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1. • CommentRowNumber1.
   • CommentAuthorMarc Hoyois
   • CommentTimeJul 21st 2013
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   Author: Marc Hoyois
   Format: MarkdownItexThe [[nPOV]] holds that [[cohomology]] is given by connected components of mapping spaces in $(\infty,1)$-topoi. Many instances of cohomology are special cases of this construction. Unfortunately, this POV does not adequately capture the notions of cohomology theories for schemes introduced by Voevodsky, e.g., in [[motivic homotopy theory]]. On [[motivic cohomology]] the nLab used to say (I just changed it): > The proposal due to Morel and Voevodsky, based on Morel-Voevodsky's [[A1-homotopy theory]] effectively identifies motivic cohomology with the cohomology as given in the [[(∞,1)-topos]] $\mathbf{H}_{Nis}$ of [[∞-stack]]s on the [[Nisnevich site]], following the general notion of [[cohomology]] (as described there): for $X$ a [[scheme]] and $A \in \mathbf{H}_{Nis}$ some coefficient object, the motivic cohomology of $X$ with coefficients in $A$ is the connected components of the [[derived hom space|(∞,1)-categorical hom-space]] of morphisms from $X$ to $A$: This was wrong on two levels. First (and this is not the point of this post), it should at least say <b>generalized</b> motivic cohomology, since motivic cohomology has a very specific meaning: it refers to <b>ordinary</b> cohomology as opposed to generalized. For example, algebraic K-theory is not motivic cohomology with coefficients in anything. But more importantly, saying that it is cohomology in the Nisnevich $(\infty,1)$-topos is a bit like saying that étale cohomology is cohomology in the presheaf $(\infty,1)$-topos on schemes. One should really say that it is cohomology in the $\mathbb{A}^1$-localization of the Nisnevich $(\infty,1)$-topos. This localization is a [[locally cartesian closed]] locally presentable $(\infty,1)$-category, but not an $(\infty,1)$-topos. In some sense this is an intrinsic feature of the motivic world: motivic cohomology itself was needed because there was no topology on schemes whose associated cohomology had the desired ``motivic'' properties, so it's non-toposic *by design*. Should the nLab relax its view on cohomology to include this case? Perhaps the page on [[cohomology]] could allow, say, locally cartesian closed $(\infty,1)$-categories in the general definition?

   The nPOV holds that cohomology is given by connected components of mapping spaces in $(\infty,1)$-topoi. Many instances of cohomology are special cases of this construction. Unfortunately, this POV does not adequately capture the notions of cohomology theories for schemes introduced by Voevodsky, e.g., in motivic homotopy theory.

   On motivic cohomology the nLab used to say (I just changed it):

   The proposal due to Morel and Voevodsky, based on Morel-Voevodsky’s A1-homotopy theory effectively identifies motivic cohomology with the cohomology as given in the (∞,1)-topos $\mathbf{H}_{Nis}$ of ∞-stacks on the Nisnevich site, following the general notion of cohomology (as described there): for $X$ a scheme and $A \in \mathbf{H}_{Nis}$ some coefficient object, the motivic cohomology of $X$ with coefficients in $A$ is the connected components of the (∞,1)-categorical hom-space of morphisms from $X$ to $A$:

   This was wrong on two levels.

   First (and this is not the point of this post), it should at least say generalized motivic cohomology, since motivic cohomology has a very specific meaning: it refers to ordinary cohomology as opposed to generalized. For example, algebraic K-theory is not motivic cohomology with coefficients in anything.

   But more importantly, saying that it is cohomology in the Nisnevich $(\infty,1)$-topos is a bit like saying that étale cohomology is cohomology in the presheaf $(\infty,1)$-topos on schemes. One should really say that it is cohomology in the $\mathbb{A}^1$-localization of the Nisnevich $(\infty,1)$-topos. This localization is a locally cartesian closed locally presentable $(\infty,1)$-category, but not an $(\infty,1)$-topos. In some sense this is an intrinsic feature of the motivic world: motivic cohomology itself was needed because there was no topology on schemes whose associated cohomology had the desired “motivic” properties, so it’s non-toposic by design.

   Should the nLab relax its view on cohomology to include this case? Perhaps the page on cohomology could allow, say, locally cartesian closed $(\infty,1)$-categories in the general definition?

