I will begin with some context; the question itself is highlighted below. This is all for some notes I am writing personally on homological algebra, amongst other things.
To construct derived functors, it seems to me that the process is typically divided into three steps. Let $F\!:\mathcal{C}\to\mathcal{D}$ be some functor between homotopical categories (so, categories adorned with some reasonable class of weak equivalences).
- Find some (full) subcategory $\mathcal{C}'$ of $\mathcal{C}$ on which $F$ is well-behaved, i.e. preserves the weak equivalences.
- For each object in $\mathcal{C}$, find a replacement in terms of an object in $\mathcal{C}'$. So, find, for all $x\in\mathcal{C}$, a weak equivalence $x\to x'$ or $x'\to x$ with $x'\in\mathcal{C}'$ (depending on whether you want a right or left derived functor).
- Plug this into your choice of derived functor machine.
There are numerous choices one can make at each stage described above, each requiring slightly different conditions to work. Three choice examples I can think of:
- Assuming one can arrange for $\mathcal{C}$ and $\mathcal{D}$ to model categories for which the (co)fibrant objects of $\mathcal{C}'$ are basically given by $\mathcal{C}'$, if $F$ preserves trivial (co)fibrations between the (co)fibrant objects, one can construct a derived functor.
- If the weak equivalences on $\mathcal{C}$ form a (left or right) multiplicative system, then following Kashiwara–Schapira's book Categories and Sheaves you can produce a (left or right) derived functor of $F$ by finding exactly the data in 1 and 2.
- If you find the data in 1, and a functorial way to present the data in 2, then you can apply the theory of deformable functors (see e.g. Riehl's Categorical Homotopy Theory, chapter 2) to produce derived functors.
All three of these produce absolute total derived functors, meaning the functors $ho(\mathcal{C})\to ho(\mathcal{D})$ one obtains constitute absolute Kan extensions. Here are some thoughts about them:
- The model category approach is systematic, has nice properties, etc., but troublesome because constructing model structures is typically very hard, especially in "exotic" situations (for example, where you have to guarantee that the (co)fibrant objects are designed in advance in some less-than-nice way).
- The approach given by Kashiwara–Schapira is the one that is probably most "practical" (at least for homological algebra), as in practice one is never working with a class of weak equivalences that doesn't form a multiplicative system (both left and right). On the other hand, the proofs are a bit involved, using properties of cofinality which I find a bit cumbersome. In addition, while it is always practically satisfied, the multiplicative system assumption doesn't feel "nice" to me.
- The construction via deformations is the most aesthetically elegant, in my view. The proofs become very simple, and largely consist of "making use of what is in front of you". They also have the benefit of yielding a "point-set" level functor $\mathcal{C}\to\mathcal{D}$ instead of a functor merely on the level of homotopy categories.
I'm interested in the deformations approach, for the reasons outlined above. It has a very big downside: in practice, satisfying the requirement of having functorial replacements (as demanded by the left/right deformation) is hard. In softest possible terms, my question is
What methods are there to produce (left/right) deformations, given the data in 1 and 2 above? To be self-contained, this consists of: a functor $Q\!:\mathcal{C}\to\mathcal{C}$ whose image is contained in $\mathcal{C}'$, along with either a natural weak equivalence $Q\Rightarrow\mathbf{1}$ or $\mathbf{1}\Rightarrow Q$.
This cannot be expected to have a reasonable answer in arbitrary settings, but I feel strongly that there ought to be some machine one can stick the provided data into (in the presence of some additional assumptions, perhaps) to get a deformation. For example, I would want it to be applicable to the situation of doing homological algebra with a Grothendieck Abelian category.
I would want to avoid assumptions on the category $\mathcal{C}'$ as much as possible, although the cases I would want to apply whatever machine can be cooked up to are pretty much the standard ones (say, the construction of the derived Hom by K-injectives, and the construction of the derived tensor product by K-flats, both in the context of something like the derived category on a nice but general scheme).
For "moral" reasons, I'd also like to avoid assuming that $\mathcal{C}$ comes from some Abelian situation, but this is flexible. In the situation where this is taken to be an assumption, I'd vastly prefer it if one assumes a general triangulated category (with some conditions, perhaps), rather than something explicit like the category of chain complexes (up to homotopy).
In terms of answers, references to the literature would be very helpful, and also understandably appropriate given the broadness of the question.
There have been a few questions on here about similar topics, but they seem to have had more limited scope (for example, one of them was about the existence of K-flat resolutions; the answers there are probably helpful to people with more expertise than me).
There are also connections between this question and the existence of model structures on chain complexes in Abelian categories, and more broadly, just methods for constructing model structures at all (particularly ones with functorial factorizations; I'm aware the method for doing this is via the small object argument).
Edit: I want to clarify the question I'm asking, as it is maybe not so clear what I'm hoping for (with a lot of wishful thinking). Essentially, I'm dreaming of a scenario wherein one can check 1 and 2 in the list from the start, and in the presence of some assumptions on the category $\mathcal{C}$, deduce that one may "rigidify" everything such that it may be passed into 3.
To give a not-quite-faithful example of what I mean, if $\mathcal{C} = \mathbf{K}(\mathcal{A})$ and $\mathcal{C}'$ are the K-injectives (or K-projectives), then having resolutions by these is enough to guarantee that the resolutions can be done functorially, and form a deformation. The essence of this can be found in Spaltenstein's Resolutions of unbounded complexes (Props. 1.4 & 1.5). By generalizing what one means by K-injectives, one can also do something similar in any triangulated category (as seen here). The linked answer above about K-flat resolutions suggests that something like that can be done for those too.
Now, these examples aren't quite faithful because they rely on properties of the category $\mathcal{C}'$, and I want to avoid this as much as possible. Instead, I'd wish one could impose some practical condition on $\mathcal{C}$ (in the context of the linked answer, that assumption is set-theoretical, by assuming the chosen Abelian category is Grothendieck) which implies that given any input data 1 and 2, it can be turned into a deformation and passed into the deformations-based derived functor machine. In hindsight, it's probably very unreasonable to assume nothing about $\mathcal{C}'$, but lacking a uniform method is also very unsatisfying, and a uniform way of attacking this kind of problem is really what I'm after.
