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. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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.
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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.
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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/ ]
[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 (33) 3 20 43 42 33 (sécrétariat) (33) 3 20 43 48 50 (sécrétariat) Fax (33) 3 20 43 43 02 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-- [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (7)
-
Andrew Salch -
Andrew Stacey -
Fred E.J. Linton -
Johannes Huebschmann -
Michael Barr -
Prof. Peter Johnstone -
Steve Vickers