2. • CommentRowNumber2.
   • CommentAuthorTobyBartels
   • CommentTimeJul 21st 2013
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   Author: TobyBartels
   Format: MarkdownItexNot knowing anything, that sounds reasonable. The mentions of motivic cohomology at [[cohomology]] refer to using only $\mathbb{A}^1$-invariant objects; this sounds more like a subcategory than a localization, so is that wrong or do they end up being equivalent?

   Not knowing anything, that sounds reasonable. The mentions of motivic cohomology at cohomology refer to using only $\mathbb{A}^1$-invariant objects; this sounds more like a subcategory than a localization, so is that wrong or do they end up being equivalent?

3. • CommentRowNumber3.
   • CommentAuthorMarc Hoyois
   • CommentTimeJul 21st 2013
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   Author: Marc Hoyois
   Format: MarkdownItexThanks for the support, Toby! ;) The category of $\mathbb{A}^1$-invariant objects is at the same time a subcategory and a localization, with the inclusion functor being right adjoint to the localization functor. Such is the magic of [reflexive localization](http://ncatlab.org/nlab/show/localization#reflective_localization_8). This is now made clear (hopefully) at [[motivic homotopy theory]].

   Thanks for the support, Toby! ;)

   The category of $\mathbb{A}^1$-invariant objects is at the same time a subcategory and a localization, with the inclusion functor being right adjoint to the localization functor. Such is the magic of reflexive localization. This is now made clear (hopefully) at motivic homotopy theory.

4. • CommentRowNumber4.
   • CommentAuthorDavidRoberts
   • CommentTimeJul 22nd 2013
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   Author: DavidRoberts
   Format: MarkdownItexIs the localisation anything like an $(\infty,1)$-$\Pi$-[[pretopos]]?

   Is the localisation anything like an $(\infty,1)$-$\Pi$-pretopos?

5. • CommentRowNumber5.
   • CommentAuthorMarc Hoyois
   • CommentTimeJul 22nd 2013
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   Author: Marc Hoyois
   Format: MarkdownItexI'm not sure. It should be [[extensive category|extensive]] in the sense that $E/a\times E/b\simeq E/(a+b)$, but I guess it may not be [[exact category|exact]]. One way in which motivic homotopy theory fails to be a topos is that discrete group objects are not equivalent to connected 1-truncated objects: not all discrete groups are loops (see Remark 3.5 [here](http://arxiv.org/pdf/1008.4915v1.pdf)). Is this possible in a $\Pi$-pretopos? Given this, I expect that many of the generalities at [[cohomology]] do not apply to motivic homotopy. For now I'll add a remark at [[cohomology]] that there are notions of cohomology that are not subsumed by the toposic definition.

   I’m not sure. It should be extensive in the sense that $E/a\times E/b\simeq E/(a+b)$, but I guess it may not be exact.

   One way in which motivic homotopy theory fails to be a topos is that discrete group objects are not equivalent to connected 1-truncated objects: not all discrete groups are loops (see Remark 3.5 here). Is this possible in a $\Pi$-pretopos?

   Given this, I expect that many of the generalities at cohomology do not apply to motivic homotopy. For now I’ll add a remark at cohomology that there are notions of cohomology that are not subsumed by the toposic definition.

6. • CommentRowNumber6.
   • CommentAuthorDavid_Corfield
   • CommentTimeJul 22nd 2013
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   Author: David_Corfield
   Format: MarkdownItexWe must remind Urs when he returns from his holiday to look at this thread, if he doesn't notice it himself.

   We must remind Urs when he returns from his holiday to look at this thread, if he doesn’t notice it himself.

