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June 26, 2009

Cohomology and Homotopy

Posted by David Corfield

In posts and this nnLab entry, Urs has been promoting his view of cohomology as about Hom spaces between objects in certain settings, where the unknown space is on the left. Similarly homotopy is where the unknown space is on the right. This got me thinking the following thoughts during some quiet moments in a conference this morning.

  • Cohomology of XX concerns Hom(X,A)Hom(X, A), for different choices of AA.
  • Homotopy of XX concerns Hom(B,X)Hom(B, X), for different choices of BB.

We often choose AA to be homotopically simple, i.e., we want Hom(S m,A)Hom(S^m, A) to be simple, where S mS^m is cohomologically simple. E.g., AA could be an Eilenberg-MacLane space K(G,n)K(G, n).

We often choose BB to be cohomologically simple, i.e., we want Hom(B,K(G,n))Hom(B, K(G, n)) to be simple, where K(G,n)K(G, n) is homotopically simple. E.g., BB could be a sphere S mS^m.

These choices work because Hom(S m,K(G,n))Hom(S^m, K(G, n)) is a kind of pairing where the result is trivial for nonzero mm unless m=nm = n.

On the one side we have cell complexes built up from attaching copies of spheres; on the other we have spaces built up as Postnikov towers out of Eilenberg-MacLane spaces.

Do we understand why Eckmann-Hilton duality breaks down?

The situation is a bit like where we have a basis for a vector space paired with the dual basis for the dual vector space. Here, however, the spheres have 1 parameter where the E-M spaces have 2 parameters. Is this why cohomology feels simpler than homotopy? Or is this also to do with the complexity of cell attachment versus complexity of building of Postnikov tower?

Just as there is arbitrariness on the choice of a basis in a vector space, could there be another pair of collections of spaces which could form a ‘duality’? In a quotation here Hatcher talks about

…what would happen if we dualized the notion of a Postnikov tower of principal fibrations, where a space is represented as an inverse limit of a sequence of fibers of maps to Eilenberg–MacLane spaces. In the dual representation, a space would be realized as a direct limit of a sequence of cofibers of maps from Moore spaces.

A Moore space is one with only nontrivial homology GG in dimension nn. Was homology rather than cohomology forced here? Will that blow the chance of a pairing with homotopically simple spaces? Or would it pair with cohomotopically simple spaces?

But might there not be a very different looking pairing between sets of spaces, such that Hom spaces between them are only nontrivial if parameters match?

Better get back to my conference now.

Posted at June 26, 2009 1:08 PM UTC

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Re: Cohomology and Homotopy

I once asked a question like this, moore spaces are just two spheres attached by the appropriate maps, i believe. This is why cohomotopy is not the most interesting, i am told. not sure if that is helpful.

Posted by: sean on June 26, 2009 2:37 PM | Permalink | Reply to this

Re: Cohomology and Homotopy

In posts and this nLab entry, Urs has been promoting his view of cohomology as about Hom spaces between objects in certain settings

While I do admit to be fond of this fact, it would be too much of an honor to call this “my” point of view.

This fact was established 35 years ago in

Kenneth Brown, [[Abstract homotopy theory and generalized sheaf cohomology]] and apparently known in one form or other before that.

This article establishes that

all of [[(ordinary, abelian) sheaf cohomology]] (with all its special cases like, say, [[Deligne cohomology]]),

all [[generalized Eilenberg-Steenrod (abelian) cohomology]] (with simple things like [[“ordinary cohomology”]] and more intricate things like [[K-theory]] and [[tmf]]),

as well as [[nonabelian cohomology]] (classifying [[gerbes]] and [[principal \infty-bundle]])

such as all the more mundane special cases of this like [[group cohomology]] and, yes, cohomology of cochain complexes itself

are naturally special cases of one single concept: that of hom-sets

H(X,A):=Ho SSh(X,A) H(X,A) := Ho_{SSh}(X,A)

in the [[homotopy category]] of [[\infty-groupoid]] valued [[sheaves]].

The only fundamental new addition to this insight that we have now is that

These categories H=Ho SShH = Ho_{SSh} are precisely the [[hom-wise decategorification]] of [[(,1)(\infty,1)-categories of (,1)(\infty,1)-sheaves]] otherwise known as the [[(,1)(\infty,1)-toposes]] of [[\infty-stacks]].

This is propositon 6.5.2.1 in Jacob Lurie’s [[Higher Topos Theory]] and builds on the fundamental work by K. Brown, Joyal and Jardine and others on the [[model structure on simplicial presheaves]].

For a motivation of these definitions from the point of view of cohomology as a homotopy hom-set of \infty-stacks see for instance the introductory pages of

Dan Dugger, Sheaves and homotopy theory .

