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added a cool reference by Brian Conrad to cohomology, which was mentioned at MathOverflow
added Jacob Lurie’s statement that on paracompact spaces nonabelian sheaf cohomology with constant coefficients coincides with the corresponding nonabelian cohomology in Top
to cohomology here
and to nonabelian cohomology here
By the way: that result is a simple consequence, of the theory of locally infinity-connected (infinity,1)-toposes if instead of looking at $X$ and the coefficients $A$ in $Sh_{(\infty,1)}(X)$ we look at them in $Sh_{(\infty,1)}(CartSp)$:
there the result follows simply by the adjunction
$Sh_{CartSp}(X, Lconst A) \simeq \infty Grpd(\Pi(X), A)$and the observation that because the nerve theorem applies on paracompact spaces, $\Pi(X)$ is $\simeq Sing(X)$, as discuss on my web at geometric realization.
Maybe I add that remark somewhere…
Maybe I add that remark somewhere…
okay, I have expanded the section a bit more.
I think I make this a blog-post now.
Zoran,
you have a query box at cohomology in the section on abelian sheaf cohomology.
Let’s sort this out here so that we can improve the entry:
I think of a derived-functor definition as a technique because it models an oo-categorical functor in terms of 1-categorical data. The entry is about hom-spaces in oo-categories and derived functors are one way to compute these, but not their fundamental definition. But we argued about this before. If we can’t agree on it, I’ll rephrase the assertion in the entry that your complaint is about.
When you say that abelian sheaf cohomology IS a derived functor you are somehow ignoring the whole point of the entire entry. The point is that there is an oo-topos such that abelian sheaf cohomology is just given by its hom-spaces, and that in the light of the long list of examples and from the nPOV, this is what the cohomology really IS.
You say it is not clear if the entry is about oo-sheaves on topological spaces or more generally: where is that not clear? It starts out talking about general oo-toposes and has an extensive list of examples over general sites. Please add a remark to the entry at the spot where you think this generality needs to be amplified further.
You see I find the universal property an easy to state property. The derived functors satisfy such property and do not need to exist in general; for example you may not have sufficiently many injective objects! Now in some situations there is an origin of their existence theorem: that is the infinity-topos origin which you discuss in the cases where it applies. The argument that there is such an origin in cases when it applies, is a very fundamental indeed, but still it is a realization of the existence problem of something what is in easy terms given by a universal property. There are generalizations of this universal property in nonabelian homological algebra of Bourn, Janelidze and others.
On the other hand, there is a minor issue about the non-paracompact case. My understanding (maybe wrong) is that the topos theoretic hom is the Čech cohomology, not the derived functor cohomology in that case, though both may exist and be different; the latter one is what is called sheaf cohomology. What do you do when the derived functor does not exist ?
For part 3: this was written many months ago when the entry was different.
My understanding (maybe wrong) is that the topos theoretic hom is the Čech cohomology, not the derived functor cohomology in that case, though both may exist and be different; the latter one is what is called sheaf cohomology.
There is the Cech oo-topos and the hypercomplete oo-topos and they give both notions of cohomologvy.
For part 3: this was written many months ago when the entry was different.
Okay, I only saw it now. All I want is that if there is something to be improved that we just imporve it and then remove these query boxes. I think query boxes are there to be reacted to and not to hang around forever.
Maybe this is not an important question, but I’m a little confused as to why it is important in the nPOV on cohomology that the infinity categories be infinity topoi. A priory it seems that for any (infinity,1)-category one may take $\pi_0$ of some hom space and call it a cohomology. In other words what properties of a higher topos make this construction have a richer or more interesting structure?
Maybe this is not an important question, but I’m a little confused as to why it is important in the nPOV on cohomology that the infinity categories be infinity topoi.
Yes, this is a good question. We have discussed this before elsewhere. Of course it is true that just forming derived hom-spaces is something available in all (oo,1)-categories.
The reasons I see for restricting the notion of cohomology to toposes are these:
This is what people mean in all examples that I know of: most definitions of cohomology that have ever been proposed can be seen to be about connected components of hom-spaces in an oo-topos. So this seems to be what “we mean” by “cohomology”.