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeJul 27th 2013
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   Author: Urs
   Format: MarkdownItexHi, indeed, I am on vacation and with no official online time granted. So just very briefly, some comments: 1. of course all definitions are subject to modifications. If most theories of cohomology are about mapping spaces in an $\infty$-topos, then clearly every mapping space in any other $\infty$-category is a kind of "generalized" cohomology. 1. So the question is: which properties do we demand of a theory of cohomology? The thing about $\infty$-toposes is that here cohomology has the crucial property that it _classifies_ something. Namely the classification theory of [[principal infinity-bundles]] in an $\infty$-toposes intimately uses the Giraud-Rezk-Lurie characterization of $\infty$-toposes -- all except the "coproducts are disjoint"-part. Therefore if one wants cohomology to be such that it has the expected classification properties, then something at least awefully close to $\infty$-toposes is required. 1. But maybe most important for the present discussion is that _in_ a given $\infty$-topos we are of course free to restrict attention to certain especially nice coefficient objects. This happens all the time. The fully general [[nonabelian cohomology]] which allows any coefficient object whatsoever is not often considered. Instead in _ordinary_ cohomology we take the coefficients to be Eilenberg-MacLane objects, in Eilenberg-Steenrod-style generalized cohomology we take them to be things in the image of $\Omega^\infty$, and so forth. This last example is good to keep in mind: Eilenberg-Steenrod style cohomology is often considered as happening in the $\infty$-category of spectra, which is not a topos. But I think it is actually a more truthful perspective to see it as happening in the $\infty$-topos of $\infty$-groupoids under the $\Omega^\infty$-adjunction. And this last perspective seems to also apply to motivic cohomology, I would say, under the $\mathbb{A}^1$-stabilization reflection. So in conclusion I think evidence is still all pointing to the statement that cohomology wants to happen in an ambient $\infty$-topos, where we need to understand that this is fully general "nonabelian" cohomology and that we often want to restrict the choice of coefficient objects. That said, of course all this or something similar might be worthwhile to say more explicitly in the nLab entry on cohomology. And now I need to go back to vacation. :-)

   Hi,

   indeed, I am on vacation and with no official online time granted. So just very briefly, some comments:

   1. of course all definitions are subject to modifications. If most theories of cohomology are about mapping spaces in an $\infty$-topos, then clearly every mapping space in any other $\infty$-category is a kind of “generalized” cohomology.

   2. So the question is: which properties do we demand of a theory of cohomology? The thing about $\infty$-toposes is that here cohomology has the crucial property that it classifies something. Namely the classification theory of principal infinity-bundles in an $\infty$-toposes intimately uses the Giraud-Rezk-Lurie characterization of $\infty$-toposes – all except the “coproducts are disjoint”-part. Therefore if one wants cohomology to be such that it has the expected classification properties, then something at least awefully close to $\infty$-toposes is required.

   3. But maybe most important for the present discussion is that in a given $\infty$-topos we are of course free to restrict attention to certain especially nice coefficient objects. This happens all the time. The fully general nonabelian cohomology which allows any coefficient object whatsoever is not often considered. Instead in ordinary cohomology we take the coefficients to be Eilenberg-MacLane objects, in Eilenberg-Steenrod-style generalized cohomology we take them to be things in the image of $\Omega^\infty$, and so forth. This last example is good to keep in mind: Eilenberg-Steenrod style cohomology is often considered as happening in the $\infty$-category of spectra, which is not a topos. But I think it is actually a more truthful perspective to see it as happening in the $\infty$-topos of $\infty$-groupoids under the $\Omega^\infty$-adjunction.

      And this last perspective seems to also apply to motivic cohomology, I would say, under the $\mathbb{A}^1$-stabilization reflection.

   So in conclusion I think evidence is still all pointing to the statement that cohomology wants to happen in an ambient $\infty$-topos, where we need to understand that this is fully general “nonabelian” cohomology and that we often want to restrict the choice of coefficient objects.

   That said, of course all this or something similar might be worthwhile to say more explicitly in the nLab entry on cohomology.

   And now I need to go back to vacation. :-)

8. • CommentRowNumber8.
   • CommentAuthorjim_stasheff
   • CommentTimeJul 27th 2013
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   Author: jim_stasheff
   Format: TextSomewhere in this discussion or the related stuff on n-lab is there an emphasis of the analog of the Eilenberg Steenrod axioms as telling us when a maaping space is a cohomology theory??