In our work that we used to discuss here we traditionally considered categories of \infty-groupoids [[internal]] to [[diffeological spaces]] and using the somewhat ideosyncratic term “[[anafunctors]] ” for the morphisms between them, in particular for talking about non-abelian cohomology in terms of such anafunctors.

Since diffeological spaces are just sheaves (concret ones, but that’s not important here) and since \infty-groupoids internal to sheaves are just sheaves with values in \infty-groupoids, and since finally “anafunctors” from XX to AA are precisely the representatives of the elements in Ho SSh(X,A)Ho_{SSh}(X,A), this is the same in slightly different words.

So not only is this point of view not one I made up, but it is actually a point of view that used to be at the center of attention here, if maybe in somewhat different words than in other schools of research out there.

Posted by: Urs Schreiber on June 26, 2009 4:09 PM | Permalink | Reply to this

Re: Cohomology and Homotopy

The n-Lab and the n-Cafe have a nice population of “experts” now that is pretty amazing to observe for a non-expert like myself. The n-Lab, in particular, is quickly becoming a valuable reference.

One side effect that hasn’t been too visible aside from my occasional piping in is that the totally public forum means that there are likely curious high school students and certainly undergraduate students watching from the sidelines.

This perspective that Urs is promoting of something that has been known in closed circles for decades is very pretty. I think that I might even be able to understand it! I bring this up because if there are curious high school students, undergrads, etc watching this, I found a hidden gem on the n-Lab that you might enjoy as well

motivation for sheaves, cohomology and higher stacks

I thought the section on “The Basic Idea of Sheaves” was particularly helpful.

Posted by: Eric on June 26, 2009 5:28 PM | Permalink | Reply to this

Re: Cohomology and Homotopy

Eric wrote:

One side effect that hasn’t been too visible aside from my occasional piping in is that the totally public forum means that there are likely curious high school students and certainly undergraduate students watching from the sidelines.

There certainly are! Thanks for the link, it was very interesting. What other common categories have the property that looking very carefully at their arrows leads naturally to a larger category? The other example I can think of is commutative rings, but the sheaf construction subsumes that one, doesn’t it?

Posted by: Qiaochu Yuan on June 27, 2009 11:47 AM | Permalink | Reply to this

Re: Cohomology and Homotopy

Speaking of helpful gems, this one is a must read

Lectures On n-Categories and Cohomology

It brings together in one place (with clear motivations) a lot of the stuff discussed around here and the nLab.

Posted by: Eric Forgy on July 12, 2009 3:55 PM | Permalink | Reply to this

Re: Cohomology and Homotopy

By the way, Carlso Simpson recently also wrotte a brief account of this kind of history of the development of the notion of \infty-stack and cohomology.

A hyperlinked version of this text is at [[Carlos Simpson]] in the section Simpson on descent.

Posted by: Urs Schreiber on June 26, 2009 6:41 PM | Permalink | Reply to this

Re: Cohomology and Homotopy

and builds on the fundamental work by K. Brown, Joyal and Jardine and others on the [[model structure on simplicial presheaves]].

By the way, I am aware of extensive literature (e.g. by Blander and Dugger-Hollander-Isaksen) on the left Bousfield localization of the global projective model structure on simplicial (pre)sheaves.

I am interested in the right Bousfield localization of the projective global model structure.

Is there anything at all in the literature on that? Or elsewhere?

(An explicit description and analysis of the cofibrations? Analogous to what Dugger-Hollander-Isaksen did for the fibrations in the left Bousfield localization.)

Posted by: Urs Schreiber on July 1, 2009 2:24 PM | Permalink | Reply to this

Re: Cohomology and Homotopy

I wrote a minute ago:

I am interested in the right Bousfield localization of the projective global model structure.

Hm, maybe I am being stupid here. The explicit description should just be verbatim but dual to that of the left localization of the injective structure… With the descent condition on the fibrations becoming a codescent condition on the cofibrations. Hm.

Posted by: Urs Schreiber on July 1, 2009 2:57 PM | Permalink | Reply to this

Re: Cohomology and Homotopy

Not sure if you maybe wanted something deeper than this, but here’s an explanation that Mike Hopkins gave me for why Eckmann-Hilton duality breaks down. A pullback of a cofibration by a fibration is a cofibration, but a pushforward of a fibration by a cofibration is not a fibration. If one of the spaces is a point, this is simpler: the fiber map of a fibration is a cofibration, but the cofiber map of a cofibration is not a fibration. The former is the key to the construction of the Serre spectral sequence, which allows computation of cohomology in fibrations. Because of the latter failure, you can’t do something similar to compute homotopy in cofibrations.

