Topology on cohomology groups

Topology on cohomology groups

[Steve Vickers]

A cohomology group can easily be an infinite power of the coefficient
group. But such a group has a natural non-discrete topology, namely the
compact-open (which in this case is also the product topology).

Are there approaches to cohomology that, as part of the process, also
supply topologies on the cohomology groups?

[I'm trying to understand the topos-theoretic account of cohomology as
in Johnstone's "Topos Theory". But it looks heavily dependent on having
a classical base topos, since it uses the classical proof of sufficiency
of injectives (together with the existence of Barr covers) to deduce the
same property internally in any Grothendieck topos. For a more fully
constructive theory I wonder if one needs to take better care of the
topologies.]

Steve Vickers.


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Re: Topology on cohomology groups

[Johannes Huebschmann]

[From moderator: This issue is fun, but off-topic... so it should be
closed. Categories posting will be intermittent until July 7, after
CT2009.]

Dear All

To add to the confusion:

There is a difference between skeleton and polygon:

skeletos, etc. is a participle

polygon is a noun

polygonon in ancient Greek
polygono in modern Greek

plural form polygona in ancient Greek

>From my recollections:
as a participle (I would have to check this):
skeletos, skeletae, skeleton etc.,

the neutrum  participle "skeleton" also has plural forms:

skeleta (nominativ)
skeleton (genitiv) (long o, i.e. omega)
skeletois (dativ)
skeleta (accusativ)

I cannot check details right now since I
cannot chek my ancient Greek sources right now to confirm.

Best regards

Johannes



HUEBSCHMANN Johannes
Professeur de Mathématiques
USTL, UFR de Mathématiques
UMR 8524 Laboratoire Paul  Painlevé
59 655 VILLENEUVE d'ASCQ Cédex/France
http://math.univ-lille1.fr/~huebschm

TEL. (33) 3 20 43 41 97
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Johannes.Huebschmann@math.univ-lille1.fr







On Tue, 23 Jun 2009, Fred E.J. Linton wrote:

> On Mon, 22 Jun 2009 09:17:05 AM EDT, "Prof. Peter Johnstone"
> <P.T.Johnstone@dpmms.cam.ac.uk>
> in response to: Andrew Stacey <andrew.stacey@math.ntnu.no> wrote, in part:
>
>> On Fri, 19 Jun 2009, Andrew Stacey wrote:
>>
>>> ... over the finite skeleta.
>>
>> Not really a contribution to the mathematical question, but I'm struck by
>> the fact that both Andrew Salch and Andrew Stacey, in their replies to
>> Steve Vickers, use the plural "skeleta". I used to do that when I was a
>> student, as a way of winding-up my teachers, but it isn't justifiable.
>>
>> The English word "skeleton" is indeed derived from a Greek root (the
>> past participle of the verb "skellein", to wither or dry up), but it
>> doesn't exist as a noun in Greek. There is therefore no justification
>> for giving it an imagined Greek plural. Having in my time devoted some
>> effort to fighting the bogus (but in fact more justifiable) Greek
>> plural "topoi", I feel bound to protest against this one too. ...
>
> The generic-seeming example "phenomenon/phenomena" certainly *suggests*
> a parallel "skeleton/skeleta" -- but it would also suggest "polygon/polyga",
> which I think we all would agree is nonsense. Peter is merely (justifiably)
> pointing out that "skeleton/skeleta" is as much nonsense as "polygon/polyga",
> and I'm with him 100% on that score.
>
> [As for the plural of "topos", I guess I'm in the mugwump camp that would
> *write* it as "topoi" (pace Peter), but *pronounce* it as "toposes" :-) .
> English was never very strong at phonetic consistency of pronunciation;
> witness GBShaw's "phonetic" spelling of FISH: "ghotip".]
>
> Cheers, -- Fred
>
> PS: "ghotip"? "gh" as in COUGH, "o" as in WOMEN,
> "ti" as in NATION, and "p" (silent) as in PNEUMONIA.
>
>
>
>
> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
>
---1463771056-1253283172-1245762566=:5534--


