Derived functors and functorial resolutions/(co)fibrant replacements

Question

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).

1. Find some (full) subcategory $\mathcal{C}'$ of $\mathcal{C}$ on which $F$ is well-behaved, i.e. preserves the weak equivalences.
2. 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).
3. 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.

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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).

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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).

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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.

Answer 1

Let me expand on my comments. Functorial deformations for $\mathbf{K} (\mathcal{A})$ (the chain homotopy category) are much easier to obtain in practice than for $\textbf{Ch} (\mathcal{A})$ because K-injective / K-projective resolutions have a genuine universal property in $\mathbf{K} (\mathcal{A})$. This is easily generalised to the non-additive situation: given a category $\mathcal{C}$ with weak equivalences $\mathcal{W}$, say an object $Z$ in $\mathcal{C}$ is $\mathcal{W}$-local iff, for every morphism $f : X \to Y$ in $\mathcal{W}$ and every morphism $h : X \to Z$ in $\mathcal{C}$, there is a unique morphism $g : Y \to Z$ in $\mathcal{C}$ such that $h = g \circ f$.

For example, a K-injective chain complex is precisely a local object in $\mathbf{K} (\mathcal{A})$ (with respect to quasi-isomorphisms). The choice of ambient category matters: there are fewer local objects in $\textbf{Ch} (\mathcal{A})$ (with respect to quasi-isomorphisms) than in $\mathbf{K} (\mathcal{A})$, whereas every object in $\mathbf{D} (\mathcal{A})$ is local (because we have already forced all quasi-isomorphisms to be isomorphisms).

As a non-additive example, consider the category $\mathbf{K} (\textbf{sSet})$ of simplicial sets modulo simplicial homotopy: then Kan complexes are local with respect to weak homotopy equivalences. (Note: the converse is false. Clearly, any simplicial set that is isomorphic in $\mathbf{K} (\textbf{sSet})$ to a Kan complex is also local with respect to weak homotopy equivalences, but such a simplicial set need not be itself a Kan complex. This is akin to the difference between K-injective chain complexes and dg-injective chain complexes.)

It is unclear to me how far we can usefully generalise this, but formally, if $\mathcal{M}$ is a model category and $\mathcal{V}$ is the class of trivial fibrations in $\mathcal{M}$, then fibrant objects are local in $\mathcal{M} [\mathcal{V}^{-1}]$ with respect to weak equivalences. Anyway, the point is that one applies this theory not to $\mathcal{M}$ (i.e. the analogue of $\textbf{Ch} (\mathcal{A})$ or $\textbf{sSet}$) itself but some category halfway between $\mathcal{M}$ and the (free) localisation of $\mathcal{M}$ with respect to weak equivalences.

Anyway, let me get back to the theory. A $\mathcal{W}$-local resolution of an object $X$ in $\mathcal{C}$ is a $\mathcal{W}$-local object $\hat{X}$ with a $h : X \to \hat{X}$ in $\mathcal{W}$. $\mathcal{W}$-local resolutions are unique up to unique isomorphism when they exist. More precisely:

Proposition. Let $X$ be an object in $\mathcal{C}$. If $X$ admits a $\mathcal{W}$-local resolution $(\hat{X}, i)$, then:

• $(\hat{X}, i)$ is an initial object in the full subcategory of ${}^{X /} \mathcal{C}$ spanned by the $\mathcal{W}$-local objects.

• The full subcategory of ${}^{X /} \mathcal{C}$ spanned by the $\mathcal{W}$-local resolutions of $X$ is a contractible groupoid, i.e. equivalent to $\mathbf{1}$.

We therefore get functoriality "for free". Even better:

Proposition. If every object in $\mathcal{C}$ admits a $\mathcal{W}$-local resolution, then the full subcategory of $\mathcal{W}$-local objects is a reflective subcategory of $\mathcal{C}$.

What is the significance of this reflective subcategory? Well, say a morphism $f : X \to Y$ is a $\mathcal{W}$-local equivalence iff, for every $\mathcal{W}$-local object $Z$ and every morphism $h : X \to Z$, there is a unique morphism $g : Y \to Z$ such that $h = g \circ f$. Clearly, every morphism in $\mathcal{W}$ is a $\mathcal{W}$-local equivalence, and every $\mathcal{W}$-local object in $\mathcal{C}$ is also local with respect to $\mathcal{W}$-local equivalences. Moreover:

Proposition. If every object in $\mathcal{C}$ admits a $\mathcal{W}$-local resolution, then the full subcategory of $\mathcal{W}$-local objects is (equivalent to) the (free) localisation of $\mathcal{C}$ with respect to $\mathcal{W}$-local equivalences.

Proposition. Every (full and replete) reflective subcategory of $\mathcal{C}$ arises as the full subcategory of $\mathcal{W}$-local objects for some $\mathcal{W}$. Furthermore, given the reflective subcategory, $\mathcal{W}$ can be chosen so that:

• A morphism is in $\mathcal{W}$ if and only if it is a $\mathcal{W}$-local equivalence.

• Every object in $\mathcal{C}$ has a $\mathcal{W}$-local resolution.

So what are deformations really about? I would say that a right deformation is a generalised reflector, and (therefore) a right deformation retract is a generalised reflective subcategory. The natural affinity between deformations and absolute Kan extensions generalises the fact that adjoints are absolute Kan extensions. Once you realise this, it is not so hard to figure out how to remove the functoriality requirements in the DHKS definition.

In the homological algebra setting where $\mathbf{D} (\mathcal{A})$ is often a (co)reflective subcategory of $\mathbf{K} (\mathcal{A})$, deformations are still useful because they enable us to lift the (co)reflective subcategory structure up to $\textbf{Ch} (\mathcal{A})$. I have the impression that until the late 1960s or so, $\mathbf{K} (\textbf{sSet})$ was also often used, but these days it is barely mentioned and the fact that $\mathbf{K} (\textbf{sSet}) \to \operatorname{Ho} \textbf{sSet}$ is a left adjoint (with fully faithful right adjoint) seems rarely remarked upon.