Posted by: Eric on June 27, 2009 8:06 AM | Permalink | Reply to this

Re: Cohomology and Homotopy

cf cohomotopy groups defined in a range of dimensions

you can start to apply E-H duality and begin a spectral sequence but it doesn’t go all the way

cf. work of Michael Barrett
I think the magic words are Track groups

jim

Posted by: jim stasheff on June 27, 2009 2:42 PM | Permalink | Reply to this

Re: Cohomology and Homotopy

A pullback of a cofibration by a fibration is a cofibration, but a pushforward of a fibration by a cofibration is not a fibration.

Interesting. So another lack of symmetry. Are there model categories for which the symmetry holds?

I see on p. 86 of Handbook of Algebraic Topology Dwyer and Spalinski note the self-duality of the axioms of model categories, and explain how the failure of Eckmann-Hilton duality for ordinary homotopy theory shows that

…there are interesting facts about ordinary homotopy theory which cannot be derived from the model category axioms.

Posted by: David Corfield on June 29, 2009 9:43 AM | Permalink | Reply to this

Re: Cohomology and Homotopy

Thanks Eric (here) and David (here) for amplifying this point.

I am wondering if it would be justified to summarize the lesson learned here very unspecifically simply as follows:

Let H\mathbf{H} be an (,1)(\infty,1)-topos and XHX \in \mathbf{H}. Then even though the homotopy of XX in H\mathbf{H} is the cohomology of XX in H op\mathbf{H}^{op}, both in general behave quite differently to the extent that H\mathbf{H} is quite different to H op\mathbf{H}^{op}.

Not that this is supposed to be deep, but it might maybe help see what’s going on with respect to the phenomenon that it seems David C. is wondering about:

even though two definitions may be related by [[abstract duality]], the objects defined by these definitions may not exhibit any duality to the extent that the ambient category CC fails to be equivalent to its opposite.

So in a simpler example: even though the notions of limit and colimit in Set is related by abstract duality, they behave very differently, as differently as SetSet is different from Set opSet^{op}.

That example, simple minded as it may be, is possibly not so far from the one that we are interested in once we realize that SetSet is the canonical (1,1)(1,1)-topos just as TopTop is the canonical (,1)(\infty,1)-topos.

Posted by: Urs Schreiber on June 29, 2009 10:45 AM | Permalink | Reply to this

Re: Cohomology and Homotopy

TopTop is the canonical (,1)(\infty,1)-topos.

So how much broken symmetry comes from the property of being a (,1)(\infty, 1)-topos, and how much from being in addition specifically TopTop?

Is there always something playing the role of the spheres within an (,1)(\infty, 1)-topos?

Is there a reason why for cohomology one needs at least the structure of an (,1)(\infty, 1)-topos?

A possibly relevant remark by Peter May:

The theories of cofiber and fiber sequences illustrate an important, but informal, duality theory, known as Eckmann-Hilton duality. It is based on the adjunction between Cartesian products and function spaces. (A Concise Course in Algebraic Topology, p. 43)

Posted by: David Corfield on June 29, 2009 11:34 AM | Permalink | Reply to this

Re: Cohomology and Homotopy

Is there always something playing the role of the spheres within an (,1)(\infty,1)-topos?

Yes, at least if we restrict attention to to the “sheaf (,1)(\infty,1)-toposes”, which are the only ones that have been discussed (i.e. the higher analog of Grothendieck toposes as opposed to all toposes).

All these (,1)(\infty,1)toposes X\mathbf{X} are modeled by presheaves with values in simplicial sets, and the presheaf constant on the [[simplicial nn-sphere]] represents the nn-sphere S nS^n in X\mathbf{X}.

And indeed, that object is used to define homotopy groups of objects in X\mathbf{X}.

This is definition 6.5.1.1 page 521 in Higher Topos Theory.

Posted by: Urs Schreiber on June 29, 2009 6:14 PM | Permalink | Reply to this

Re: Cohomology and Homotopy

And you get Eilenberg-MacLane space objects?

Posted by: David Corfield on June 30, 2009 8:35 AM | Permalink | Reply to this

Re: Cohomology and Homotopy

And you get Eilenberg-MacLane space objects?

Yes.

This is actually no news, once one unwraps what’s going on here. Like this:

- We are talking about sheaf (,1)(\infty,1)-topoi/toposes/toponten/whatever

- by definition these are (,1)(\infty,1)-categories of \infty-stacks

- by a theorem of Lurie these are [[presented]] by the standard Brown-Joyal-Jardine [[model]] given by [[simplicial sheaves]];

- by the [[Dold-Kan correspondence]] ordinary complexes of abelian sheaves embed into that

- (leading to the embedding of ordinary [[abelian sheaf cohomology]] into all of [[nonabelian]] [[cohomology]])

- so in particular the standard Eilenberg-Mac Lane sheaves each model an \infty-stack

- indeed, the classical prescription for computing sheaf cohomology is secretly nothing but computing the [[infinity-stackification]] of these complexes of abelian sheaves under this Dold-Kan embedding into simplicial sheaves.