   Somewhere in this discussion or the related stuff on n-lab is there an emphasis of the analog of the Eilenberg Steenrod axioms as telling us when a maaping space is a cohomology theory??
9. • CommentRowNumber9.
   • CommentAuthorMarc Hoyois
   • CommentTimeAug 2nd 2013
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   Author: Marc Hoyois
   Format: MarkdownItex> And this last perspective seems to also apply to motivic cohomology, I would say, under the $\mathbb{A}^1$-stabilization reflection. I wouldn't say so. Nisnevich excision and $\mathbb{A}^1$-invariance are both equally important features of motivic spaces, and if you remove either you are left with a completely different theory. The Nisnevich topos by itself barely has anything to do with motivic homotopy theory. Of course cohomology in the motivic homotopy category classifies interesting things, even if it doesn't have all the properties you expect. For example, $BGL_n$ classifies rank $n$ vector bundles over smooth affine schemes -- this is not true in the Nisnevich topos. A non-topological localization seems unavoidable for this type of results, so I think this is evidence that sometimes cohomology does *not* want to happen in a topos.

   And this last perspective seems to also apply to motivic cohomology, I would say, under the $\mathbb{A}^1$-stabilization reflection.

   I wouldn’t say so. Nisnevich excision and $\mathbb{A}^1$-invariance are both equally important features of motivic spaces, and if you remove either you are left with a completely different theory. The Nisnevich topos by itself barely has anything to do with motivic homotopy theory.

   Of course cohomology in the motivic homotopy category classifies interesting things, even if it doesn’t have all the properties you expect. For example, $BGL_n$ classifies rank $n$ vector bundles over smooth affine schemes – this is not true in the Nisnevich topos. A non-topological localization seems unavoidable for this type of results, so I think this is evidence that sometimes cohomology does not want to happen in a topos.

10. • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeAug 4th 2013
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    Author: Urs
    Format: MarkdownItexAs I said in point one, please feel free to add a comment to the entry. But concerning this point: whether the localization is topological or not, as long as it is reflective the homs are those of the ambient $\infty$-topos. To decide whether a cocycle $\infty$-groupoid is more naturally regarded in some $\infty$-topos or one of its reflective subcategories one needs to state the desired constraints.

    As I said in point one, please feel free to add a comment to the entry.

    But concerning this point: whether the localization is topological or not, as long as it is reflective the homs are those of the ambient $\infty$-topos. To decide whether a cocycle $\infty$-groupoid is more naturally regarded in some $\infty$-topos or one of its reflective subcategories one needs to state the desired constraints.

11. • CommentRowNumber11.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 4th 2013
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    Author: David_Corfield
    Format: MarkdownItexGood to have you back, Urs. On the subject of cohomology, I've been [chatting](http://golem.ph.utexas.edu/category/2013/07/minhyong_kim_in_the_reasoner.html) with Minhyong Kim, and noted in some [slides](http://minhyongkim.files.wordpress.com/2013/07/busan.pdf) of his that he claims to be doing >non-abelian gauge theory in arithmetic topology, on the basis of mapping into some cohomological target.

    Good to have you back, Urs.

    On the subject of cohomology, I’ve been chatting with Minhyong Kim, and noted in some slides of his that he claims to be doing

    non-abelian gauge theory in arithmetic topology,

    on the basis of mapping into some cohomological target.

12. • CommentRowNumber12.
    • CommentAuthorMarc Hoyois
    • CommentTimeAug 6th 2013
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    Author: Marc Hoyois
    Format: MarkdownItex> But concerning this point: whether the localization is topological or not, as long as it is reflective the homs are those of the ambient $\infty$-topos. To decide whether a cocycle $\infty$-groupoid is more naturally regarded in some $\infty$-topos or one of its reflective subcategories one needs to state the desired constraints. Well, yeah, but with this perspective you could as well rewrite the whole page on [[cohomology]] using presheaf (∞,1)-categories instead of (∞,1)-topoi. Generally speaking, the fact that MHT is not an (∞,1)-topos makes the theory more difficult, but that's something we have to live with and invoking a containing (∞,1)-topos doesn't help. Morel's connectivity theorem and his theory of A1-coverings are examples where one must work really hard to prove that the A1-homotopy category has some basic topos-like features. I'll add that the cohomology-only-happens-in-a-topos POV does not account for basic things like cohomology of spectra that are not suspension spectra! After taking a closer look, it seems to me that everything at [[cohomology]] makes sense in any (∞,1)-category (with existence of some (co)limits assumed here and there), and much of it is still relevant. The obvious exception is the section [Relation to (∞,1)-topos theory](http://ncatlab.org/nlab/show/cohomology#ToposTheory), but the "Historical aspect" part is only supporting the idea that cohomology is given by mapping spaces, not that it should be given by mapping spaces *in an (∞,1)-topos*, and the "Abstract aspect" part is only about the classication theorem for BG-cohomology in an (∞,1)-topos (which actually holds in MHT, provided that G has a delooping). Given all this, would you be opposed to a minor rewriting of the page to include the more general definition, with the "Abstract aspect" section promoted to a section about the specifics of cohomology in an (∞,1)-topos?

