Famously, there are exactly nine model structures on the category of sets, which are detailed here. In this case, one can exhaustively determine all six weak factorization systems and then see which ones fit together into model structures.

Question 1: Are there other categories for which this program can be carried out? How many model structures does one have in these cases?

For instance,

• pointed sets?

• Some simple presheaf categories like the category of maps of sets? Or $G$-sets for a group $G$?

• Vector spaces over a division ring $k$?

• Chain complexes over $k$ (of fixed length? Bounded above / below? Unbounded?)

• variants on the two previous where $k$ is a bit more complicated? Maybe abelian groups?

• …

Question 2: Is there an example of a complete and cocomplete category for which it’s possible to enumerate all model structures, but not feasible to enumerate all weak factorization systems?

Famously, there are exactly nine model structures on the category of sets, which are detailed here. In this case, one can exhaustively determine all six weak factorization systems and then see which ones fit together into model structures.

Question 1: Are there other categories for which this program can be carried out? How many model structures does one have in these cases?

For instance,

• pointed sets?

• Some simple presheaf categories like the category of maps of sets? Or $G$-sets for a group $G$?

• Vector spaces over a division ring $k$?

• Chain complexes over $k$ (of fixed length? Bounded above / below? Unbounded?)

• variants on the two previous where $k$ is a bit more complicated? Maybe abelian groups?

• …

Question 2: Is there an example of a complete and cocomplete category for which it’s possible to enumerate all model structures, but not feasible to enumerate all weak factorization systems?