There are some definitions of cohomology that are not equivalent to hom-spaces in an (oo,1)-topos. For instance the naive definition of Lie and topological group cohomology. But in these cases, the definitions is also not hom-spaces in a more general (oo,1)-category, either. Instead, I think these definitions as are in fact “broken” and should be discarded. For instance the wrong definition of Lie group cohomology should be replaced with the right one by Segal-Brylinski. That turns out to be given by homs in an oo-topos. (as described at group cohomology).
More abstractly, one should think about what one wants cohomology to be and to do.
One thing we want is that cohomology classes on some $X$ classify certain things over $X$. This is true in an oo-topos, as described in detail at principal infinity-bundle. For that classification to work, however, one needs pullbacks, colimits and universal colimits in the ambient oo-1-category (universality of the colimits is used in the step of the proof that every internal G-principal bundle does come from some cocycle). Now, by the oo-Giraud theorem, oo-toposes do have all limits, all colimits and universal colimits.
Another thing we want cohomology classes to be is to have a grading. As described in the entry, in the abstract topos-language this grading is induced by forming deloopings of coefficient group objects, of course. So we want a context in which all such deloopings exist. In other words, that “all groupoid objects are effective”. This is another one of the characteristics of oo-toposes by the oo-Giraud theorem.
The last of the Giraud axioms says that coproducts in an oo-topos are disjoint. I am not sure yet if we strictly need that to get a good notion of cohomology. Maybe not. But we do need all the above Giraud axioms. This is what makes oo-topos the right ambient context to define cohomology in terms of (oo,1)-categorical hom-spaces.
(This kind of discussion here should eventually be polished and moved into the entry. I might do that later, right now I am bit busy with something else).
Great answer. Thanks! At a first reading and with my limited understanding you even seem to give the raison d’être of higher topoi. At least for someone who comes with an interest in cohomology.
You see Urs when I was writing the query box my complaint was in fact different in my memory. I was in fact pointing mainly to the fact that cohomology is a derived functor of Hom, while there are also other derived functors, regardless what your view on derived functors is. Now I am more worrying about the fact that to raise the whole picture of cohomology to infinite categorical realization you need to assume some assumptions on the abelian category (in the easiest case enough injectives) what is far from generic.
Old, likely obsolete, and in particular longer query boxes could be cut and paste into an existing or new forum discussion, which can stay open, not to clutter the entry with or without a link from the nlab page.
Old, likely obsolete, and in particular longer query boxes could be cut and paste into an existing or new forum discussion, which can stay open, not to clutter the entry with or without a link from the nlab page.
Yes, so I would eventually like to remove the query box from cohomology, since I want that page to look nice.
But I also want it to be good and want to address your concerns.
You write
I was in fact pointing mainly to the fact that cohomology is a derived functor of Hom,
Okay, so that’s essentially just another way of saying (oo,1)-categorical hom-space. So there is no real diagreement here.
You continue:
while there are also other derived functors
Not sure what you are getting at here. If you are thinking of the global sections functor: that’s just the derived hom out of the terminal object, and hence it computes the cohomology of the terminal object in the oo-topos, which is the underlying space (if any), which is indeed what is going on in the traditional treatment of abelian sheaf cohomology.
You write:
you need to assume some assumptions on the abelian category
What is “the” abelian category here? Which abelian category? In abelian sheaf cohomology, if that’s what you are thinking of, the abelian category in question is that of chain complexes of sheaves, which by Dold-Kan is a subcategory of all simplical sheaves. This is a key step in BrownAHT and in some regard the whole entry cohomology is just an afterthought of this seminal article.
I added to the end of the Idea-section at cohomology the following paragraphs:
Finally we discuss why the notion of cohomology is related to that of (∞,1)-toposes in
Essentially nothing about this perspective on cohomology is really new, many aspects of it have been made explicit in the literature here and there. In fact, to some extent everything here is just an afterthought of the old seminal article
in the light of fully fledged (∞,1)-topos theory, of which it is the effectively the seed, by noticing that this article secretly discusses precisely the homotopy categories of hypercomplete (∞,1)-toposes. At the same time, to some extent everything here is also an afterthought of the theory of cohomology in 1-categorical topos theory as reviewed for instance in
- Ieke Moerdijk, Classifying Spaces and Classifying Topoi , section I.4
by noticing that the constructions on simplicial objects in toposes used there secretly precisely compute the (∞,1)-categorical hom-objects of an (∞,1)-topos as presented by the model structure on simplicial sheaves on the underlying site.
