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The following leftover discussion was still sitting at derived functor. I have removed it there to copy it here. I think the main point of the discussion (that the correct way to think of derived functors is as ordinary functors applied to resolutions) is by now well represented in the entry in the section that discusses the relation to -functors. If anyone finds the discussion below contains an aspect that needs to be included in the main entry text outside a query box– then please do so!!
Mike: I personally hold the opinion that it is better to define a derived functor to be the result of applying to a fibrant and/or cofibrant replacement of some sort. I have come to view the fact that sometimes (when a fibrant or cofibrant replacement alone suffices) this happens to be a left or right Kan extension as merely an accident, which is of little practical or conceptual use. I think this is borne out by the fact that these Kan extension are not “pointwise,” that they are basically the only non-pointwise Kan extensions that I have seen arise anywhere in mathematics, and that many of the nice properties of Kan extensions apply only to pointwise ones (in fact, Kelly defines Kan extension to refer only to pointwise ones). But I would be very interested to hear alternate points of view.
Zoran Skoda: This is very interesting; maybe it suggests that one needs to choose sort of factorization system or something, so that still thing has some universal property: among functors of special kind with respect to the factorization system of a sort. I would like to hear your examples and insight in more detail. Surely in homological algebra osmetimes one does not have sufficiently many projective or injectives so there is an appeal in something universal. Most interestingly, satelites make sense sometimes when derived functors don’t but still satelites in very general (nonabelian) context are themselves still sort of Kan extensions. I do not understand why total derived functor would have kind of universality as one step in forming it has.
Tim: The ’homotopy Kan extensions’ are weighted Kan extensions and can handle quite a few of the ’derived functor’ cases. As they are enriched versions they seem to generalise the pointwise case, yet manage to do more. Perhaps that neaeds airing as an idea. I suspect it does not handle all the derived functors we might want.
Mike: It’s not so much that I have lots of examples of derived functors that are not Kan extensions, as that I think the fact that they are Kan extensions is irrelevant and misleading. Can you point to any situation in which the fact that a derived functor is a Kan extension is important or useful? In my experience whenever you want to prove something about a derived functor, you need to invoke its construction as a composite of the original functor with fibrant and/or cofibrant replacement.
(The observation that sometimes the functor you are taking a derived functor of is itself a “Kan extension” functor (that is, a functor on a diagram category which takes a diagram to its Kan extension), so that the derived functor is a “homotopy Kan extension” functor, is a completely different question from whether the derived functor is a Kan extension along the localization functor. But maybe I misunderstood what Tim said.)
That said, here is a general situation in which you get derived functors that are not necessarily Kan extensions. Let be a monoidal model category and be -model categories. Then any -enriched functor , whether or not it has an adjoint or preserves cofibrations or fibrations, has a “derived functor” represented by the composite , where is a cofibrant and fibrant replacement functor on . This is because in a -model category, weak equivalences between fibrant-cofibrant (“bifibrant”?) objects are necessarily “-equivalences,” which are preserved by all -functors. (When is , , or then -equivalences are homotopy equivalences, simplicial homotopy equivalences, or chain homotopy equivalences, respectively.)
Such “derived functors” are not Kan extensions because does not come equipped with a single weak equivalence to or from the identity functor, but rather a zigzag (or ). If you generalize the definition of “derived functor” to allow replacement by such zigzags, there is no longer any universal property or uniqueness, since left and right derived functors are now both examples and they don’t need to be the same. But for that very reason, this generalization can be important in proving commutation relations between left and right derived functors. See my paper, sections 4 and 17-18, and May-Sigurdsson for some applications (in more classical language).
I don’t know what a “satellite” is.
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continuation from previous entry
Zoran Cartan-Eilenberg book has the whole chapter on satelites. If you have derivd functors then they can be expressed via satelites; but satelites may exist even when the functor is not half exact. It is also instructive to look at satelites in nonabelian setup, for eample the small note by Janelidze, from 1970-s reprinted on the arXiv.
Mike: Thanks! Looking at the definition in Cartan-Eilenberg, it seems that the left satellites of an additive functor are the homology of applied to a projective resolution, and the right satellites are the homology of applied to an injective resolution. These make sense and are homotopy invariant for any additive functor , and apparently if is ’half exact’ (preserves exactness in the middle of short exact sequences) then its left and right satellites fit together into a bi-infinite long exact sequence, generalizing the more familiar cases of left and right derived functors.
Now there are two (well, more than that, but two that seem most relevant) model structures on the category of chain complexes: the projective one, where projective resolutions are cofibrant and every object is fibrant, and the injective one, which is dual (although harder to construct, because of the non-duality between cofibrant generation and fibrant generation). Thus, it seems that the left (right) satellites are the homology of the generalized derived functor with respect to the projective (injective) model structure, which turns out to actually be a left (right) derived functor since every object is fibrant (cofibrant). Does that make sense?
Toby: Excuse me, but maybe this discussion could use a link: satellite.
end of old query box content
I added a section In homological algebra with more details on how the use of terminology of derived functors in homological algebra relates to the general use.