    But concerning this point: whether the localization is topological or not, as long as it is reflective the homs are those of the ambient $\infty$-topos. To decide whether a cocycle $\infty$-groupoid is more naturally regarded in some $\infty$-topos or one of its reflective subcategories one needs to state the desired constraints.

    Well, yeah, but with this perspective you could as well rewrite the whole page on cohomology using presheaf (∞,1)-categories instead of (∞,1)-topoi.

    Generally speaking, the fact that MHT is not an (∞,1)-topos makes the theory more difficult, but that’s something we have to live with and invoking a containing (∞,1)-topos doesn’t help. Morel’s connectivity theorem and his theory of A1-coverings are examples where one must work really hard to prove that the A1-homotopy category has some basic topos-like features.

    I’ll add that the cohomology-only-happens-in-a-topos POV does not account for basic things like cohomology of spectra that are not suspension spectra!

    After taking a closer look, it seems to me that everything at cohomology makes sense in any (∞,1)-category (with existence of some (co)limits assumed here and there), and much of it is still relevant. The obvious exception is the section Relation to (∞,1)-topos theory, but the “Historical aspect” part is only supporting the idea that cohomology is given by mapping spaces, not that it should be given by mapping spaces in an (∞,1)-topos, and the “Abstract aspect” part is only about the classication theorem for BG-cohomology in an (∞,1)-topos (which actually holds in MHT, provided that G has a delooping).

    Given all this, would you be opposed to a minor rewriting of the page to include the more general definition, with the “Abstract aspect” section promoted to a section about the specifics of cohomology in an (∞,1)-topos?

13. • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeAug 6th 2013
    • (edited Aug 6th 2013)
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    Author: Urs
    Format: MarkdownItexI am not opposed to you editing the page! Let me briefly recall what I am saying: the properties of what one takes to be the ambient $\infty$-category determine what properties one's cohomology theory has. $\infty$-Toposes as homes for cohomology are singled out by the fact that among all presentable $\infty$-categories, they provide almost exactly the structure that makes the classification of principal $\infty$-bundles come out as expected: because that involves the axioms "universal colimits" and "groupoid objects are effective".

    I am not opposed to you editing the page!

    Let me briefly recall what I am saying: the properties of what one takes to be the ambient $\infty$-category determine what properties one’s cohomology theory has. $\infty$-Toposes as homes for cohomology are singled out by the fact that among all presentable $\infty$-categories, they provide almost exactly the structure that makes the classification of principal $\infty$-bundles come out as expected: because that involves the axioms “universal colimits” and “groupoid objects are effective”.

14. • CommentRowNumber14.
    • CommentAuthorMarc Hoyois
    • CommentTimeAug 6th 2013
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    Author: Marc Hoyois
    Format: MarkdownItexRight, but this classification result only concerns cohomology with coefficients in a very specific object, and moreover that object BG and the notion of "principal ∞-bundle" already require something close to an ∞-topos to be defined. It makes more sense to me to view this result as a feature of ∞-topoi rather than as a feature of cohomology.

    Right, but this classification result only concerns cohomology with coefficients in a very specific object, and moreover that object BG and the notion of “principal ∞-bundle” already require something close to an ∞-topos to be defined. It makes more sense to me to view this result as a feature of ∞-topoi rather than as a feature of cohomology.

15. • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeAug 6th 2013
    • PermaLink
    Author: Urs
    Format: MarkdownItex> only concerns cohomology with coefficients in a very specific object, Usually people consider coefficients much more restrictive than any $\mathbf{B}G$. General $\mathbf{B}G$ yields general "[[nonabelian cohomology]]", wheras typically people impose strong extra conditions on $G$. But, yeah, please feel invited to edit the entry as you find appropriate. I am sure I won't have substantial disagreement. Maybe to put this in perspective: when I wrote that entry back then, there was -- as far as I am aware -- nothing even close to this kind of statement "all theories of cohomologies are about mapping spaces in $\infty$-categories" made explicit anywhere in the literature. In fact several people expressed doubt when I first mentioned this statement to them.. With hindsight of course it is obvious, and now that it is obvious, there is room for some fine-tuning, I guess.

    only concerns cohomology with coefficients in a very specific object,

    Usually people consider coefficients much more restrictive than any $\mathbf{B}G$. General $\mathbf{B}G$ yields general “nonabelian cohomology”, wheras typically people impose strong extra conditions on $G$.

    But, yeah, please feel invited to edit the entry as you find appropriate. I am sure I won’t have substantial disagreement.

    Maybe to put this in perspective: when I wrote that entry back then, there was – as far as I am aware – nothing even close to this kind of statement “all theories of cohomologies are about mapping spaces in $\infty$-categories” made explicit anywhere in the literature. In fact several people expressed doubt when I first mentioned this statement to them.. With hindsight of course it is obvious, and now that it is obvious, there is room for some fine-tuning, I guess.

16. • CommentRowNumber16.
    • CommentAuthorMarc Hoyois
    • CommentTimeAug 6th 2013
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    Author: Marc Hoyois
    Format: MarkdownItexAh yes, $\mathbf{B} G$ is more general than I realized above (I was thinking of discrete $G$, I guess). Anyway, I'll make some edits later. I'm quite sold on the mapping space idea, of course. Another example of essentially non-toposic cohomology that just came to me is the cohomology of pro-spaces: it doesn't even take place in an accessible ∞-category, but it still has interesting classifying features.

    Ah yes, $\mathbf{B} G$ is more general than I realized above (I was thinking of discrete $G$, I guess). Anyway, I’ll make some edits later. I’m quite sold on the mapping space idea, of course.

    Another example of essentially non-toposic cohomology that just came to me is the cohomology of pro-spaces: it doesn’t even take place in an accessible ∞-category, but it still has interesting classifying features.

17. • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeAug 6th 2013
    • (edited Aug 6th 2013)
    • PermaLink
    Author: Urs
    Format: MarkdownItexJust one random thought: When (re)writing the entry, maybe good to keep in mind how the cycles of abstraction and generalization work in mathematics. For instance here is an example of a somewhat similar flavor: first we realize that traditional logic is equivalently the theory of certain posets (abstraction). Then we turn this around and assign to _every_ category its internal logic (generalization). The resulting logics are way more general than what was originally thought of such, but now they are naturally seen as generalizations. Same here: first we see realize that most traditional notions of cohomology are about mapping spaces in something close to $\infty$-toposes (abstraction). Then we can turn this around and declare that cohomology theory is really nothing but the theory of hom-spaces in $\infty$-categories whatsoever (generalization). Incidentally, taken together both thense abstractions+generalization say that at this level of generality there is no real difference between a) $\infty$-categories, b) $\infty$-logic, c) cohomology.

    Just one random thought: When (re)writing the entry, maybe good to keep in mind how the cycles of abstraction and generalization work in mathematics.

    For instance here is an example of a somewhat similar flavor: first we realize that traditional logic is equivalently the theory of certain posets (abstraction). Then we turn this around and assign to every category its internal logic (generalization). The resulting logics are way more general than what was originally thought of such, but now they are naturally seen as generalizations.

    Same here: first we see realize that most traditional notions of cohomology are about mapping spaces in something close to $\infty$-toposes (abstraction). Then we can turn this around and declare that cohomology theory is really nothing but the theory of hom-spaces in $\infty$-categories whatsoever (generalization).

    Incidentally, taken together both thense abstractions+generalization say that at this level of generality there is no real difference between a) $\infty$-categories, b) $\infty$-logic, c) cohomology.

18. • CommentRowNumber18.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 6th 2013
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    Author: David_Corfield
    Format: MarkdownItexHence we should expect [Cohomology Detects Failures of Classical Mathematics](http://golem.ph.utexas.edu/category/2013/07/cohomology_detects_failures_of.html)?

