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1. • CommentRowNumber1.
   • CommentAuthorMike Shulman
   • CommentTimeAug 20th 2012
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   Author: Mike Shulman
   Format: MarkdownItexIs there a reason for the page [[cohomology theory]] to exist independently, rather than as a redirect to [[generalized (Eilenberg-Steenrod) cohomology]]?

   Is there a reason for the page cohomology theory to exist independently, rather than as a redirect to generalized (Eilenberg-Steenrod) cohomology?

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeAug 20th 2012
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   Author: Urs
   Format: MarkdownItexNot a good reason, at least. I guess we should merge then, yes.

   Not a good reason, at least. I guess we should merge then, yes.

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeAug 20th 2012
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexok, done. Cache bug exists.

   ok, done. Cache bug exists.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeAug 20th 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks. I have removed the cache file now.

   Thanks. I have removed the cache file now.

5. • CommentRowNumber5.
   • CommentAuthorjim_stasheff
   • CommentTimeAug 21st 2012
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   Author: jim_stasheff
   Format: TextWhat about cohomology theory for groups, algebras, etc?

   What about cohomology theory for groups, algebras, etc?
6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeAug 21st 2012
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   Author: Mike Shulman
   Format: MarkdownItexIs there an abstract notion of "a cohomology theory" for those, akin to the Eilenberg-Steenrod axioms? We do have the general page [[cohomology]].

   Is there an abstract notion of “a cohomology theory” for those, akin to the Eilenberg-Steenrod axioms? We do have the general page cohomology.

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeAug 21st 2012
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   Author: Urs
   Format: MarkdownItexGroup cohomology of $G$ is just cohomology of $B G$. In this sense [[generalized (Eilenberg-Steenrod) cohomology]] subsumes bare group cohomology. For structured group cohomology the same remains true after internalization over the relevant site. For instance Segal-Brylinski's refined Lie group cohomology is just the ordinary cohomology of the moduli stack $\mathbf{B}G$ (at least for compact $G$).

   Group cohomology of $G$ is just cohomology of $B G$. In this sense generalized (Eilenberg-Steenrod) cohomology subsumes bare group cohomology. For structured group cohomology the same remains true after internalization over the relevant site. For instance Segal-Brylinski’s refined Lie group cohomology is just the ordinary cohomology of the moduli stack $\mathbf{B}G$ (at least for compact $G$).

8. • CommentRowNumber8.
   • CommentAuthorjim_stasheff
   • CommentTimeAug 22nd 2012
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   Author: jim_stasheff
   Format: TextTrue but very misleading. Which algebraists would want to learn group cohomology via BG? and for algebras, Lie algebras,...

   True but very misleading. Which algebraists would want to learn group cohomology via BG? and for algebras, Lie algebras,...
9. • CommentRowNumber9.
   • CommentAuthorjim_stasheff
   • CommentTimeAug 22nd 2012
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   Author: jim_stasheff
   Format: TextMike is asking about axioms for cohomology in the hyperactive sense of the general page cohomology. Anything even attempted?

   Mike is asking about axioms for cohomology in the hyperactive sense of the general page cohomology.
   Anything even attempted?
10. • CommentRowNumber10.
    • CommentAuthorDavidRoberts
    • CommentTimeAug 23rd 2012
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    Author: DavidRoberts
    Format: MarkdownItexUrs says that cohomology is just equivalence classes of 1-arrows in $(\infty,1)$-categories, but then that's perhaps a little too general... :-)

    Urs says that cohomology is just equivalence classes of 1-arrows in $(\infty,1)$-categories, but then that’s perhaps a little too general… :-)

11. • CommentRowNumber11.
    • CommentAuthorTim_Porter
    • CommentTimeAug 23rd 2012
    • (edited Aug 23rd 2012)
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    Author: Tim_Porter
    Format: MarkdownItexAlgebraic cohomology does fit in neatly into the homotopical algebra scene of course, and although that can be seen as part of the study of (∞,1)-categories, perhaps some discussion should be attempted so as to link in more to both `topological' cohomology and to keep that origins in view. The non-abelian cohomology entries already have some attempt to provide this but they tend to get swamped by the (∞,1)-category approach (which is fine when you understand it but daunting if you don't).