This and a list of other releated references and historical developments is given at
Then I added that section Relation to (oo,1)-topos theory, essentially using my reply to Michael in #10 above. This is still a bit stubby, but I thought these comments are worth including.
The derived functor of the tensor by a (bi)module, for example….
The derived functor of the tensor by a (bi)module, for example….
Heh? For example of what?
Are we still talking about cohomology theories?
I was thinking of this quote by Zoran
…there are also other derived functors, regardless what your view on derived functors is.
And it would be some sort of homology theory, I imagine. But you’re right, I’m straying beyond cohomology. It would be interesting to see what other derived functors turn out to be, in the context of (oo,1)-categories.
Hi David,
okay, so somehow this conversation dropped out of its original context.
It would be interesting to see what other derived functors turn out to be, in the context of (oo,1)-categories.
Okay, if you want to list examples of derived functors on (oo,1)-categories, that would be a good thing to add to the entry derived functors. But let’s not do it here in this thread, because I am hoping we can sort out some concrete issues concerning the entry “cohomology” here.
(Just to make sure: there are gazillions of examples of derived functors. It sounds a bit as if you are saying it would be interesting to see more than two examples.)
Any ideas how to respond to that point made in the final discussion about : Crystalline cohomology, rigid cohomology, syntomic cohomology? Perhaps we could find a friendly algebraic geometer to help out.
David,
we should eventually add some comments on this, but in principle there is no mystery here: all these different “cohomology theories” used in algebraic geometry are all abelian sheaf cohomology, just for different sites.
Crystalline cohomology, for instance, is obtained by replacing the ordinary site of a scheme by that of its de Rham space.
It’s this somewhat peculiar way of talking in algebraic geometry: for instance threy will say that “Grothendieck invented etale cohomology” to solve this and that conjecture. What it really means is that he cooked up a Grothendieck topology and then considered its abelian sheaf cohomology. So the general notion of cohomology here is always the same, it’s always the intrinsic cohomology of some (oo,1)-topos. What changes is which (oo,1)-topos one chooses.
created stub for crystalline cohomology
created stub for syntomic cohomology
What is “the” abelian category here? Which abelian category? In abelian sheaf cohomology, if that’s what you are thinking of, the abelian category in question is that of chain complexes of sheaves, which by Dold-Kan is a subcategory of all simplical sheaves.
No, it is not always, abelian sheaves can be sheaves with values in various abelian categories and in some of them even sheafification functor does not exist, I alude to the axioms from Osdol etc. For me abelian sheaves are not just categories of sheaves of abelian groups but categories of sheaves with values in more general abelian categories. Many abelian categories of sheaves on the other hand do not have enough injective objects. For example it is not known for a general scheme if the category of quasicoherent sheaves has enough injective objects. So it is not that clear how generally you can realize the picture with what you say. You are looking at an ideal and easy situation just abelian groups. Then you have everything. But most of the abelian sheaf cohomology is used for more complicated categories of abelian sheaves.
Zoran,
okay, it would be great if we could look at concrete examples. Let’s list cohomology theories that might not fit the pattern, and see if they really don’t.
Give me some concrete pointers to the literature, specific cohomology theories that you think cannot be realized as homs in an (oo,1)-topos. Then I can look at it.
So for the abelian examples, what do you do for abelian categories without sufficiently many injectives for example ? How do you get the infinity category ?
I miss now the last bus…
For nonabelian case get the K-theory of a Quillen exact category as a derived functor of a functor from some category of exact categories.
A classical example of a cohomology in additive but not abelian context which has not being so far successfully treated with the derived functor approach is the study of cohomologies for Banach algebras, and similar classes of algebras in functional analysis. The main work in this area is due Helemski. He had to resort to concrete resolutions to work, but never any universality property has not being successfully defined.
I should also point out that there are Čech cohomologies of quasicoherenet sheaves on noncommutative algebras, used in practice and no derived functor approach has reproduced those. There are papers for example of van Oystaeyen on those.