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Re: Topology on cohomology groups

[Fred E.J. Linton]

On Mon, 22 Jun 2009 09:17:05 AM EDT, "Prof. Peter Johnstone"
<P.T.Johnstone@dpmms.cam.ac.uk> 
in response to: Andrew Stacey <andrew.stacey@math.ntnu.no> wrote, in part:

> On Fri, 19 Jun 2009, Andrew Stacey wrote:
> 
> > ... over the finite skeleta.
> 
> Not really a contribution to the mathematical question, but I'm struck by
> the fact that both Andrew Salch and Andrew Stacey, in their replies to
> Steve Vickers, use the plural "skeleta". I used to do that when I was a
> student, as a way of winding-up my teachers, but it isn't justifiable.
> 
> The English word "skeleton" is indeed derived from a Greek root (the
> past participle of the verb "skellein", to wither or dry up), but it
> doesn't exist as a noun in Greek. There is therefore no justification
> for giving it an imagined Greek plural. Having in my time devoted some
> effort to fighting the bogus (but in fact more justifiable) Greek
> plural "topoi", I feel bound to protest against this one too. ...

The generic-seeming example "phenomenon/phenomena" certainly *suggests*
a parallel "skeleton/skeleta" -- but it would also suggest "polygon/polyga",
which I think we all would agree is nonsense. Peter is merely (justifiably)
pointing out that "skeleton/skeleta" is as much nonsense as "polygon/polyga",
and I'm with him 100% on that score.

[As for the plural of "topos", I guess I'm in the mugwump camp that would
*write* it as "topoi" (pace Peter), but *pronounce* it as "toposes" :-) .
English was never very strong at phonetic consistency of pronunciation;
witness GBShaw's "phonetic" spelling of FISH: "ghotip".]

Cheers, -- Fred

PS: "ghotip"? "gh" as in COUGH, "o" as in WOMEN, 
"ti" as in NATION, and "p" (silent) as in PNEUMONIA.




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Re: Topology on cohomology groups

[Prof. Peter Johnstone]

On Fri, 19 Jun 2009, Andrew Stacey wrote:

> I'm not sure if this is quite what you are looking for, but the topology on
> cohomology theories is given as an inverse limit (if I have my limits the
> correct way round) over the finite skeleta.

Not really a contribution to the mathematical question, but I'm struck by
the fact that both Andrew Salch and Andrew Stacey, in their replies to
Steve Vickers, use the plural "skeleta". I used to do that when I was a
student, as a way of winding-up my teachers, but it isn't justifiable.

The English word "skeleton" is indeed derived from a Greek root (the
past participle of the verb "skellein", to wither or dry up), but it
doesn't exist as a noun in Greek. There is therefore no justification
for giving it an imagined Greek plural. Having in my time devoted some
effort to fighting the bogus (but in fact more justifiable) Greek
plural "topoi", I feel bound to protest against this one too.

Peter Johnstone



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Re: Topology on cohomology groups

[Michael Barr]

Very tentatively, I have a memory that Lefschetz in the 30s invented
linearly compact vector spaces (and proved that the "category" of them
was dual to the "category" of discrete vector space (this was before
categories) for the purpose of making cohomology more closely dual to
homology.

Michael

On Fri, 19 Jun 2009, Steve Vickers wrote:

> A cohomology group can easily be an infinite power of the coefficient
> group. But such a group has a natural non-discrete topology, namely the
> compact-open (which in this case is also the product topology).
>
> Are there approaches to cohomology that, as part of the process, also
> supply topologies on the cohomology groups?
>
> [I'm trying to understand the topos-theoretic account of cohomology as
> in Johnstone's "Topos Theory". But it looks heavily dependent on having
> a classical base topos, since it uses the classical proof of sufficiency
> of injectives (together with the existence of Barr covers) to deduce the
> same property internally in any Grothendieck topos. For a more fully
> constructive theory I wonder if one needs to take better care of the
> topologies.]
>
> Steve Vickers.
>


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Re: Topology on cohomology groups

[Andrew Salch]

This certain does happen--if E^* is a generalized cohomology theory, and X
is an infinite complex, we often want to topologize E^*(X) by the inverse
limit of the E-cohomology of the finite skeleta of X. This is why, for
example, if E is a complex-oriented cohomology theory, the E-cohomology of
infinite-dimensional complex projective space is a power series ring (and
not merely a polynomial ring), in one variable, over the coefficient ring
E^*.

