Categories on which one can determine all model structures?

Question

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?

Answer 1

If I remember correctly, about ten years ago I calculated the following: if $A$ is an Artinian commutative ring in which the square of each maximal ideal is zero, then the category of $A$-modules has exactly $5^x 11^y$ model structures, where $x$ is the number of reduced points in $Spec\ A$, and $y$ is the number of non-reduced points in $Spec\ A$.

I am Andrew Salch, who David mentioned in his answer. (Hi David, thanks for the mention.) I think the result I described in the previous paragraph is the one David is thinking of. About ten years ago I went through a phase of calculating all possible model structures on various categories each day during my daily bus ride to work. The calculation I described above was probably the most interesting one of that kind that I made. It's made by reducing the question of calculating all the pairs of compatible factorization systems on $Mod(A)$ to the question of showing the existence of a certain kind of generalized Smith normal form over $A$, then figuring out what the atomic blocks in the generalized SNF can be.

Joey Beauvais-Feisthauer has been looking over my old notes on this calculation, and maybe he and I will polish them up and turn them into something fit for public consumption.

Answer 2

Yes, this has been done in other settings. For example, Scott Balchin, Kyle Ormsby, Angélica M. Osorno, and Constanze Roitzheim wrote a paper, Model structures on finite total orders, that enumerates all model structures on a finite total order [n]. In fact, this paper is one in a series of recent papers on problems like this, usually with at least one of those four authors involved. For instance, the paper Composition closed premodel structures and the Kreweras lattice counts premodel structures and identifies model structures with certain tricolored trees. This area is starting to be called homotopical combinatorics, and at that link you can read a nice description by Kyle Ormsby about the connection between lattices and weak factorization systems.

I believe Inna Zakharevich had done earlier work counting model structures on posets, but I need to take some time to search for it. And, I think Andrew Salch told me about some work of his related to the nine model structures problem, perhaps extending it to vector spaces. I will look into my notes when I have time, and might edit this answer with more references if I find them.

EDIT: I found another paper in homotopical combinatorics. In Self-duality of the lattice of transfer systems via weak factorization systems the authors prove a correspondence between $G$-transfer systems (where $G$ is a finite group) and weak factorization systems on the poset category of subgroups of $G$. I also wanted to point out that a lot of this work has been done by undergraduates at Reed College under the supervision of Ormsby and Osorno, and I find that super impressive.

Also, I found the paper of Inna Zakharevich (and Jean-Marie Droz it turns out): Extending to a model structure is not a first-order property. As the abstract says, they "characterize all model structures where $\mathcal{C}$ is a partial order." That paper builds upon an earlier one by the same authors, Model categories with simple homotopy categories. Related work of Droz (Quillen model structures on the category of graphs) proves that there are uncountably many model structures on the particular category of graphs she works with.

Andrew Salch made me aware of these papers, and also told me that there are five model structures on the category of vector spaces (a fact that was also known to Tom Goodwillie, as the comments below this answer make clear). In the notes of my conversation with Andrew, I find several more interesting statements of the type the OP is looking for. However, looking through Andrew's webpage and arXiv, it seems he hasn't published these statements yet. Therefore, I encourage the OP, and anyone else interested in these kinds of questions, to email Andrew.

EDIT 2: Just today, Andrew Blumberg, Mike Hill, Kyle Ormsby, Angélica Osorno, and Constanze Roitzheim published a paper in the AMS Math Monthly, summarizing the new field of homotopical combinatorics. In the section Model Structures on Posets, they discuss how lattices parameterize weak factorization systems and allow for the counting of model structures. That article will surely be the most complete literature survey available at present.