    Algebraic cohomology does fit in neatly into the homotopical algebra scene of course, and although that can be seen as part of the study of (∞,1)-categories, perhaps some discussion should be attempted so as to link in more to both ‘topological’ cohomology and to keep that origins in view. The non-abelian cohomology entries already have some attempt to provide this but they tend to get swamped by the (∞,1)-category approach (which is fine when you understand it but daunting if you don’t).

12. • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeAug 23rd 2012
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    Author: Urs
    Format: MarkdownItex> in (∞,1)-categories, No, in $(\infty,1)$-toposes > but then that's perhaps a little too general... :-) No, it's tradtional equivariant non-abelian sheaf hyper-cohomology over arbitrary sites.

    in (∞,1)-categories,

    No, in $(\infty,1)$-toposes

    but then that’s perhaps a little too general… :-)

    No, it’s tradtional equivariant non-abelian sheaf hyper-cohomology over arbitrary sites.

13. • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeAug 23rd 2012
    • (edited Aug 23rd 2012)
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    Author: Urs
    Format: MarkdownItexI have touched the Idea-section of _[[cohomology]]_ once more: I have removed the standout-box that used to be there and replaced it with > This captures the generality of what traditionally would be called _[[equivariant cohomology|equivariant]] [[non-abelian cohomology|non-abelian]] [[generalized (Eilenberg-Steenrod) cohomology|generalized]] [[sheaf cohomology|sheaf]] [[hypercohomology]] over a [[simplicial site]]_. Maybe the red standout box came across as "hyperactivism", I don't know. On the other hand, given that the entry's main point was [[BrownAHT|known in 1973]] and since used by authors who you probably regard as authorities, if that is what is needed to convince (K. Brown, Moerdijk, Lurie, all represented with commented references at the entry), the reactions it still receives can make one feel like posting more such boxes all over the place. ;-) By the way, a fairly comprehensive account of the perspective of the entry, showing that it's definition has exactly the right level of generality is in our _[[schreiber:Principal ∞-bundles -- theory, presentations and applications]]_.

    I have touched the Idea-section of cohomology once more: I have removed the standout-box that used to be there and replaced it with

    This captures the generality of what traditionally would be called equivariant non-abelian generalized sheaf hypercohomology over a simplicial site.

    Maybe the red standout box came across as “hyperactivism”, I don’t know. On the other hand, given that the entry’s main point was known in 1973 and since used by authors who you probably regard as authorities, if that is what is needed to convince (K. Brown, Moerdijk, Lurie, all represented with commented references at the entry), the reactions it still receives can make one feel like posting more such boxes all over the place. ;-)

    By the way, a fairly comprehensive account of the perspective of the entry, showing that it’s definition has exactly the right level of generality is in our Principal ∞-bundles – theory, presentations and applications (schreiber).

14. • CommentRowNumber14.
    • CommentAuthorTim_Porter
    • CommentTimeAug 23rd 2012
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    Author: Tim_Porter
    Format: MarkdownItexTypo corrected.

    Typo corrected.

15. • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeAug 23rd 2012
    • (edited Aug 23rd 2012)
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    Author: Urs
    Format: MarkdownItexThanks. By the way, I have also expanded the next paragraph in the Idea-section of _[[cohomology]]_, which now reads as follows: > This notion accurately captures the general classification and extension problems as expected: For $A = \mathbf{B}G$ a [[pointed object|pointed]] [[connected object in an (∞,1)-topos|connected object]], hence the [[delooping]] of an [[∞-group]] $G$ the cohomology $H^1(X,G) := H(X, \mathbf{B}G)$ classifies $G$-[[principal ∞-bundles]] in $\mathbf{H}$ over $X$. If $X = \mathbf{B}H$ itself then this classifies [[group extensions]], being a non-abelian generalization of the [[Ext]]-functor in [[abelian sheaf cohomology]]. If $G = \mathbf{Aut}(V)$ is the [[automorphism ∞-group]] of some object $V$, then this classifies equivalently $V$-[[fiber ∞-bundles]]. Moreover, the full [[cocycle]] [[∞-groupoid]] $\mathbf{H}(X, \mathbf{B}G)$ is [[natural equivalence|naturally]] [[equivalence in an (∞,1)-category|equivalent]] to the $\infty$-groupoid $G Bund(X)$ of $G$-principal $\infty$-bundles on $X$.