So, yes, every (“Grothendieck-Rezk-Lurie”-) (,1)(\infty,1)-topos has the expected [[Eilenberg-Mac Lane objects]].

Posted by: Urs Schreiber on June 30, 2009 9:12 PM | Permalink | Reply to this

Re: Cohomology and Homotopy

Is there a reason why for cohomology one needs at least the structure of an (,1)(\infty,1)-topos?

That’s a good question. By itself the definition of cohomology as the hom-set in the homotopy category of an (,1)(\infty,1)-catgeory needs no further assumptions.

It seems more that when the (,1)(\infty,1)-category here happens to be an (,1)(\infty,1)-topos, that we recognize the corresponding hom-thing as something that deserves to be called cohomology.

I’d be very interested in hearing other people’s comment on this.

Posted by: Urs Schreiber on June 29, 2009 6:20 PM | Permalink | Reply to this

Re: Cohomology and Homotopy

If one observes the way that cohomology is used, there is a lot of extra structure (cup products, etc.) that comes `into the action’ (in the non-technical sense). This is due to the nice structure on the `coefficients’ being reflected in the cohomology objects that you get out at the end. This gives nice pairings etc. and this is one reason why cohomology tends to be more used than `mere’ homotopy. This then raises the problem of finding in the (,1)(\infty,1)-topos case, some good coefficient objects, that are natural to consider and endow the resulting cohomology with nice structure. This would be great if the structure was somehow `new’.

Posted by: Tim Porter on June 30, 2009 12:59 PM | Permalink | Reply to this

Re: Cohomology and Homotopy

This may be off topic.

Can I think of cohomology with coefficients in a field as a some sort of a monoidal functor from manifolds to graded rings?

The multiplication on manifolds is just the Cartesian product.

Posted by: Eugene Lerman on June 30, 2009 4:07 PM | Permalink | Reply to this

Re: Cohomology and Homotopy

Not just manifolds, but spaces generally. That is, the cohomology ring functor

H *(,k):Top opGradedAlg kH^*(-, k): Top^{op} \to Graded Alg_k

takes products in TopTop to graded tensor products. For the codomain category, we actually mean the one consisting of graded algebras which are commutative in the graded sense, and graded tensor products there are the coproducts in that category. So cohomology (coefficients in a field) takes coproducts in Top opTop^{op} to coproducts in commutative graded algebras. Thinking of coproducts as examples of monoidal products, cohomology is thus an example of a strong monoidal functor.

For closed oriented manifolds, cohomology becomes even more pleasant by virtue of Poincaré duality.

Posted by: Todd Trimble on June 30, 2009 6:31 PM | Permalink | Reply to this

Re: Cohomology and Homotopy

That is exactly what I was meaning by saying that the coefficients in cohomology give a lot of extra structure to the values taken by the cohomology. If I ask a naive question, … is it clear for what objects of coefficients the cohomology is, say, a graded algebra, (or more general).

Posted by: Tim Porter on June 30, 2009 9:23 PM | Permalink | Reply to this

Re: Cohomology and Homotopy

I wrote:

By itself the definition of cohomology as the hom-set in the homotopy category of an (,1)(\infty,1)-catgeory needs no further assumptions.

It seems more that when the (,1)(\infty,1)-category here happens to be an (,1)(\infty,1)-topos, that we recognize the corresponding hom-thing as something that deserves to be called cohomology.

I’d be very interested in hearing other people’s comment on this.

Tim Porter suggested

If one observes the way that cohomology is used, there is a lot of extra structure (cup products, etc.) that comes ‘into the action’ (in the non-technical sense). This is due to the nice structure on the ‘coefficients’ being reflected in the cohomology objects that you get out at the end.

On the other hand, as you say, this structure is all in the coefficients.

I’d expect the reason why we want to be looking for cohomology inside (,1)(\infty,1)-topoi H\mathbf{H} instead of more general (,1)(\infty,1)-categories must be hidden in the (,1)(\infty,1)-categorical analogs of Giraud’s axioms (page 427):

i) H\mathbf{H} is presentable;

ii) colimits in H\mathbf{H} are universal;

iii) coproducts in H\mathbf{H} are disjoint;

iv) every groupoid object in H\mathbf{H} “is effective”.

I suspect iv) is the crucial one…

This then raises the problem of finding in the (,1)(\infty,1)-topos case, some good coefficient objects, that are natural to consider and endow the resulting cohomology with nice structure. This would be great if the structure was somehow ‘new’.

One “new” (not for you, though! :-) coefficient structure to be found is nonabelian cohomology groups.

Apart from new coefficient structure we also get “correct” cohomologies with familiar structure.

