The category $\mathbf{Top}$ of compactly generated weak Hausdorff spaces has a model structure:

• Weak equivalences: Weak homotopy equivalences ($\pi_n$-isomorphisms for all $n$ and all basepoints).
• Fibrations: Serre fibrations (RLP with respect to $D^n \hookrightarrow D^n \times [0,1]$).
• Cofibrations: Retracts of relative CW inclusions.

Every object is fibrant. Cofibrant objects are retracts of CW complexes. The homotopy category is the classical homotopy category of CW complexes.

The category $\mathbf{sSet}$ has the Kan--Quillen model structure:

• Weak equivalences: Maps inducing isomorphisms on all homotopy groups (after geometric realization).
• Fibrations: Kan fibrations (RLP with respect to horn inclusions $\Lambda^n_k \hookrightarrow \Delta[n]$).
• Cofibrations: Monomorphisms (levelwise injections).

Every object is cofibrant. Fibrant objects are Kan complexes.

The category $\mathbf{Ch}_{\geq 0}(R)$ of non-negatively graded chain complexes of $R$-modules has a model structure:

• Weak equivalences: Quasi-isomorphisms (maps inducing isomorphisms on homology).
• Fibrations: Epimorphisms in positive degrees (surjective on $C_n$ for $n \geq 1$).
• Cofibrations: Monomorphisms with projective cokernel in each degree.

This is connected to $\mathbf{sSet}$ via the Dold--Kan correspondence: $\mathbf{Ch}_{\geq 0}(\mathbb{Z}) \simeq \mathbf{sAb}$ (simplicial abelian groups).

The category $\mathbf{Cat}$ of small categories has the Thomason model structure:

• Weak equivalences: Functors $F: \mathcal{C} \to \mathcal{D}$ such that $|N(F)|: |N(\mathcal{C})| \to |N(\mathcal{D})|$ is a weak homotopy equivalence.
• The fibrations and cofibrations are determined by the adjunction with $\mathbf{sSet}$.

This model structure makes $\mathbf{Cat}$ Quillen equivalent to $\mathbf{sSet}$ and $\mathbf{Top}$. Every homotopy type is represented by a category.

Any bicomplete category has a trivial model structure: weak equivalences are isomorphisms, every morphism is both a fibration and a cofibration. The homotopy category is the category itself.

This shows the axioms are satisfiable but the resulting homotopy theory is uninteresting. The choice of weak equivalences is what makes a model structure meaningful.

In $\mathbf{Top}$, the factorization axiom gives:

• Any map $f: X \to Y$ factors as $X \hookrightarrow Z \xrightarrow{\sim} Y$ (cofibration followed by trivial fibration). This is the mapping cylinder construction: $Z = M_f = (X \times [0,1]) \sqcup_f Y$.
• Any map $f: X \to Y$ factors as $X \xrightarrow{\sim} P \twoheadrightarrow Y$ (trivial cofibration followed by fibration). This is the path space construction: $P = X \times_Y Y^{[0,1]}$ (the homotopy fiber).

In $\mathbf{sSet}$, factorizations are constructed by the small object argument:

• For cofibration + trivial fibration: attach cells to kill all horn-lifting obstructions.
• For trivial cofibration + fibration: attach cells to fill all horns simultaneously.

The small object argument produces functorial factorizations, which is technically important for many constructions.

In $\mathbf{sSet}$, every simplicial set $X$ needs a fibrant replacement -- a Kan complex weakly equivalent to $X$. Kan's $\operatorname{Ex}^\infty$ functor provides one: $\operatorname{Ex}^\infty(X) = \operatorname{colim}_n \operatorname{Ex}^n(X)$, where $\operatorname{Ex}$ is the right adjoint of the barycentric subdivision $\mathrm{sd}$.

The map $X \to \operatorname{Ex}^\infty(X)$ is a weak equivalence, and $\operatorname{Ex}^\infty(X)$ is a Kan complex. Another approach uses the small object argument to fill all horns.

In $\mathbf{Ch}_{\geq 0}(R)$, a cofibrant replacement of a chain complex $C$ is a quasi-isomorphism $P \xrightarrow{\sim} C$ where $P$ is a complex of projective modules. This is precisely a projective resolution of $C$.

This shows that the classical notion of projective resolution is a special case of cofibrant replacement in a model category.

In a model structure on $\mathbf{Ch}(R)$ where fibrations are epimorphisms, a fibrant replacement of $C$ is a quasi-isomorphism $C \xrightarrow{\sim} I$ where $I$ is a complex of injective modules. This is an injective resolution.

Derived functors in homological algebra are thus derived functors in the model categorical sense.

In $\mathbf{sSet}$, the cylinder object for $X$ is $X \times \Delta[1]$. The two inclusions $X \hookrightarrow X \times \Delta[1]$ (at vertices $0$ and $1$) give the cofibration $X \sqcup X \hookrightarrow X \times \Delta[1]$, and the projection $X \times \Delta[1] \to X$ is a weak equivalence (in fact a simplicial homotopy equivalence).