K-theory of Quillen exact categories is still the most interesting case which is yet not known if it is a derived functor in some context. It is true for special case of algebraic K-theory of rings (at least 2 formalisms do it).
what do you do for abelian categories without sufficiently many injectives for example ? How do you get the infinity category?
One obvious way to get an $\infty$-category is to take the category of chain complexes and homotopically invert the quasi-isomorphisms. Without enough injectives or projectives that might not be realizable as a model category, but you can still do it. Of course then there is the question of whether that realizes whatever you might mean by (co)homology. Although without enough injectives it’s not immediately obvious to me how to even define cohomology in the abelian way.
It’s this somewhat peculiar way of talking in algebraic geometry: for instance threy will say that “Grothendieck invented etale cohomology” to solve this and that conjecture. What it really means is that he cooked up a Grothendieck topology and then considered its abelian sheaf cohomology.
Although I agree with the general point, my understanding of the history of this particular example (which could be off base) is that Grothendieck essentially invented Grothendieck topologies in order to be able to then “invent” etale cohomology as the abelian sheaf cohomology of a particular Grothendieck topology. So in the particular case of Grothendieck inventing etale cohomology, the word “invented” may be more justified than in other cases! (-:
Although without enough injectives it’s not immediately obvious to me how to even define cohomology in the abelian way.
Well, there are situations where you can prove the existence of a derived functor of some particular functor by using the notion of an adapted class in triangulated category. If one has injectives, they form a class which is adapted to any functor, but the convenient thing is to go beyond it.
Some people also work with nonadditive versions of triangulated categories; I do not know if there is a way to enhance those infinitely categorically, it would be quite interesting.
threy will say that “Grothendieck invented etale cohomology” to solve this and that conjecture. What it really means is that he cooked up a Grothendieck topology and then considered its abelian sheaf cohomology
Well, the genius is not only in inventing the Grothendieck topology but in deep motivation for it exactly from that case: the insight that the theory of Galois extensions of fields and rings and the theory of covering spaces have the same common generalization; this then forced the consideration of “multivalued open subsets” that is maps from more general sets then subsets. The crudeness of Zariski topology was then naturally refined to get amazing similarity of properties of varieties in etale topology to complex analytic topology. These insights are far deeper than the very formalism of Grothendieck topology.
A classical example of a cohomology in additive but not abelian context which has not being so far successfully treated with the derived functor approach is the study of cohomologies for Banach algebras, and similar classes of algebras in functional analysis. The main work in this area is due Helemski. He had to resort to concrete resolutions to work, but never any universality property has not being successfully defined.
Okay, one day I try to track down references for this and to see what’s going on.
K-theory of Quillen exact categories is still the most interesting case which is yet not known if it is a derived functor in some context. It is true for special case of algebraic K-theory of rings (at least 2 formalisms do it).
Algebraic K-theory is about turning a stable oo-category (which may or may not come from an abelian category or Quillen exact category) into a spectrum (of which the K-group is the $\pi_0$) and that is then fed into the cohomology formalism that we are talking about.
This is incidentally the topic of the second part of BrownAHT.
For instance when describing ordinary topological K-theory in the oo-topos framework, you apply the K-functor $K : Stable (\infty,1)Cat \to Spectra$ objectwise to the $(\infty,2)$-sheaf $Vect$ on Diff that assigns the stable (oo,1)-category corresponding to the Quillen exact category $Vect(X)$ to a given $X \in Diff$ and then produces the corresponding algebraic K-spectrum. Then the corresponding K-cohomology is once again the sheaf cohomology with values in that specturm-valued sheaf.
So algebraic K-theory fits very nicely into the (oo,1)-topos picture, though it is true that the entry at the moment does not say anything about this.
Concerning the “invention” of etale cohomology:
the history notwithstanding, it deserves to be pointed out, as that last query box discussion that David C. referred to exemplfied, that all these “cohomology theories” that have and keep being “invented” in algebraic geometry are each and every one abelian sheaf cohomology, just for different sites.
Algebraic K-theory is about turning a stable oo-category (which may or may not come from an abelian category or Quillen exact category) into a spectrum (of which the K-group is the π 0) and that is then fed into the cohomology formalism that we are talking about.
