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1. • CommentRowNumber1.
   • CommentAuthorDavid_Corfield
   • CommentTimeJun 20th 2012
   • (edited Jun 26th 2012)
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   Author: David_Corfield
   Format: MarkdownItexUrs has started a page on [[obstruction]]. Is there a dual story to tell in terms of homotopy measuring obstruction to certain extension problems? Is there some connection between these problems of lifting and extension, and those dealt with by Kan lifts and extensions? In the latter case, it's not just a question of being obstructed. Rather, one finds the best approximations. Or would the better analogy be with [[(infinity,1)-Kan extension]]s?

   Urs has started a page on obstruction. Is there a dual story to tell in terms of homotopy measuring obstruction to certain extension problems?

   Is there some connection between these problems of lifting and extension, and those dealt with by Kan lifts and extensions? In the latter case, it’s not just a question of being obstructed. Rather, one finds the best approximations.

   Or would the better analogy be with (infinity,1)-Kan extensions?

2. • CommentRowNumber2.
   • CommentAuthorTim_Porter
   • CommentTimeJun 20th 2012
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   Author: Tim_Porter
   Format: MarkdownItexPerhaps there is a need for an entry that summarises the way that, say, Spanier handles obstruction theory. That looks at extension problems as the main example if I remember correctly. It then translates them into lifting problems in Postnikov towers. Sometimes the classical case of a problem gives some insights that the more modern treatments assume as given.

   Perhaps there is a need for an entry that summarises the way that, say, Spanier handles obstruction theory. That looks at extension problems as the main example if I remember correctly. It then translates them into lifting problems in Postnikov towers. Sometimes the classical case of a problem gives some insights that the more modern treatments assume as given.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJun 20th 2012
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   Author: Urs
   Format: MarkdownItex> Is there a dual story to tell in terms of homotopy measuring obstruction to certain extension problems? As we have noticed before, people don't usually speak of the "nonabelian homotopy" that would appear in the formal dual of this statement. But in parts this is simply because classifying spaces play the role of both: coefficient objects for nonabelian cohomology, as well as domain objects for "nonabelian homotopy", namely for [[universal characteristic classes]]. In this sense, the formal dual of the obstruction theory discussed so far is the following extension problem: we start with a universal characteristic map $$ \mathbf{B}G \stackrel{\mathbf{c}}{\to} \mathbf{B}^n A $$ representing a class $[\mathbf{c}] \in H^n(\mathbf{B}G, A)$ in the $A$-cohomology of $\mathbf{B}G$. Then given a morphism $\phi : \mathbf{B}G \to \mathbf{B}H$ we may ask for the obstruction to extending $\mathbf{c}$ along it. Now the statement is: if $\phi$ is a homotopy cofiber, then there is a good obstruction theory to answer this question. Namely in that situation we are looking at a diagram of the form $$ \array{ \mathbf{B}Q &\stackrel{f}{\to}& \mathbf{B}G &\stackrel{\mathbf{c}}{\to}& \mathbf{B}^n A \\ \downarrow && \downarrow^{\mathrlap{\phi}} & \nearrow_{\mathrlap{\hat \mathbf{c}}} \\ * &\to& \mathbf{B}H } $$ where the left square is an $\infty$-pushout. By its universal property, the extension $\hat {\mathbf{c}}$ of $\mathbf{c}$ exists as indicated precisely if the class $$ [f^* \mathbf{c}] \in H^n(\mathbf{B}G, A) $$ is trivial.

   Is there a dual story to tell in terms of homotopy measuring obstruction to certain extension problems?

   As we have noticed before, people don’t usually speak of the “nonabelian homotopy” that would appear in the formal dual of this statement. But in parts this is simply because classifying spaces play the role of both: coefficient objects for nonabelian cohomology, as well as domain objects for “nonabelian homotopy”, namely for universal characteristic classes.

   In this sense, the formal dual of the obstruction theory discussed so far is the following extension problem:

   we start with a universal characteristic map

   $$\mathbf{B}G \stackrel{\mathbf{c}}{\to} \mathbf{B}^n A$$

   representing a class $[\mathbf{c}] \in H^n(\mathbf{B}G, A)$ in the $A$-cohomology of $\mathbf{B}G$. Then given a morphism $\phi : \mathbf{B}G \to \mathbf{B}H$ we may ask for the obstruction to extending $\mathbf{c}$ along it.