    Thanks.

    By the way, I have also expanded the next paragraph in the Idea-section of cohomology, which now reads as follows:

    This notion accurately captures the general classification and extension problems as expected: For $A = \mathbf{B}G$ a pointed connected object, hence the delooping of an ∞-group $G$ the cohomology $H^1(X,G) := H(X, \mathbf{B}G)$ classifies $G$-principal ∞-bundles in $\mathbf{H}$ over $X$. If $X = \mathbf{B}H$ itself then this classifies group extensions, being a non-abelian generalization of the Ext-functor in abelian sheaf cohomology. If $G = \mathbf{Aut}(V)$ is the automorphism ∞-group of some object $V$, then this classifies equivalently $V$-fiber ∞-bundles. Moreover, the full cocycle ∞-groupoid $\mathbf{H}(X, \mathbf{B}G)$ is naturally equivalent to the $\infty$-groupoid $G Bund(X)$ of $G$-principal $\infty$-bundles on $X$.

16. • CommentRowNumber16.
    • CommentAuthorMike Shulman
    • CommentTimeAug 23rd 2012
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    Author: Mike Shulman
    Format: MarkdownItexIt's a little redundant to say both "generalized" and "hyper-", isn't it? Being as how chain complexes are just particular kinds of spectra? Also, I feel that in that sentence some mention should be made of looping and delooping. A traditional cohomology theory isn't just a bunch of independent groups; there are relationships between them. So it seems to me that hom-spaces in an $(\infty,1)$-topos only really include traditional cohomology when you combine them with the notions of looping and delooping. (Plus, to get equivariant Bredon cohomology, you need a fancier kind of looping and delooping.)

    It’s a little redundant to say both “generalized” and “hyper-“, isn’t it? Being as how chain complexes are just particular kinds of spectra?

    Also, I feel that in that sentence some mention should be made of looping and delooping. A traditional cohomology theory isn’t just a bunch of independent groups; there are relationships between them. So it seems to me that hom-spaces in an $(\infty,1)$-topos only really include traditional cohomology when you combine them with the notions of looping and delooping. (Plus, to get equivariant Bredon cohomology, you need a fancier kind of looping and delooping.)

17. • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeAug 23rd 2012
    • (edited Aug 23rd 2012)
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    Author: Urs
    Format: MarkdownItex> It's a little redundant to say both "generalized" and "hyper-", isn't it? True. On the other hand, the way these terms are traditionally used, neither is a special of the other. But okay, I have changed it in the entry to [[generalized (Eilenberg-Steenrod) cohomology|generalized]]/[[hypercohomology|hyper]]. How is that? > Also, I feel that in that sentence some mention should be made of looping and delooping. A traditional cohomology theory isn't just a bunch of independent groups; there are relationships between them. If the loopings/delooping exist. If not, then not. Nonabelian cohomology $H^1(X,G)$ has no further degrees, and is still a respected "cohomology theory". More generally, this is true for all twisted cohomologies (I have introduced the adjective [[twisted cohomology|twisted]] to the list, too, now), where you can't deloop arbitrarily anymore because the twists change. In the $\infty$-topos language this is reflected in the fact that twisted cohomology is the hom-space in a slice, and even if the fibers of a bundle are stable, the bundle is in general not as an object in the slice. But okay, there should be some hint of that in the sentence. I have further edited it now to read as follows: > This definition unwinds to what traditionally would be called _[[equivariant cohomology|equivariant]] [[twisted cohomology|twisted]] [[non-abelian cohomology|non-abelian]] [[generalized (Eilenberg-Steenrod) cohomology|generalized]]/[[hypercohomology|hyper]] [[sheaf cohomology]] over a [[simplicial site]]_ in all degrees in which this makes sense for the given coefficient (if $A$ is $n$-fold [[delooping|deloopable]], then $A$-cohomology exists in the first $n$ degrees).