For instance, as was mentioned in another comment # , the ordinary definition of group cohomology for GG a compact simple, simply connected Lie group fails to see what the (,1)(\infty,1)-topos tells us is there: a degree 3 integral cohomology class.

So this is with coefficients as boring old as could be. Still, it improves on something.

Posted by: Urs Schreiber on June 30, 2009 10:21 PM | Permalink | Reply to this

Re: Cohomology and Homotopy

Although the structure is in the coefficients, the resulting cohomology gives useful information about the space being studied. That is somehow strange perhaps even wonderful and mysterious.

That raises two problems:

(i) For a given setting, what sorts of fairly general coefficients give good information on the spaces;

(ii) is there a costructure on homotopy (yes in many cases) sort of generalising cogroups etc.

I have looked at the possible Eilenberg-Zilber type structures on simplicial groupoids and this suggests there may be some general principles behind the known cases of nicely structured cohomology. Those structures seem to be linked to structural information encoded in the homotopy types, but how to describe this neatly and categorically still seems a long way off. (As a `for instance’ the Whitehead product mechanisms and other homotopy operations `must’ be there in the weak infinity groupoid structures, but it is not clear how to get at them.)

Posted by: Tim Porter on July 1, 2009 9:10 AM | Permalink | Reply to this

Re: Cohomology and Homotopy

I have looked at the possible Eilenberg-Zilber type structures on simplicial groupoids and this suggests there may be some general principles behind the known cases of nicely structured cohomology.

Can you give me a reference or otherwise more details on this?

Posted by: Urs Schreiber on July 1, 2009 2:03 PM | Permalink | Reply to this

Re: Cohomology and Homotopy

Section 4b on page 33 of Baues’ Algebraic Homotopy discusses E-H duality breaking down in Top.

Posted by: David Corfield on June 29, 2009 4:43 PM | Permalink | Reply to this

Re: Cohomology and Homotopy

One author I read noted an ontological difference. I hope this is at least mildly interesting to you, or maybe only to others like me, neophytes.

I was glancing through “What is Category Theory” by Sica and
came across an interesing quote taken from the paper listed below.

http://www.math.purdue.edu/~gottlieb/Papers/papers.html
6) A History of Duality in Algebraic Topology ( with James C. Becker),
Chapter 25 of HISTORY OF TOPOLOGY, I. M. James, ed., Elsevier, Amsterdam, 1999,
pp. 725 - 745 (This versions differs slightly from the published version.)

2. Categorical Points of View. [page 3 and 4]
“There are two major groupings of dualities in algebraic topology:
Strong duality and Eckmann–Hilton duality. Strong duality was first
employed by Poincare (1893) in a note in which “Poincare duality” was
used without proof or formal statement. The various instances of
strong duality (Poincare, Lefschetz, Alexander, Spanier–Whitehead,
Pontrjagin, cohomology–homology), seemingly quite different at first,
are intimately related in a categorical way which was finally made
clear only in 1980. Strong duality depends on finiteness and compactness.
On the other hand, Eckmann–Hilton duality is a loose collection of useful
dualities which arose from categorical points of view first put forward
by Beno Eckmann and P.J. Hilton in Eckmann (1956).

Eckmann–Hilton duality was first announced in lectures by Beno Eckmann
and also by Hilton, see Eckmann (1956), (1958). A very good description
of how this duality works, and some eyewitness history is given in Hilton
(1980). Instead of being a collection of theorems, Eckmann–Hilton duality
is a principle for discovering interesting concepts, theorems, and questions.
It is based on the dual category, that is, on the duality between the target
and source of a morphism; and also on the duality between functors and their
adjoints.
In fact it is a method wherein interesting definitions or theorems are given
a description in terms of a diagram of maps, or in terms of functors. Then
there is a dual way to express the diagram, or perhaps several different dual
ways. These lead to new definitions or conjectures. Some, not all, of these
definitions turn out to be very fruitful and some of the conjectures turn out
to be important theorems. … Eckmann–Hilton duality was conceived as a method
based on a categorical point of view in the early 1950’s. The challenge was to
use the point of view to generate interesting results.”


Posted by: Stephen Harris on June 29, 2009 2:45 PM | Permalink | Reply to this

Re: Cohomology and Homotopy

My train comes any minute, but when I have a bit more time I should add something like the following discussion to the entry on group cohomology:

namely when one passes from plain groups to groups with extra structure, such as smooth groups, etc, one sees that the usual definition of group cohomology as maps out of the standard bar construction B×GG\cdots \to B \times G \to G is wrong.

The point is that this works for groups in sets because there (G ×n)(G^{\times n}) is a good enough resolution (is cofibrant). For Lie groups that’s no longer the case. There one needs much finer resolutions.

Often this is stated as saying that for instance “smooth group cohomology” is trivial, in interesting cases.