Urs, you are talking about a recipe, or intuition. But the whole composition is presumably a natural construction which is not only a construction, procedure or trickery, but also a solution to a universal problem. While cyclic homology is a derived functor, Hochschild homology is a derived functor, then don’t you expect the algebraic K-theory to be as well ?
Passing to spectra is the Quillen trick which makes many top mathematicians whom I know very unhappy.
So far infinity topoi may reword taking the homotopy groups of K-theory space or K-theory spectrum if you wish, but they do not help yet grasp K-functor for exact categories as a single functor. Don’t you see what the real question is ? Not realizing something unique by nice steps which have interpretation but seeing that natural steps realize a universal thing which is fundamental in itself. Cyclic homology can also be divided in two steps (you can form a space or spectrum as an intermediate step; many people think in terms of that space) but still it is just a natural thing like Hochschild homology is. So what about K ?
that all these “cohomology theories” that have and keep being “invented” in algebraic geometry are each and every one abelian sheaf cohomology, just for different sites.
Surely they are all just partial information on motives.
that have and keep being “invented” in algebraic geometry
You are not saying anything what is not assumed in algebraic geometry from the start. All cohomologies in Grothendieck school are viewed as computing derived functors of global sections for sheaves on some site.
Rosenberg’s setup for nonabelian homological algebra is very simple and extremely general setup for right derived functors (you just have a (say regular) category with a fixed choice of subcanonical singleton Grothendieck pretopology and possibly with existance of some limits needed for various constructions). Can you explain his derived functors (the setup is so categorical and simple) by $(\infinity,1)$-categories ? The case of Banach algebras is likely to correspond to a specific choice of Grothendieck pretopology of that kind.
In the dual of the same formalism he can do the algebraic K-theory as a derived functor. But it is not known if his universal definition of K-theory indeed agrees with Quillen’s for Quillen exact categories (this depends on the vanishing of Quillen’s functor on certain resolutions, nobody done the computation so far).
All cohomologies in Grothendieck school are viewed as computing derived functors of global sections for sheaves on some site.
Having said that, one should point out that the additional structures some cohomologies posses for good projective varieties do not ocme from such easy general nonsense. For example, the Hodge structures and nonabelian Hodge structures…
Well, there are situations where you can prove the existence of a derived functor of some particular functor by using the notion of an adapted class in triangulated category. If one has injectives, they form a class which is adapted to any functor, but the convenient thing is to go beyond it.
Yes, certainly. That’s true in topology too; people talk sometimes about “deformations” or “derivable functors”. You have all sorts of classes of objects that are not quite cofibrant or fibrant, but are (co)fibrant enough for a particular functor. But I think all of those situations can still be viewed as constructing an $(\infty,1)$-functor.
But I think all of those situations can still be viewed as constructing an (∞,1)-functor.
Yes, I believe so, eventually. The real problem is with nonsymmetric situations like in Rosenberg’s work: if there were a infinity setup there in the usual sense, then one would have both left and right derived functors making sense; but his setup enables situations in which just left or just right derived make sense, while the rest of the structure is not provided. So it is morally less than infinity category in the usual sense.
if there were a infinity setup there in the usual sense, then one would have both left and right derived functors making sense
Whether or not I agree with that depends on what you mean by “making sense.” (-: Can you give a reference to the work you’re talking about?
Algebraic K-theory is about turning a stable oo-category (which may or may not come from an abelian category or Quillen exact category) into a spectrum (of which the K-group is the π 0) and that is then fed into the cohomology formalism that we are talking about.
Urs, you are talking about a recipe, or intuition.
No, I am talking about a precise stament. It’s summarized at K-theory.
But the whole composition is presumably a natural construction which is not only a construction, procedure or trickery, but also a solution to a universal problem. While cyclic homology is a derived functor, Hochschild homology is a derived functor, then don’t you expect the algebraic K-theory to be as well ?
Why do you amplify derived functors so much? The notion of derived functor is more restrictive than that of (oo,1)-functor. Not every (oo,1)-functor needs to come from a derived functor. At K-theory is recalled the entirely (oo,1)-categorical procedure for extracting the algebraic K-theory spectrum from a stable (oo,1)-category.
