint. $\operatorname{Ex}^\infty(X)$ is always a Kan complex.
2. Small object argument: Iteratively fill all horns.
3. Localization: For quasi-categories, the maximal sub-Kan-complex $X^{\simeq} \hookrightarrow X$ extracts the underlying $\infty$-groupoid (space of objects with equivalences).

Each method has its uses: $\operatorname{Ex}^\infty$ is functorial and explicit, the small object argument is general, and localization methods are conceptually clean.