But this is really a consequence of mixing up the general notion of cohomology with the specific realization in the case of group cohomology in terms of maps out of G ×nG^{\times n}.

For instance for GG a finite group and

μ:BGB 3U(1) \mu : \mathbf{B}G \to \mathbf{B}^3 U(1)

a group 3-cocycle, we get the corresponding 2-group G μG_\mu as the homotopy fiber of this map

BG μ * BG μ B 3U(1). \array{ \mathbf{B}G_\mu &\to& {*} \\ \downarrow && \downarrow \\ \mathbf{B}G &\stackrel{\mu}{\to}& \mathbf{B}^3 U(1) } \,.

Standard lore had it that there is no corresponding globally smooth 3-cocycle on a compact simple simply connected Lie group GG that would make the analogous statement true for G μG_\mu the String 2-group.

but this is true only with the “wrong” definition of group cohomology. With the right definition in terms of maps out of general resolutions, it does still work.

see here

Posted by: Urs Schreiber on June 29, 2009 11:12 AM | Permalink | Reply to this

Re: Cohomology and Homotopy

Urs wrote:

Standard lore had it that there is no corresponding globally smooth 3-cocycle on a compact simple simply connected Lie group G that would make the analogous statement true for G μG_\mu the String 2-group.

I proved a theorem to this effect: it’s Theorem 59 on page 70 of HDA5.

There’s a little wiggle room left in this theorem. It would be nice to rule out certain remaining possibilities. But it’s not that important: I completely agree with you that the naive definition of group cohomology using smooth or continuous cocycles (often called ‘van Est cohomology’) is ‘bad’, for the reasons you mention.

Posted by: John Baez on June 30, 2009 12:56 PM | Permalink | Reply to this

van Est

On the other hand, there are lots of ways in which it is `good’ or, at least, beautiful.

Posted by: jim stasheff on June 30, 2009 1:33 PM | Permalink | Reply to this

Higher Coboundaries

Over at [[cohomology]], it says the higher morphisms of H(X,A)H(X,A) are the higher coboundaries, or coboundaries of coboundaries.

I thought coboundaries of coboundaries were always zero (???) If not, that would imply you have boundaries of boundaries that are not zero *confused*

Posted by: Eric Forgy on July 11, 2009 12:02 AM | Permalink | Reply to this

Re: Higher Coboundaries

or coboundaries of coboundaries.

Or ghosts of ghosts, speaking gauge-theory-wise.

I thought coboundaries of coboundaries were always zero (???)

Yes, no worries.

“Higher coboundaries” refers to their degree.

Cocycles are points.

Coboundaries are edges between points.

Coboundaries of coboundaries are surfaces between edges.

Etc.

In one word: degree nn-coboundaries of AA-cocycles on XX are nothing but the nn-morphisms in the \infty-groupoid H(X,A)\mathbf{H}(X,A).

Cohomology classes are the connected components of that \infty-groupoid

H(X,A)=π 0H(X,A). H(X,A) = \pi_0 \mathbf{H}(X,A).

(I hope the boldface comes out right.)

Posted by: Urs Schreiber on July 11, 2009 11:35 AM | Permalink | Reply to this

Re: Higher Coboundaries

I think the wording “coboundaries of coboundaries” is a bit confusing for beginners, for the reason you mention. In many important and simple cases of cohomology, it’s indeed true that that coboundary of a coboundary is zero.

Someone should write a nice response to your plea on the nLab: “Could someone explain how this relates to the version of cohomology I am familiar with? Where is the coboundary?”

In fact your question is answered on the nnLab, directly below your question — but perhaps in terms too erudite for you to recognize it as an answer!

The section “ordinary cohomology of cochain complexes” starts by taking a cochain complex V V^\bullet and turning it into a chain complex XX, and taking the complex numbers (or your favorite field) and turning it into a chain complex AA, and then constructing a gizmo H(X,A)H(X,A) that’s very closely related to what you would call ‘the nnth cohomology of V V^\bullet’.

This gizmo H(X,A)H(X,A) is in fact a cochain complex whose 0th cohomology is the nnth cohomology of your cochain complex V V^\bullet.

As the article says, “The cocycles are the chain maps from XX to AA. The coboundaries are the chain homotopies between these chain maps.” If we take cocycles mod coboundaries, we get the nnth cohomology of V V^\bullet.

This section doesn’t mention “coboundaries between coboundaries”. These would be chain homotopies between chain homotopies. They’re not needed for computing the good old fashioned nnth cohomology of V V^\cdot.

WARNING: this portion of the article does not actually use the terms XX and AA. Instead of XX it says V V_\bullet. And instead of AA it says four different things: K(n,I)K(n,I), and B nIB^n I, and B nIB^{-n} I, and I[n] I[n]_\bullet. These four things are all notations for something incredibly simple: a chain complex that’s zero except in degree nn (or n-n), where it’s equal to your favorite field. I’m not sure why four different notations were used: it’s not necessary. Perhaps it’s the result of various people not having made up their minds on how to explain this stuff, or whether they need nn or n-n.