And apart from that, saying “comes from a derived functor” seems to be way too unspecific to me. I think in order to make progress in this discussion here, it would be good if you could be more specific. It’s very hard to reply constructively to vague statements.
The real problem is with nonsymmetric situations like in Rosenberg’s work: if there were a infinity setup there in the usual sense, then one would have both left and right derived functors making sense; but his setup enables situations in which just left or just right derived make sense,
Potentially this kind of situation is an argument in favor of a more abstract (oo,1)-categorical description than against it: the (oo,1)-functor may exist without coming from a derived functor.
Regarding algebraic K-theory, I’m surprised that its recently discovered universal property hasn’t been mentioned.
My previous long reposnse has been eaten by the computer, as the ocnnection has been lost. in nlab taht never happens as you can go page back. But if I reload page back in nForum I get empty form, what in mozilla does not happen with nlab.
the (oo,1)-functor may exist without coming from a derived functor.
The argument is that there is not enough data to make any kind of stable infinity category, while it is enough to make a good formalism of derived functors. So the wishful thinking does not help. If there is some sort of less symmetruc setup than stable maybe it is possible.
Can you give a reference to the work you’re talking about?
Again A. Rosenberg, Homological algebra of noncommutative ’spaces’ I, 199 pages, preprint Max Planck, Bonn: MPIM2008-91.
The notion of derived functor is more restrictive than that of (oo,1)-functor. Not every (oo,1)-functor needs to come from a derived functor
You see in the (oo,1)-categories the (oo,1)-functors do not need as you say coming from 1-categorical-with-structure notions of derived functor. If you are in 1-categorical with structure situations then it is possible in some cases other way around that some derived functors do not come from infinity-categorical picture (especially not the stable one). There is a big common intersection for sure, but looking from infinity world you just recognize whatmakes sense there and say, not everything here is from outside. Similar viewpoint can have somebody who looks from the point of view of structured 1-categories, not everthing they see comes from infinite categories. Now there is strong evidence that model categories are less natural while essentially equivalent to a subcase of infinity,1-categories. But if you look at more nonsymmetric setups like the nonadditive generalization of triangulated categories, or the derived functros of Rosenberg, than it seems that the notion of (infinity,1)-category should be extended in unknown direction to have a similar capacity to absorb those in full generality.
Regarding algebraic K-theory, I’m surprised that its recently discovered universal property hasn’t been mentioned.
I was not aware of this, though I would expect a similar statement to hold. Still, Rosenberg’s universality is much more general, I mean it is for functor extended from Quillen exact categories to general categories with “right exact structure” as he would put it and the universality is without any additional property; but with K_0 specified. It can be further generalized to a universal K-theory defined a class of rather general fibered categories (unpublished), without further structure but with some exactness properties.
I’m surprised that its recently discovered universal property hasn’t been mentioned.
Well, I am also surprised that I find the time to chat about K-theory at all at the moment! :-)
But thanks for the reference. I looked at Barwick’s analog a while ago, but not at this one.
Added the reference with a brief remark to K-theory and algebraic K-theory.
The argument is that there is not enough data to make any kind of stable infinity category, while it is enough to make a good formalism of derived functors.
But stable $(\infty,1)$-categories are only the place where abelian cohomology lives. Nonabelian cohomology lives in an $(\infty,1)$-topos directly, or even in a more general $(\infty,1)$-category.
My previous long reposnse has been eaten by the computer, as the ocnnection has been lost. in nlab taht never happens as you can go page back. But if I reload page back in nForum I get empty form, what in mozilla does not happen with nlab.
Until and unless Andrew manages to fix this, you could prevent it from happening in the future by composing your comments (especially long ones) in a text editor, saving as a local file, and only pasting them into the comment form when finished.
How best to relate the nPOV on cohomology with the idea described by Tao as
Global structure / local structure = cohomology ?
But stable (∞,1)-categories are only the place where abelian cohomology lives.