   Now the statement is: if $\phi$ is a homotopy cofiber, then there is a good obstruction theory to answer this question. Namely in that situation we are looking at a diagram of the form

   $$\array{ \mathbf{B}Q &\stackrel{f}{\to}& \mathbf{B}G &\stackrel{\mathbf{c}}{\to}& \mathbf{B}^n A \\ \downarrow && \downarrow^{\mathrlap{\phi}} & \nearrow_{\mathrlap{\hat \mathbf{c}}} \\ * &\to& \mathbf{B}H }$$

   where the left square is an $\infty$-pushout. By its universal property, the extension $\hat {\mathbf{c}}$ of $\mathbf{c}$ exists as indicated precisely if the class

   $$[f^* \mathbf{c}] \in H^n(\mathbf{B}G, A)$$

   is trivial.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJun 20th 2012
   • (edited Jun 20th 2012)
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   Author: Urs
   Format: MarkdownItex> That looks at extension problems as the main example if I remember correctly. It then translates them into lifting problems in Postnikov towers Just take $F \to A$, in the notation of [[obstruction|the entry]], to be a Postnikov stage, then you get that theory.

   That looks at extension problems as the main example if I remember correctly. It then translates them into lifting problems in Postnikov towers

   Just take $F \to A$, in the notation of the entry, to be a Postnikov stage, then you get that theory.

5. • CommentRowNumber5.
   • CommentAuthorTim_Porter
   • CommentTimeJun 20th 2012
   • (edited Jun 20th 2012)
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   Author: Tim_Porter
   Format: MarkdownItexThat is put in a 'nutshell' as they say. I like what you have written there. A more chatty discussion entry may still be useful. I will see if I can put one together if I have time, but my list of new entries 'to do' is getting longer.

   That is put in a ’nutshell’ as they say. I like what you have written there. A more chatty discussion entry may still be useful. I will see if I can put one together if I have time, but my list of new entries ’to do’ is getting longer.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeJun 20th 2012
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   Author: Urs
   Format: MarkdownItexI like to distinguish between the theory and its presentations. Obstruction theory _is that simple_ and it is a pity that this is not made clearer in the literature. This is not just a nutshell version, this is the full theory. What can get more intricate is presenting the various structures involved by various models, and this is what you have in mind. But it should not obfuscate the plain theory.

   I like to distinguish between the theory and its presentations. Obstruction theory is that simple and it is a pity that this is not made clearer in the literature. This is not just a nutshell version, this is the full theory.

   What can get more intricate is presenting the various structures involved by various models, and this is what you have in mind. But it should not obfuscate the plain theory.

7. • CommentRowNumber7.
   • CommentAuthorTim_Porter
   • CommentTimeJun 20th 2012
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   Author: Tim_Porter
   Format: MarkdownItexBy saying that you had put it in a nutshell, I was paying you a complement. :-) You said all that was needed to describe the theory in a few words. The simplicity of the idea is not clear in the literature and I was always wary of obstruction theory since it seemed complex. My point however is that the gap between the literature and the simple description that you give may need bridging so as to wean users off the more obscure version. Once used to the simple version they can throw away the bridge.

   By saying that you had put it in a nutshell, I was paying you a complement. :-) You said all that was needed to describe the theory in a few words. The simplicity of the idea is not clear in the literature and I was always wary of obstruction theory since it seemed complex. My point however is that the gap between the literature and the simple description that you give may need bridging so as to wean users off the more obscure version. Once used to the simple version they can throw away the bridge.

8. • CommentRowNumber8.
   • CommentAuthorDavid_Corfield
   • CommentTimeJun 20th 2012
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   Author: David_Corfield
   Format: MarkdownItexRe #3, does this extension situation crop up much? If it's worth including in the [[obstruction]] entrty, then note $$ [f^* \mathbf{c}] \in H^n(\mathbf{B}G, A) $$ should be "in $H^n(\mathbf{B}Q, A)$".