This is sometimes important and useful in topology (for example, in the
situation above, where one uses the above description of E^*(CP^{\infty})
to associate a 1-dimensional formal group law to E), and sometimes it's
more just a hassle: for example, the early papers on the Adams-Novikov
spectral sequence used MU-cohomology (i.e., complex cobordism), and this
necessarily meant keeping track of the topology on MU^*(X) of various
spectra E, since for example MU^*(MU), the ring of stable natural
transformations of MU^*, has infinite homogeneous sums, and one had to
handle completed tensor products of MU^*(MU)-modules; the modern way is to
use generalized homology instead of generalized cohomology for these
generalized Adams spectral sequences, which does away with the topologies
and the need for completed tensor products (of course, the price one pays
is that one is then, in the case of MU, dealing with MU_*(MU)-comodules
rather than (topological) MU^*(MU)-modules, and computing Cotor rather
than Ext; but this seems to be worth it).

There's some discussion of this in Ravenel's green book. The paper on
unstable operations by Boardman, Johnson, and Wilson in the Handbook of
Algebraic Topology also includes some discussion and some nice
manipulations of topologies (again, coming from the finite skeleta of an
infinite complex) on some generalized cohomology rings and modules.

There are also generalized homology theories, like the Morava E-theories,
which occur as completions of other generalized homology theories, and so
E_*(X) naturally has a topology (coming from the completion) when E is one
of these theories; recent developments in stable homotopy theory make it
seem likely that there will be more such theories in our future.

Hope this is useful to you,
Andrew S.


On Fri, 19 Jun 2009, Steve Vickers wrote:

> A cohomology group can easily be an infinite power of the coefficient
> group. But such a group has a natural non-discrete topology, namely the
> compact-open (which in this case is also the product topology).
>
> Are there approaches to cohomology that, as part of the process, also
> supply topologies on the cohomology groups?
>
> [I'm trying to understand the topos-theoretic account of cohomology as
> in Johnstone's "Topos Theory". But it looks heavily dependent on having
> a classical base topos, since it uses the classical proof of sufficiency
> of injectives (together with the existence of Barr covers) to deduce the
> same property internally in any Grothendieck topos. For a more fully
> constructive theory I wonder if one needs to take better care of the
> topologies.]
>
> Steve Vickers.
>

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Re: Topology on cohomology groups

[Andrew Stacey]

On Fri, Jun 19, 2009 at 10:26:13AM +0100, Steve Vickers wrote:
> A cohomology group can easily be an infinite power of the coefficient
> group. But such a group has a natural non-discrete topology, namely the
> compact-open (which in this case is also the product topology).
>
> Are there approaches to cohomology that, as part of the process, also
> supply topologies on the cohomology groups?
>

I'm not sure if this is quite what you are looking for, but the topology on
cohomology theories is given as an inverse limit (if I have my limits the
correct way round) over the finite skeleta.  This has an impact, for example,
in the correct statement of the Kunneth theorem on cohomology of products (one
has to complete the tensor product with respect to the topology).

A fairly comprehensive and detail account is in Boardman and
Boardman+Johnson+Wilson in the Handbook of Algebraic Topology: MR1361889 and
MR1361900 (though it was known well before that).  These papers are available
online from Steve Wilson's homepage:

http://www.math.jhu.edu/~wsw/

(scan right down to the bottom).

Sarah Whitehouse and I also look at this in our paper 'The Hunting of the Hopf
Ring' (arxiv:0711.3722, to appear in HHA).

Andrew Stacey

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