I was aware of the relevance of that remark. I really mean that not only the stable categories but most of the “Quillen’s formalisms” like Quillen exact categories, model categories, and now the general formalism of categories with weak equivalences, are very symmetric in a sense. Thus they allow simulteneously homology and cohomology, left and right derived functors. I advocate here, with principal example being Rosenberg’s work, for one-sided examples. While cohomology is related to a Grothendieck (pre)topology in the picture, the homology is to a copretopology. More generally, the covers form a fiber of a fibered category and one can do diagram chasing with cartesian squares; in dual picture one does cofibered (you like to say opfibered) categories with diagram chasing for cocartesian squares. A notion of weak equivalence is too symmetric to sense that distinction, I suppose. I may be wrong eventually, but so far I have not seen an argument how to treat this with weak equivalences only.
I still don’t understand what you mean by “allowing” both sides. In what sense does Rosenberg’s work not “allow” both sides? (I don’t have time to read 199 pages right now!) There are certainly functors between categories-with-weak-equivalences that only have a left derived functor, or only a right derived functor.
The important thing is after the preliminaries which you can skip in only several simple pages.
He defines a category with right exact structure, as a category with a choice of subcanonical singleton pretopology. For a functors whose domain is such a category one can define a sequence of notions completely analogous to Tohok – universal delta functors and alike.
A dual notion is a category with left exact structure.
While one of the two yields a good notion of left derived functors, another does right derived functors. It is not an existence statement. Simply the structure mentioned allows a natural notion of a left derived functor while not the right derived functor. The definition makes sense in one case and not in another. The existence is a separate question (and is solved like in classical case: for example if one has sufficiently many “projectives” in certain sense or adapted class etc).
A sentence on the page cohomology:
There is an equivalence between $(\infty,1)$-sheaves on $X$ and topological spaces over $X$, as described in detail at (∞,1)-sheaves and over-spaces
But there is no such entry. Where is that detailed description given?
This is discussed somewhere towards the end of “Higher Topos Theory”.
Presumably the author of that sentence had a location on nLab in mind when they wrote it. Perhaps they meant at (infinity,1)-category of (infinity,1)-sheaves.
I think it is really meant to point to relation to “over spaces”, since that’s the special aspect of this particular statement. (I may have written that, way back, I forget and don”t have time to dig through the entry’s history now, which is always tedious.) Best way to fix it for the time being would be to add pointer to that section in Lurie’s HTT. It should be somewhere in section 7 (sorry, no time to check).
You mentioned elsewhere that this view is contested. Of course the flip is always possible, where, if someone present a cohomology which can’t be seen in this way, we declare it to be not a cohomology, or we improve it so as to fit. The $\infty$-topos account becomes constitutive of what it is to be a cohomology.
[Such a flip is described here.]
At an earlier stage, Čech homology was declared no longer a homology theory for failing the Eilenberg-Steenrod exactness axiom. We have strong homology claiming to be its improvement. But on presenting this case of Čech homology to Barry Mazur once, he remarked “Yes, but it has made its comeback in étale cohomology”. Does anyone have a view on that?
I don’t know that it “is contested”. It has been contested, in that a person on MO somewhere may have said he isn’t sure if it’s right.
Zoran also objected, in discussions here. But two people is not overwhelming opposition.
Well…
The slogan doesn’t say that all notions of cohomology are hom-spaces in an $\infty$-topos, only that “thousands” of them are, so I think it’s okay. Although are there really literally “thousands” of kinds of cohomology? I doubt I could come up with more than “dozens” myself, and I might have to stretch to reach the plural.
Etale cohomology, various versions of algebraic K-theory, the concept of ”arithmetic vs. geometric” cohomology theories, absolute Hodge cohomology, Hodge cohomology, Amitsur cohomology, archimedean cohomology, Andre-Quillen cohomology, Betti cohomology, Borel-Moore homology, cdh cohomology, Cech cohomology, Chow groups, arithmetic Chow groups, Arakelov Chow groups, group cohomology and continuous group cohomology, crystalline cohomology, crystalline Deligne cohomology, de Rham cohomology, Deligne cohomology, Deligne- Beilinson cohomology, smooth Deligne cohomology, Eichler cohomology, elliptic Bloch groups, equivariant Deligne cohomology, etale K-theory, etale motivic cohomology, flat cohomology, Fontaine-Messing cohomology, Friedlander- Suslin cohomology, Galois cohomology, Hyodo-Kato cohomology, Lawson homology, cohomology of Lie algebras, ”log” versions of Betti, de Rham, crystalline and etale cohomology, Milnor K-theory, Kato homology, Monsky- Washnitzer cohomology, morphic cohomology, motivic cohomology, nonabelian cohomology, Nisnevich cohomology, p-adic etale cohomology, parabolic cohomology, rigid cohomology, syntomic cohomology, rigid syntomic cohomology, relative log convergent cohomology, Rost’s cycle modules, singular cohomology of arithmetic schemes, Suslin homology, Tate cohomology, unramified cohomology, Weil-etale cohomology, Zariski cohomology, and various theories with compact support. Also, various notions of motives and of mixed motives, and various other kinds of algebraic cycle groups. In addition, many of the theories come with a choice of coefficients. One could also extend the list to theories occurring in other areas of mathematics, there would then be at least a few hundreds of them. (Andreas Holmstrom, Questions and speculation on cohomology theories in arithmetic geometry)
Seriously, algebraic geometers commonly speak of one cohomology theory per site, and algebraic topologists speak of one cohomology theory per spectrum. In this sense there are uncountably infinitely many cohomology theories, all subsumed by the slogan.