    Hence we should expect Cohomology Detects Failures of Classical Mathematics?

19. • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeAug 6th 2013
    • (edited Aug 6th 2013)
    • PermaLink
    Author: Urs
    Format: MarkdownItexHi David, yes, that's actually an example. This aspect I had made a comment on recently at _[[discrete object]]_ in the section _[In infinity-toposes](http://ncatlab.org/nlab/show/discrete%20object#ExamplesInInfinityToposes)_. And, yes, by this general identification of cohomology with mapping spaces I find that "cohomology can detect failures of classical mathematics" is almost a tautology. But it's one of those general abstract tautologies that are useful to make explicit.

    Hi David,

    yes, that’s actually an example. This aspect I had made a comment on recently at discrete object in the section In infinity-toposes.

    And, yes, by this general identification of cohomology with mapping spaces I find that “cohomology can detect failures of classical mathematics” is almost a tautology. But it’s one of those general abstract tautologies that are useful to make explicit.

20. • CommentRowNumber20.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 7th 2013
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    Author: David_Corfield
    Format: MarkdownItexI [asked](http://golem.ph.utexas.edu/category/2013/07/cohomology_detects_failures_of.html#c044333) Mike whether the failure of Whitehead's theorem was part of the same story. He said he wondered about this, but couldn't see how.

    I asked Mike whether the failure of Whitehead’s theorem was part of the same story. He said he wondered about this, but couldn’t see how.

21. • CommentRowNumber21.
    • CommentAuthorMarc Hoyois
    • CommentTimeAug 7th 2013
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    Author: Marc Hoyois
    Format: MarkdownItexI made some minimal changes at [[cohomology]] to account for the more general definition, but there are still some inconsistencies to address. I also wrote anew the section about [exotic gradings](http://ncatlab.org/nlab/show/cohomology#Grading), following [our discussion](http://nforum.mathforge.org/discussion/2258/question-by-visitor-about-bigrading-in-motivic-cohomology/#Item_18) from yesterday.

    I made some minimal changes at cohomology to account for the more general definition, but there are still some inconsistencies to address.

    I also wrote anew the section about exotic gradings, following our discussion from yesterday.

22. • CommentRowNumber22.
    • CommentAuthorUrs
    • CommentTimeAug 7th 2013
    • PermaLink
    Author: Urs
    Format: MarkdownItexThanks!

    Thanks!

23. • CommentRowNumber23.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 7th 2013
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexThe trouble with modifying one page is that others might need it too. E.g., at [[relative cohomology]] >Recall the general abstract definition of [[cohomology]], as discussed there... which is in terms only of $(\infty, 1)$-toposes.

    The trouble with modifying one page is that others might need it too. E.g., at relative cohomology

    Recall the general abstract definition of cohomology, as discussed there…

    which is in terms only of $(\infty, 1)$-toposes.

24. • CommentRowNumber24.
    • CommentAuthorUrs
    • CommentTimeAug 7th 2013
    • PermaLink
    Author: Urs
    Format: MarkdownItexWe could eventually adapt, but there is not much harm done in either case.

    We could eventually adapt, but there is not much harm done in either case.

25. • CommentRowNumber25.
    • CommentAuthorMarc Hoyois
    • CommentTimeAug 8th 2013
    • PermaLink
    Author: Marc Hoyois
    Format: MarkdownItex> After taking a closer look, it seems to me that everything at [[cohomology]] makes sense in any (∞,1)-category (with existence of some (co)limits assumed here and there), and much of it is still relevant. The obvious exception is the section [Relation to (∞,1)-topos theory](http://ncatlab.org/nlab/show/cohomology#ToposTheory), but the "Historical aspect" part is only supporting the idea that cohomology is given by mapping spaces, not that it should be given by mapping spaces *in an (∞,1)-topos*, and the "Abstract aspect" part is only about the classication theorem for BG-cohomology in an (∞,1)-topos (which actually holds in MHT, provided that G has a delooping). I split the section "Relation to (∞,1)-topos theory" into two sections, "About the nPOV on cohomology" and "Cohomology in (∞,1)-topoi" (both could use some more material). I think the whole page makes sense on its own now.