Also added to more items in the Examples-section
I have expanded the In homological algebra section a bit, to include a discussion of the long exact sequences that are traditionally used to introduce derived functors.
The section previously contained the assertion that if is left exact, then is right Quillen, but I don’t see why that should be — in particular, I don’t see why would necessarily have a left adjoint. So I tried to explain why I think still has derived functors even without this conclusion.
I have touched the formatting at derived functor a little.
Since the Idea-section was (and is) long, with plenty of information, I divided it into three subsections now, to make information easier to find.
The main deficit of the page is that amongst the discussion of derived functors in the general sense, it will leave the standard consument of homological algebra literature a bit unsatisfied. So I am now splitting off an entry derived functor in homological algebra for dedicated discussion of that case.
Shouldn’t the -categorical derived functor also be a Kan extension, with homotopy categories replaced by presented -categories?
A homotopy Kan extension, you mean? Good question, I don’t know.
A naïve question, perhaps: Why don’t we require that total derived functors (in the sense of Quillen and Verdier) be absolute Kan extensions? After all, this is something we get for free whenever we have a model category with functorial factorisation.
But not in general right? The point of derived functors is that their definition and existence does not depend on whether or not the category with weak equivalences is or can be equipped with a suitable model category theory.
And also, as I think discussed in the entry, the fact that derived functors are Kan extensions at all is kind of a red herring. This is not really what captures their conceptual meaning, and it is amost never referred to/used in practice.
Urs, I think that the Lin’s question was about the difference between the Kan extension and absolute Kan extension. It is far easier to do proofs with absolute Kan extensions, and virtually all known examples of derived functors a la Kan extensions are absolute.
But not all of them are, are they?
The question was why we don’t make that the definition of derived functors. And my answer was that it not only misses cases but also is not a natural conceptual characterization.
I think the Kan extension definition is perfectly serviceable conceptual characterisation, but then again I am a mere mortal uninitiated in -thinking. Anyway, I am specifically asking about total derived functors in sense of Quillen and Verdier. I know there are other definitions, but I want to start with the one that requires the least machinery. (I suppose the next one up in the simplicity hierarchy would be the DHKS definition via homotopical Kan extensions.)
It appears to me that total derived functors computed using not necessarily functorial deformation retracts are always absolute Kan extensions. This includes the classical case of Quillen adjunctions with old-style model categories as well as closely related things like two-variable Quillen adjunctions. Once we know that they are absolute Kan extensions, the existence of derived adjunctions is just abstract nonsense – no need to muck about with bijections of homotopy classes of morphisms as in [Dwyer and Spaliński].
Actually, there’s also an abstract nonsense reason why derived functors of Quillen functors are absolute Kan extensions, but I haven’t figured out a proof that doesn’t use deformation retracts or Ken Brown’s lemma. It comes down to this. Let be a model category and let be the class of trivial fibrations in . Then there is a canonical comparison functor , and of course, any right Quillen functor factors through in a unique way. Moreover, the 2-dimensional universal property of guarantees that any (left or right) Kan extension along of a functor that inverts trivial fibrations is also a (left or right) Kan extension along of the corresponding functor . Now something amazing happens: the canonical comparison functor has a right adjoint, induced by fibrant replacement. So left Kan extensions along exist and are absolute.
Another abstract reason for the existence of derived adjunctions (which also uses deformation retracts) is that passage to derived functors is a double pseudofunctor.
Since I mentioned the double-categorical functoriality of derived functors as an example of quintets, I also added a section about functoriality to derived functor.
Hinich posted a new definition of derived functors on arXiv: https://arxiv.org/abs/1811.12255
This looks really interesting. The existing definitions look like computational formulas to me, except for the one with Kan extensions (which discards important information, though), but this looks very conceptual and apparently has nice properties.
Can you summarize the definition for the lazy rest of us?
It’s a 6 page paper, but luckily
Our approach can be explained in one sentence.
This is:
Convert the functor you want to derive into a (co)cartesian fibration over , then localize.
Hinich suggests to encode functors as cartesian or cocartesian fibrations over {0→1}. The domain of such a fibration is equipped with weak equivalences coming from both source and target. Then we invert these weak equivalences up to a homotopy. If the resulting object is still a cartesian or cocartesian fibration, the functor it classifies is the right or left derived functor.
added pointer to this new textbook:
and am also adding this to abelian sheaf cohomology
A comment by Cisinski:
the nLab is also a wiki: that means it is regularly outdated and needs refreshing. In particular when it comes to derived functors, nowadays.
Any ideas from the discussion as to what might be adjusted?
After the OP recalls that cohomology is the homotopy groups of a derived hom space (which is what I have been advertizing in the entry cohomology and in other writing), that MO discussion takes twists and turns that don’t make sense to me.
It starts with me not understanding the question on boldfaced “orthogonal” versus “tied together”, which would seem to make more sense on Philosophy.SE.
Sometimes reading MO is a terrible way of sorting things out. Actually MO is also a question & answer site, which means that it regularly hosts ill-formed questions and stray answers.
Yes, one might have hoped better from those concerned.
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