Now you may say: Ah, but these two infinitudes are really just two notions of cohomology. To which the slogan replies: Sure, but once you start passing from the concrete particulars to the general abstract this way, you should go all the way and realize that in this abstract sense then even these two notions unify: Both are about hom-spaces in an $\infty$-topos!
(in $T \infty Grpd$ in the second case!).
Now you may say: Ah, but these two infinitudes are really just two notions of cohomology. To which the slogan replies: Sure, but once you start passing from the concrete particulars to the general abstract this way, you should go all the way and realize that in this abstract sense then even these two notions unify: Both are about hom-spaces in an $\infty$-topos!
But that is irrelevant to the question of how to count the number of “notions of cohomology” that the one unifying perspective is being compared to, since for the slogan to have any teeth there must be more than one of them.
Why should the type of notions of cohomology be a set? [Added: Clearly it’s a contractible type, hence ’the’.]
I would argue that a “notion” of cohomology is a syntactic or linguistic construct, hence form a non-contractible set.
If we see cohomology arise from $Hom_{\mathbf{H}}(X, A)$, then there’s a single dependency, but we might reflect on different classes that the variables range over. $\mathbf{H}$ might belong to tangent $(\infty, 1)$-toposes, cohesive $(\infty, 1)$-toposes, etc. $A$ might be a general object, stable object, etc.
With the fully general definition, you might think the term ’cohomology’ would vanish. I’m unlikely to say that when we take the powerset of a set we’re considering cohomology with coefficients in the subobject classifier.
Do the chow groups in algebraic geometry have an interpretation as coming from hom groupoids in an $(\infty, 1)$-category?
I am not expert on the algebraic geometry involved, but I gather (and this seems to be confirmed by what it says in the entry motivic homotopy theory, search for “Chow group” there) that Chow groups are realized as homotopy hom-sets in the $\infty$-topos over the Nisnevich site.
(This is traditionally stated, instead, in the $\mathbb{A}_1$-localization of that $\infty$-topos, which is not itself an $\infty$-topos anymore. But since this localization is a left adjoint, it equivalently gives a corresponding statement in the Nisnevich $\infty$-topos.)
I didn’t know that the term originates with Whitney, interesting.
Trying to look at the article, I noticed that the doi link
for Whitney 37 does Not work. Even though that’s indeed the link given by Project Euclid here.
Probably there is a single wrong character in the doi. But for the moment I have fixed it by instead linking to the url of the Euclid page.
Further up, in the first line of the References-section, I changed “The original references” to
The original references on cochain cohomology and ordinary cohomology:
Finally, I made the author name Andrei Kolmogoroff of the first articles listed there redirect to the existing entry Andrey Kolmogorov.
Hoping that this is correct? The probability-theory Kolmogorov also invented (cochain) cohomology?!
The probability-theory Kolmogorov also invented (cochain) cohomology?!
Yes, and cup product too. (But not the singular cohomology, that was done by Eilenberg much later.)
I see, thanks. What Kolmogoroff/Alexander had is more like cellular (co)homology, I suppose. (Have only glanced at some of these old articles).
We should put this into an !include
-entry cohomology – early references so that it’s easer to include the pointers also at singular cohomology and elsewhere.
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