    After taking a closer look, it seems to me that everything at cohomology makes sense in any (∞,1)-category (with existence of some (co)limits assumed here and there), and much of it is still relevant. The obvious exception is the section Relation to (∞,1)-topos theory, but the “Historical aspect” part is only supporting the idea that cohomology is given by mapping spaces, not that it should be given by mapping spaces in an (∞,1)-topos, and the “Abstract aspect” part is only about the classication theorem for BG-cohomology in an (∞,1)-topos (which actually holds in MHT, provided that G has a delooping).

    I split the section “Relation to (∞,1)-topos theory” into two sections, “About the nPOV on cohomology” and “Cohomology in (∞,1)-topoi” (both could use some more material). I think the whole page makes sense on its own now.

26. • CommentRowNumber26.
    • CommentAuthorUrs
    • CommentTimeAug 8th 2013
    • (edited Aug 8th 2013)
    • PermaLink
    Author: Urs
    Format: MarkdownItexIf we are still energetic about this topic, we should really turn that section which you now moved to the very bottom into something that lists a) properties of the ambient $\infty$-category against b) properties of the induced cohomology theory . Notably it should become clear that hom-spaces in a generic $\infty$-category (which might be an $\infty$-groupoid or might be a 1-category, which might be just a poset) could still be called "cohomology" from the super asbtract perspective, but will look nothing like what one traditionally expects a cohomology theory to be like. Right now I am too busy with other things to invest much time into this. But given the changes you made, I think eventually adding such a discussion is necessary.

    If we are still energetic about this topic, we should really turn that section which you now moved to the very bottom into something that lists

    a) properties of the ambient $\infty$-category

    against

    b) properties of the induced cohomology theory .

    Notably it should become clear that hom-spaces in a generic $\infty$-category (which might be an $\infty$-groupoid or might be a 1-category, which might be just a poset) could still be called “cohomology” from the super asbtract perspective, but will look nothing like what one traditionally expects a cohomology theory to be like.

    Right now I am too busy with other things to invest much time into this. But given the changes you made, I think eventually adding such a discussion is necessary.

27. • CommentRowNumber27.
    • CommentAuthorMarc Hoyois
    • CommentTimeAug 8th 2013
    • PermaLink
    Author: Marc Hoyois
    Format: MarkdownItexWell, that section used to be at the bottom, but it should probably be moved to near the beginning as well. I think it would be difficult to populate such a list though. The concept is so general that any nontrivial property will only appear under significant extra assumptions.

    Well, that section used to be at the bottom, but it should probably be moved to near the beginning as well. I think it would be difficult to populate such a list though. The concept is so general that any nontrivial property will only appear under significant extra assumptions.

28. • CommentRowNumber28.
    • CommentAuthorUrs
    • CommentTimeAug 8th 2013
    • (edited Aug 8th 2013)
    • PermaLink
    Author: Urs
    Format: MarkdownItex> I think it would be difficult to populate such a list though. You had already mentioned one further aspect that should go into the list: if the ambient $\infty$-category is stable, then cohomology is canonically $\mathbb{Z}$-graded. So it should start out like this | ambient $\infty$-category | $\Rightarrow$ properties of cohomology | |--|--| | $\infty$-topos | equivalent to principal $\infty$-bundles | | stable $\infty$-category | $\mathbb{Z}$-grading | | $\cdots$ | $\cdots$ |

    I think it would be difficult to populate such a list though.

    You had already mentioned one further aspect that should go into the list: if the ambient $\infty$-category is stable, then cohomology is canonically $\mathbb{Z}$-graded. So it should start out like this

    ambient $\infty$-category	$\Rightarrow$ properties of cohomology
    $\infty$-topos	equivalent to principal $\infty$-bundles
    stable $\infty$-category	$\mathbb{Z}$-grading
    $\cdots$	$\cdots$

29. • CommentRowNumber29.
    • CommentAuthorUrs
    • CommentTimeAug 9th 2013
    • PermaLink
    Author: Urs
    Format: MarkdownItexI have added this remark super-briefly to a new section _[Properties -- Dependence on ambient category](http://ncatlab.org/nlab/show/cohomology#DependenceOnTheAmbientCategory)_. This is more of a reminder to comeback to it later...

    I have added this remark super-briefly to a new section Properties – Dependence on ambient category. This is more of a reminder to comeback to it later…