    It’s a little redundant to say both “generalized” and “hyper-“, isn’t it?

    True. On the other hand, the way these terms are traditionally used, neither is a special of the other.

    But okay, I have changed it in the entry to generalized/hyper. How is that?

    Also, I feel that in that sentence some mention should be made of looping and delooping. A traditional cohomology theory isn’t just a bunch of independent groups; there are relationships between them.

    If the loopings/delooping exist. If not, then not. Nonabelian cohomology $H^1(X,G)$ has no further degrees, and is still a respected “cohomology theory”. More generally, this is true for all twisted cohomologies (I have introduced the adjective twisted to the list, too, now), where you can’t deloop arbitrarily anymore because the twists change. In the $\infty$-topos language this is reflected in the fact that twisted cohomology is the hom-space in a slice, and even if the fibers of a bundle are stable, the bundle is in general not as an object in the slice.

    But okay, there should be some hint of that in the sentence. I have further edited it now to read as follows:

    This definition unwinds to what traditionally would be called equivariant twisted non-abelian generalized/hyper sheaf cohomology over a simplicial site in all degrees in which this makes sense for the given coefficient (if $A$ is $n$-fold deloopable, then $A$-cohomology exists in the first $n$ degrees).

18. • CommentRowNumber18.
    • CommentAuthorMike Shulman
    • CommentTimeAug 23rd 2012
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    Author: Mike Shulman
    Format: MarkdownItexWell, when "generalized" is combined with "sheaf" I would expect it to subsume "hyper", although probably that isn't used in traditional literature. (-: I agree about nonabelian and twisted cohomology, but I'm still not satisfied with the current version. Let me think a bit and then edit it myself, and see what you think about what I come up with.

    Well, when “generalized” is combined with “sheaf” I would expect it to subsume “hyper”, although probably that isn’t used in traditional literature. (-: I agree about nonabelian and twisted cohomology, but I’m still not satisfied with the current version. Let me think a bit and then edit it myself, and see what you think about what I come up with.

19. • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeAug 24th 2012
    • (edited Aug 24th 2012)
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    Author: Urs
    Format: MarkdownItex> Well, when "generalized" is combined with "sheaf" I would expect it to subsume "hyper", although probably that isn't used in traditional literature. Yes, I entirely agree. The sentence can't give a perfect match... simply because the traditional terminology is so imperfect. For instance also the pieces "equivariant" and "over a simplicial site" are not independent of each other, and nevertheless neither is quite a special case of the other. Also higher non-abelian cohomology can be decomposed into twisted abelian cohomology, and so there is some overlap there, too. But it's okay if these traditional terms are a mess. After all, the entry cleans up that mess. All I would like that sentence to achieve is to indicate that while intrinsic cohomology in $\infty$-toposes is very general, each and every general aspect of it is already known, if maybe in some disguise, in some form or other, so that in total this intrinsic definition just unifies the traditional cohomology mess in one single structure. > Let me think a bit and then edit it myself, and see what you think about what I come up with. Sure. Please do.

    Well, when “generalized” is combined with “sheaf” I would expect it to subsume “hyper”, although probably that isn’t used in traditional literature.

    Yes, I entirely agree. The sentence can’t give a perfect match… simply because the traditional terminology is so imperfect. For instance also the pieces “equivariant” and “over a simplicial site” are not independent of each other, and nevertheless neither is quite a special case of the other. Also higher non-abelian cohomology can be decomposed into twisted abelian cohomology, and so there is some overlap there, too.

    But it’s okay if these traditional terms are a mess. After all, the entry cleans up that mess. All I would like that sentence to achieve is to indicate that while intrinsic cohomology in $\infty$-toposes is very general, each and every general aspect of it is already known, if maybe in some disguise, in some form or other, so that in total this intrinsic definition just unifies the traditional cohomology mess in one single structure.

    Let me think a bit and then edit it myself, and see what you think about what I come up with.

    Sure. Please do.

