A (weak) factorization system on a category $\mathcal{C}$ consists of a pair of classes of morphisms in the category $(L,R)$ satisfying

1. Every morphism $f:x \to y \in \mathcal{C}$ can be factored (not necessarily uniquely) as a composition $f = gh$ where $g \in R, f\in L$.
2. The morphisms in $L$ (resp. $R$) are precisely those morphisms in $\mathcal{C}$ which have a left (resp. right) lifting property w.r.t. all the morphisms in $R$ (resp. $L$).

The category $sSet$ of simplicial sets admits many interesting factorization systems/ Many of them pop up repeatedly in applications to homotopy theory and higher category theory in particular. It would be nice to have an accessible comprehensive list of useful factorization systems along with some of their notable properties.

I will say that $(L,R)$ forms an enriched factorization system w.r.t. $(L_0,R_0)$ iff for every $A \to B \in L$ and $X \to Y \in R$ we have $X^B \to X^A \times_{Y^A} Y^B \in R_0$. Observe that if we forget the self enrichment and take the (Mono,Epi) factorization on $Set$ this recovers the usual notion (i have no idea whether this term is standard or not).

I will say that a collection of morphisms $S$ generates the class $L$ (resp. $R$) if $L$ is precisely the class of all morphisms which have left (resp. right) lifting property w.r.t. all morphisms which have right (resp. left) lifting property w.r.t. $S$.

Here are a couple of examples

1. (Mono,TFib): Where Mono stands for Monomorphism and TFib stands for Trivial Fibration Properties:

   • Coincides with $(Cof, Fib \cap W)$ for the Joyal (and Quillen) model structure on simplicial sets.
   • Enriched w.r.t. (Mono,TFib).
   • $L$ is generated by the boundary inclusions $\{ \partial \Delta^n \hookrightarrow \Delta^n\}$

2. (An,KFib): Where An stands for Anodyne and KFib stands for Kan Fibration

   • Coincides with $(Cof \cap W, Fib)$ for the Quillen model structure on simplicial sets.
   • Enriched w.r.t. (Mono,TFib).
   • $L$ is generated by the horn inclusion $\{\Lambda^n_j \hookrightarrow \Delta^n \}$

3. (RAn,RFib) (resp. (LAn,LFib)): Where RAn (resp. LAn) stands for Right Anodyne (resp. Left Anodyne) and RFib stands for Right Fibration (resp. Left Fibration).

   • Enriched w.r.t. (Mono,TFib).
   • $L$ is generated by the left (resp. right) horn inclusions $\{\Lambda^n_j \hookrightarrow \Delta^n : 0 \lt j \le n \text{ (resp. } 0 \le j \lt n \text{)} \}$

4. (InnAn,InnFib): Where InnAn stands for Inner Anodyne and InnFib stands for Inner Fibration

   • Enriched w.r.t. (Mono,TFib).
   • $L$ is generated by the left (resp. right) horn inclusions $\{\Lambda^n_j \hookrightarrow \Delta^n : 0 \lt j \le n \text{ (resp. } 0 \le j \lt n \text{)} \}$

Beyond those I don't know a great deal more. Here are a couple of factorization systems some of which I know exist but have absolutely nothing useful to say about and some of which I suspect may exist:

• The factorization system $(Cof \cap W, Fib)$ for the Joyal model structure. I think the left class should be the monomorphisms which are also categorical equivalences. Do the fibrations have a nice description in general? (or do you have to assume the target is a quasi-category?). Is there a nice class of generators for the acyclic cofibrations?

• A factorization system whose left class models $n$-cofinal (resp. $n$-final) functors between quasi-categories?

• A factorization system whose right class is the minimal containing all morphisms of (arbitrary) simplicial sets which are isofibrations - and not necessarily inner fibration! (???) (I consider an edge of a simplicial set an isomorphism if it is sent to an isomorphism in the homotopy category - i.e. the left adjoint to the nerve).

Any corrections and further suggestions are very welcome of course.

As this is a big list kind of question I think it would be better to make it Community Wiki.