In summary: the good old fashioned cohomology you know and love is being massively generalized here, so that when we restrict back to that special case the formalism seems a bit complicated and confusing… especially when the explanation hasn’t been polished yet.

But don’t feel bad. Even in the best of worlds, it takes a while to understand how the good old fashioned approach to cohomology is related to the nn-categorical approach. That’s why I wrote a long joke-filled expository paper on nn-categories and cohomology, which naturally I recommend.

Posted by: John Baez on July 11, 2009 11:47 AM | Permalink | Reply to this

Re: Higher Coboundaries

In fact your question is answered on the nLab, directly below your question — but perhaps in terms too erudite for you to recognize it as an answer!

I was in the middle of describing how the familiar abelian Cech cohomology follows from the general nonsense, when I saw Eric’s query box.

Being already under time pressure with the Cech cohomology thing and because I need to work on the entry model structure on simplicial presheaves for a talk next Monday, I took just five minutes off to hastily scribble a reply to Eric.

As I logged at latest changes:

- following Eric’s question I typed a quick reply into cohomology on how the ordinary notion of cohomology in cochain complexes is reproduced. In principle this gives all the necessary information, but I’ll try to find the time later to give a long detailed exposition of how this basic important special case arises from the very general perspective

Unless you beat me to it, of course.

Posted by: Urs Schreiber on July 11, 2009 12:33 PM | Permalink | Reply to this

Re: Higher Coboundaries

The best way to explain this would be to first describe how chain homology is really the homotopy of a chain complex.

That way all the dualizations don’t get in the way.

Then cochain cohomology is correspondingly the cohomotopy=cohomology.

Posted by: Urs Schreiber on July 11, 2009 1:04 PM | Permalink | Reply to this

Re: Higher Coboundaries

Ignoring duality, when is a complex a cochain complex? If the differential goes up? or do we mean a cochain functor? i.e.
contravariant as a functor?

Posted by: jim stasheff on July 11, 2009 2:58 PM | Permalink | Reply to this

Re: Higher Coboundaries

Ignoring duality, when is a complex a cochain complex? If the differential goes up?

Yes.

Posted by: Urs Schreiber on July 13, 2009 7:00 AM | Permalink | Reply to this

Re: Higher Coboundaries

Kowabunga this is a nice paper :)

I added it as a reference to

[[chain homology and cohomology]]

I wish Kotiuga would get into this stuff. What I’m after is something like “n-categories for scientific computation” (not computer science). I’m not likely up to the task. Cohomology is important for applied computational electromagnetics so this is getting close enough that those interested in scientific computation might start taking note (if they haven’t).

Posted by: Eric Forgy on July 11, 2009 5:16 PM | Permalink | Reply to this

Re: Higher Coboundaries

There is an extensive theory of effective homology (i.e. computational homology theory) both in algebraic topology and in homological algebra,

see:

http://www-fourier.ujf-grenoble.fr/~sergerar/Kenzo/

for the effective spatial stuff,

and the pages of Graham Ellis and others at Galway:

http://hamilton.nuigalway.ie/Hap/www/

and

http://hamilton.nuigalway.ie/CHA/

There is also the CHOMP project

http://chomp.rutgers.edu/

Posted by: Tim Porter on July 11, 2009 9:47 PM | Permalink | Reply to this

Re: Higher Coboundaries

Just a quick comment while I am at the station:

Kowabunga this is a nice paper :)

I added it as a reference to

[[chain homology and cohomology]]

Notice that we have an nnLab entry on this paper:

here.

That page 24 and onwards that you point to discusses how ordinary generalized (Eilenberg-Steenrod) cohomology may be found inside some nonabelian cohomology. Maybe the link should go there.

Posted by: Urs Schreiber on July 13, 2009 6:57 AM | Permalink | Reply to this

Re: Higher Coboundaries

I modified the reference to include a link to the nLab entry (thanks for pointing that out) and then duplicated the reference on [[generalized (Eilenberg-Steenrod) cohomology]].

Posted by: Eric Forgy on July 13, 2009 3:44 PM | Permalink | Reply to this

Re: Higher Coboundaries

Okay, so I also added some comments to that reference and also added John’s TWF 149 to generalized (Eilenberg-Steenrod) cohomology, which should be read before those lectures, by the uninitiated.

I also added the lecture reference to Postnikov system as that seems to be the central topic there.

By the way, it seems the entry on Postnikov systems needs some expansion in general and could do with some of the expositions in those lectures in particular.

Maybe somebody feels inspired enough by John’s lectures to take care of the Postnikov system entry.