   Re #3, does this extension situation crop up much? If it’s worth including in the obstruction entrty, then note

   $$[f^* \mathbf{c}] \in H^n(\mathbf{B}G, A)$$

   should be “in $H^n(\mathbf{B}Q, A)$”.

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeJun 20th 2012
   • (edited Jun 20th 2012)
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   Author: Urs
   Format: MarkdownItexDavid, right, sorry for all these typos. Yes, that extension problem does arise in practice. A central example that I happen to care about is that where one considers refined Lie group cohomology on simply connected Lie groups and is asking for ways to push it down to discrete quotients, i.e. non-simply connected Lie groups integrating the same Lie algebra. This is often phrased in terms of "multiplicative bundle gerbes" over these Lie groups, but that's just another way of talking about the corresponding cohomology. There should be many more examples, I can try to add more later when I feel that I have the leisure. Currently I am a bit short of time. Accordingly, if you feel you have the time to put more of this discussion here into the entry, please do!

   David,

   right, sorry for all these typos.

   Yes, that extension problem does arise in practice.

   A central example that I happen to care about is that where one considers refined Lie group cohomology on simply connected Lie groups and is asking for ways to push it down to discrete quotients, i.e. non-simply connected Lie groups integrating the same Lie algebra. This is often phrased in terms of “multiplicative bundle gerbes” over these Lie groups, but that’s just another way of talking about the corresponding cohomology.

   There should be many more examples, I can try to add more later when I feel that I have the leisure. Currently I am a bit short of time. Accordingly, if you feel you have the time to put more of this discussion here into the entry, please do!

10. • CommentRowNumber10.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 20th 2012
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    Author: David_Corfield
    Format: MarkdownItexOK, I've pasted what you wrote in. By the way, is "Extension through Postnikov stages" the right title for a section on lifts?

    OK, I’ve pasted what you wrote in.

    By the way, is “Extension through Postnikov stages” the right title for a section on lifts?

11. • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJun 20th 2012
    • (edited Jun 20th 2012)
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    Author: Urs
    Format: MarkdownItexThanks! True, it should better be "lift through Postnikov stages". I have fixed it in the entry. Also added some hyperlinks. Thanks again.

    Thanks!

    True, it should better be “lift through Postnikov stages”. I have fixed it in the entry. Also added some hyperlinks. Thanks again.

12. • CommentRowNumber12.
    • CommentAuthorjim_stasheff
    • CommentTimeJun 21st 2012
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    Author: jim_stasheff
    Format: TextI find the discussion very slanted by the emphasis on BG and on lifting as primary with extension secondary. Are fibrations more important than cofibrations? or is it just the language that biases? Urs has several applications in mind, but they are only (a current) part of the story. There is also obstruction theory as in deformation theory and... From an old fogey of another generation.

    I find the discussion very slanted by the emphasis on BG and on lifting as primary with extension secondary.
    Are fibrations more important than cofibrations? or is it just the language that biases?
    Urs has several applications in mind, but they are only (a current) part of the story.
    
    There is also obstruction theory as in deformation theory and...
    
    From an old fogey of another generation.
13. • CommentRowNumber13.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 21st 2012
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    Author: David_Corfield
    Format: MarkdownItexI think the page is very much 'under construction', and content and organization are still to be decided. How would the page best be introduced? What is the range of things which can be obstructed? Just lifting and extension? The link to [[deformation theory]] needs to be made, especially as it says there >Formal deformation theory studies the obstruction theory of extensions to infinitesimal thickenings. So, I linked from there to [[obstruction]].

    I think the page is very much ’under construction’, and content and organization are still to be decided. How would the page best be introduced? What is the range of things which can be obstructed? Just lifting and extension?

    The link to deformation theory needs to be made, especially as it says there

    Formal deformation theory studies the obstruction theory of extensions to infinitesimal thickenings.

    So, I linked from there to obstruction.

14. • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeJan 23rd 2013
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    Author: Urs
    Format: MarkdownItexadded to _[[obstruction]]_ a paragraph _[Examples: obstruction to quantization -- quantum anomaly](http://ncatlab.org/nlab/show/obstruction#QuantumAnomaly)_

    added to obstruction a paragraph Examples: obstruction to quantization – quantum anomaly