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1. • CommentRowNumber1.
   • CommentAuthorRyan Thorngren
   • CommentTimeJul 10th 2013
   • PermaLink
   Author: Ryan Thorngren
   Format: MarkdownItexHello, Everyone. This is my first post here, and I hope it's in the right place. This is an extension of the discussion in my mathoverflow question here: http://mathoverflow.net/questions/135882/topological-class-of-a-flat-2-bundle-with-nontrivial-extension-class I'm currently trying to calculate the cohomology of some classifying spaces of n-groups. I'd be interested to hear about the general story, but I'd like to be as concrete as possible. The first case I want to consider is a 2-group presented by a crossed module $t:H \to G$, where t has finite kernel and cokernel. I have a pretty good understanding of the classifying spaces $B(coker t)$ and $B^2(ker t)$, and it's known that the 2-group can be described using the action of the cokernel on the kernel (which is in general nontrivial) and the extension class of the crossed module in the corresponding twisted cohomology $H^3(coker t, ker t)$. I'd like to understand the classifying space of the 2-group using this data, where by understand I mean have a cell structure or understand it as some kind of n-bundle from which I can use the Hoschild-Serre spectral sequence to calculate some things in low degrees. David Roberts in his answer to my mathoverflow question mentioned--and I think he meant in the special case that the action of the cokernel is trivial--that indeed the classifying space of the 2-group is a 2-bundle with fiber $B^2(ker t)$ over $B(coker t)$. A cocycle representing the extension class $(ker t)^3 \to coker t$ I feel must have something to do with the Cech cocycle of this 2-bundle. What is the precise relationship? What happens when the action is nontrivial? Thanks.

   Hello, Everyone. This is my first post here, and I hope it’s in the right place.

   This is an extension of the discussion in my mathoverflow question here: http://mathoverflow.net/questions/135882/topological-class-of-a-flat-2-bundle-with-nontrivial-extension-class

   I’m currently trying to calculate the cohomology of some classifying spaces of n-groups. I’d be interested to hear about the general story, but I’d like to be as concrete as possible. The first case I want to consider is a 2-group presented by a crossed module $!A = \sum_{E: V} (E \to A}t:H \to G$, where t has finite kernel and cokernel. I have a pretty good understanding of the classifying spaces $B(coker t)$ and $B^2(ker t)$, and it’s known that the 2-group can be described using the action of the cokernel on the kernel (which is in general nontrivial) and the extension class of the crossed module in the corresponding twisted cohomology $H^3(coker t, ker t)$. I’d like to understand the classifying space of the 2-group using this data, where by understand I mean have a cell structure or understand it as some kind of n-bundle from which I can use the Hoschild-Serre spectral sequence to calculate some things in low degrees.

   David Roberts in his answer to my mathoverflow question mentioned–and I think he meant in the special case that the action of the cokernel is trivial–that indeed the classifying space of the 2-group is a 2-bundle with fiber $B^2(ker t)$ over $B(coker t)$. A cocycle representing the extension class $(ker t)^3 \to coker t$ I feel must have something to do with the Cech cocycle of this 2-bundle. What is the precise relationship?

   What happens when the action is nontrivial?

   Thanks.

2. • CommentRowNumber2.
   • CommentAuthorAndrew Stacey
   • CommentTimeJul 10th 2013
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   Author: Andrew Stacey
   Format: MarkdownItexAdding the link: <http://mathoverflow.net/questions/135882/topological-class-of-a-flat-2-bundle-with-nontrivial-extension-class> (they aren't automatically converted here).

   Adding the link: http://mathoverflow.net/questions/135882/topological-class-of-a-flat-2-bundle-with-nontrivial-extension-class (they aren’t automatically converted here).