20. • CommentRowNumber20.
    • CommentAuthorMike Shulman
    • CommentTimeAug 24th 2012
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    Author: Mike Shulman
    Format: MarkdownItexOkay, I did a bit of something, have to stop now. I don't really like the "annotated table of contents" — I think what the reader wants (or, at least, what _this_ reader wants) after the link to "motivation..." is to go directly into the Tour section, explaining how all the classical kinds of cohomology are special cases of this one. Also I am a big believer that making things more concise makes them more readable. So how would you feel about getting rid of that?

    Okay, I did a bit of something, have to stop now. I don’t really like the “annotated table of contents” — I think what the reader wants (or, at least, what this reader wants) after the link to “motivation…” is to go directly into the Tour section, explaining how all the classical kinds of cohomology are special cases of this one. Also I am a big believer that making things more concise makes them more readable. So how would you feel about getting rid of that?

21. • CommentRowNumber21.
    • CommentAuthorTobyBartels
    • CommentTimeAug 24th 2012
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    Author: TobyBartels
    Format: MarkdownItexI like that list in your edit #103, Mike, at least *somewhere*.

    I like that list in your edit #103, Mike, at least somewhere.

22. • CommentRowNumber22.
    • CommentAuthorMike Shulman
    • CommentTimeAug 24th 2012
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    Author: Mike Shulman
    Format: MarkdownItexHaha, so, as I was writing that list, I realized I was basically duplicating the information in the "Tour" section, and it felt like it was getting too long and detailed for an "Idea" section anyway, so I stopped. But that was what made me realize that as a reader I wanted to get that information right away. Do you (Toby) think a list format would be better than the current prose in the "Tour" section?

    Haha, so, as I was writing that list, I realized I was basically duplicating the information in the “Tour” section, and it felt like it was getting too long and detailed for an “Idea” section anyway, so I stopped. But that was what made me realize that as a reader I wanted to get that information right away. Do you (Toby) think a list format would be better than the current prose in the “Tour” section?

23. • CommentRowNumber23.
    • CommentAuthorUrs
    • CommentTimeAug 24th 2012
    • (edited Aug 24th 2012)
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    Author: Urs
    Format: MarkdownItexThanks, looks good. There is one typo-kind of thing: you have: > (1) the traditional algebraic-topology taught in algebraic topology, I have changed that to > (1) the traditional (e.g. [[singular cohomology|singular]]) cohomology of [[topological spaces]] taught in [[algebraic topology]], but feel free to edit this or undo it if you had something else in mind. Concerning lists: I am all in favor of lists. There has long been a list further down, the _[Long list of examples](http://www.ncatlab.org/nlab/show/cohomology#LongListOfExamples)_ but all this can do with plenty of polishing, brushing-up rearranging etc, certainly.

    Thanks, looks good.

    There is one typo-kind of thing: you have:

    (1) the traditional algebraic-topology taught in algebraic topology,

    I have changed that to

    (1) the traditional (e.g. singular) cohomology of topological spaces taught in algebraic topology,

    but feel free to edit this or undo it if you had something else in mind.

    Concerning lists: I am all in favor of lists. There has long been a list further down, the Long list of examples but all this can do with plenty of polishing, brushing-up rearranging etc, certainly.

24. • CommentRowNumber24.
    • CommentAuthorjim_stasheff
    • CommentTimeAug 24th 2012
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    Author: jim_stasheff
    Format: Texttrivial comment: whenever I see `hypercohomology', I have to stop and remind myself that is is usually (always?) a pseudonym for total cohomology (of a bicomplex or such). Should be at least a remark somewhere.

    trivial comment: whenever I see `hypercohomology', I have to stop and remind myself that is is usually (always?)
    a pseudonym for total cohomology (of a bicomplex or such). Should be at least a remark somewhere.
25. • CommentRowNumber25.
    • CommentAuthorTobyBartels
    • CommentTimeAug 24th 2012
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    Author: TobyBartels
    Format: MarkdownItexThe list was nice and compact, despite actually having (or so it seemed) more examples. It might be worth having a leisurely tour on a separate page!

    The list was nice and compact, despite actually having (or so it seemed) more examples. It might be worth having a leisurely tour on a separate page!