Posted by: Urs Schreiber on July 13, 2009 6:12 PM | Permalink | Reply to this

Re: Higher Coboundaries

Re-reading this discussion maybe for the record one should still say:

if V V^\bullet is a cochain complex, then a higher coboundary of degree (n+1)(n+1) is simply an element in d(V n)d(V^n).

Similarly for V V_\bullet a chain complex, an nn-dimensional boundary is an element in V n+1\partial V_{n+1}.

So the (co)boundary of a (co)boundary always vanishes, but there are nevertheless “higher” (co)boundaries.

Posted by: Urs Schreiber on July 13, 2009 7:10 AM | Permalink | Reply to this

Re: Higher Coboundaries

unless degree and V^n have some other meaning than the usual, these are just ordinary coboundaries

Posted by: jim stasheff on July 13, 2009 1:58 PM | Permalink | Reply to this

Re: Higher Coboundaries

unless degree and V nV^n have some other meaning than the usual, these are just ordinary coboundaries

Yes, and that’s all that was claimed here: the nn-cells in H(X,A)\mathbf{H}(X,A) are the degree nn coboundaries.

Okay, I changed in the entry the line

higher coboundaries or coboundaries of coboundaries

to just

higher degree coboundaries

I see now that the “coboundaries of coboundaries” was bad (did I type that? probably I did, the wiki history knows).

All I meant to say is that

- if cc and cc' are two objects in H(X,A)\mathbf{H}(X,A)

- then a morphism λ:cc\lambda : c \to c' corresponds to a coboundary c=c+dλc' = c + \mathrm{d} \lambda;

- a 2-morphism ρ:λλ\rho : \lambda \to \lambda' corresponds to a degree 2 coboundary λ=λ+dρ\lambda' = \lambda + \mathrm{d} \rho

etc., the usual picture

λ c ρ c λ \array{ & \nearrow && \searrow^{\lambda} \\ c &&\Downarrow^{\rho}&& c' \\ & \searrow &&\nearrow_{\lambda'} }

so λ\lambda and λ\lambda' are like “cobordisms” between cc and cc' and ρ\rho is like a “cobordism between these cobordisms”.

etc.

Posted by: Urs Schreiber on July 13, 2009 5:34 PM | Permalink | Reply to this

Re: Higher Coboundaries

We must have been tuned to the same wavelength. I wrote my comment about changing “coboundaries of coboundaries” to “higher coboundaries” before seeing this comment of yours.

I did go ahead and change the nLab entry even after your revision. If there are issues with my change, we can roll it back to what you had.

I think the reference to jj-morphisms and consolidating the two bullets is nice though.

Posted by: Eric Forgy on July 13, 2009 6:12 PM | Permalink | Reply to this

Re: Higher Coboundaries

On [[cohomology]], it says:

More precisely:

  • the objects cH(X,A)c\in H(X,A) are the cocycles on XX with values in AA;
  • the morphisms λ:cc\lambda:c\to c′ in H(X,A)H(X,A) are the coboundaries between cocycles;
  • the higher morphisms in H(X,A)H(X,A) are the higher coboundaries or coboundaries of coboundaries.
  • the equivalence classes [c]π 0H(X,A)[c]\in\pi_0 H(X,A) are the cohomology classes.

I think the term “coboundaries of coboundaries” is loaded and should probably be replaced with simply “higher coboundaries”.

Or perhaps even better, I think I’d like to change the paragraph to:

More precisely:

  • the objects cH(X,A)c\in H(X,A) are the cocycles on XX with values in AA;
  • the j-morphisms in H(X,A)H(X,A) are the coboundaries;
  • the equivalence classes [c]π 0H(X,A)[c]\in\pi_0 H(X,A) are the cohomology classes.

Is that acceptable?

PS: I may go ahead and change it, but we can easily roll it back if I mess something up.

Posted by: Eric Forgy on July 13, 2009 5:50 PM | Permalink | Reply to this

Re: Higher Coboundaries

Is that acceptable?

Yes, apparently our comments overlapped, see above.

Posted by: Urs Schreiber on July 13, 2009 6:06 PM | Permalink | Reply to this

Re: Higher Coboundaries

Bringing in jj-morphisms into the paragraph also brings in an interesting efficiency. Since an object is a 00-morphism, then cocycles are 00-morphisms and coboundaries are jj-morphisms for j>0j \gt 0.

That seems interesting.

Posted by: Eric Forgy on July 13, 2009 6:18 PM | Permalink | Reply to this

Re: Higher Coboundaries

Okay, I branched off an extra entry for that discussion:

[[chain homology and cohomology]].

Man, now I am gonna get into trouble, I am late for an appointment.

Exercise for the reader: finish and polish the entry.

Posted by: Urs Schreiber on July 11, 2009 1:51 PM | Permalink | Reply to this

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