Example 9.2.6.2. Let $\operatorname{\mathcal{C}}= \operatorname{Set_{\Delta }}$ be the category of simplicial sets. We have already encountered several examples of weak factorization systems $(S_{L}, S_{R} )$ on $\operatorname{\mathcal{C}}$:
We can take $S_{R}$ to be the collection of Kan fibrations and $S_{L}$ the collection of anodyne morphisms (Proposition 3.1.7.1).
We can take $S_{R}$ to be the collection of inner anodyne morphisms and $S_{L}$ the collection of inner fibrations (Proposition 4.1.3.2).
We can take $S_{R}$ to be the collection of left fibrations and $S_{L}$ the collection of left anodyne morphisms (Proposition 4.2.4.7).
We can take $S_{R}$ to be the collection of right fibrations and $S_{L}$ the collection of left anodyne morphisms (Variant 4.2.4.8).
We can take $S_{R}$ to be the collection of trivial Kan fibrations and $S_{L}$ the collection of monomorphisms (Exercise 3.1.7.11).