When a model category with prescribed homotopy category exists?

Question

My question is probably insanely hard or very well-studied, but I could not find an answer, so I will ask it here.

Assume that we have a suitably complete closed module over $\operatorname{Ho}(sSet).$ When is this category a homotopy category of a model category (with this module structure)? Clearly, there are concreteness constraints for model categories "with classifying objects" (see, e.g., this paper). However, does it mean, for example, that concrete categories can't be realized as homotopy categories? Does the problem simplify if we replace model categories with relative categories or $(\infty,1)$-categories? Maybe this is well understood for derived categories?

I understand that if we drop the assumption that there is this extra structure (like module structure or triangulation), we can take the trivial model structure. But I have no idea how to recover this extra stuff.

If this is well-studied, please point me toward some literature.

Answer 1

Let me see if I understand the question. If you start with one of the following pieces of data:

• a model category
• a relative category
• an $(\infty,1)$-category

you can extract a homotopy category plus possibly some extra data, such as

• a tensoring over the homotopy category
• a triangulated structure

and your question is about what the image of this "jazzed-up homotopy category" functor might be.

You also have some question about concreteness of homotopy categories which seems perhaps a bit orthogonal to this other family of questions, about jazzing up the homotopy category? If it's more closely related than I'm understanding, perhaps you could clarify, but for now I'll ignore this question.

Here are some thoughts:

• A relative category $C$ always presents an $(\infty,1)$-category, and every $(\infty,1)$-category arises this way. So these two versions of the question are equivalent. At this level of generality, you can talk about an enrichment of $Ho(C)$ in $Ho(sSet)$, but not a tensoring like you describe.

• A combinatorial model category always presents a presentable $(\infty,1)$-category, and every presentable $(\infty,1)$-category arises in this way. So if "model category" is taken to mean "combinatorial model category", then the answer will be the same as for "presentable $(\infty,1)$-category. If you're particularly interested in non-combinatorial model categories, then you might want to clarify some set-theoretical subtleties about what you want to ask.

• I don't know much about lifting an enrichment in the homotopy category to an $\infty$-category structure. But for the "triangulated category" version of the question, you might be interested in Muro, Schwede, and Strickland's work on triangulated categories which do not lift to the structure of stable $\infty$-category.

• You might be interested in the discussion here which I've been meaning to polish into something publishable for awhile...