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJul 10th 2013
   • (edited Jul 10th 2013)
   • PermaLink
   Author: Urs
   Format: MarkdownItexHi, first for the statement that we have a bundle of classifying spaces, as you suggest. Use (I add links just for bystanders...): 1. for $(H \stackrel{t}{\to} G \stackrel{\alpha}{\to} Aut(H))$ a [[crossed module]] corresponding to a [[2-group]] $\hat G$, there is a [[homotopy fiber sequence]] of [[delooping]] [[infinity-groupoids]] $$ \mathbf{B}^2 ker(t) \to \mathbf{B}\hat G \to \mathbf{B} coker(t) \,; $$ 1. for $A \to B \to C$ a homotopy fiber sequence in [[smooth infinity-groupoids]] such that the objects have a presentation by simplicial paracompact smooth manifolds, then [[geometric realization of cohesive infinity-groupoids]] ${\vert -\vert} \;:\; Smooth\infty Grpd \simeq Sh_\infty(SmoothMfd) \to \infty Grpd \simeq L_{whe}Top$ preserves this and yields a homotopy fiber sequence ${\vert A \vert} \to {\vert B \vert} \to {\vert C \vert}$ of topological spaces (homotopy types). To see the first, proceed as in the examples at [[homotopy fiber]]: replace the point inclusion $\ast \to \mathbf{B}coker(t)$ with the [[universal principal infinity-bundle]]-projection $\mathbf{E}coker(t) \simeq coker(t)//coker(t) \to \ast //coker(t) \simeq \mathbf{B} coker(t)$ which is a fibration resolution (say in the global projective [[model structure on simplicial presheaves]] $L_{lwhe} Funct(CartSp^{op}, sSet)$, which is sufficient for computing _finite_ homotopy limits). Then compute the ordinary fiber product of 2-groupoids of $$ coker(t)//coker(t) \to \ast // coker(t) \leftarrow \mathbf{B}\hat G \,. $$ The result is a "big" 2-groupoid which however is easily seen to receive an inclusion from $\mathbf{B}^2 ker(t)$ that is an equivalence. For the second statement: this is the content of the theorem in section 4.3.4.1 "Geometric realization of topological $\infty$-groupoids" in the pdf at [[schreiber:differential cohomology in a cohesive topos]]. So under the condition that the quotient $coker(t)$ and the kernel $ker(t)$ are nice enough to be represented by a manifold (but I guess that's what you are interested in anyway?) it follows that we have a homotopy fiber sequence of geometric realizations: $\vert \mathbf{B}^2 ker(t) \vert \to \vert \mathbf{B}\hat G\vert \to \vert \mathbf{B}coker(t)\vert$, yes. Indeed, again under the niceness assumption that the simplicial topological spaces representing these group stacks are "well-pointed" one can show (which you seem to be assuming anyway) using results by Danny Stevenson, that these geometric realizations are indeed the classifying spaces for the corresponding topological/smooth 2-bundles.

   Hi,

   first for the statement that we have a bundle of classifying spaces, as you suggest.

   Use (I add links just for bystanders…):

   1. for $(H \stackrel{t}{\to} G \stackrel{\alpha}{\to} Aut(H))$ a crossed module corresponding to a 2-group $\hat G$, there is a homotopy fiber sequence of delooping infinity-groupoids

      $$\mathbf{B}^2 ker(t) \to \mathbf{B}\hat G \to \mathbf{B} coker(t) \,;$$

   2. for $A \to B \to C$ a homotopy fiber sequence in smooth infinity-groupoids such that the objects have a presentation by simplicial paracompact smooth manifolds, then geometric realization of cohesive infinity-groupoids

      ${\vert -\vert} \;:\; Smooth\infty Grpd \simeq Sh_\infty(SmoothMfd) \to \infty Grpd \simeq L_{whe}Top$

      preserves this and yields a homotopy fiber sequence

      ${\vert A \vert} \to {\vert B \vert} \to {\vert C \vert}$ of topological spaces (homotopy types).

   To see the first, proceed as in the examples at homotopy fiber: replace the point inclusion $\ast \to \mathbf{B}coker(t)$ with the universal principal infinity-bundle-projection

   $\mathbf{E}coker(t) \simeq coker(t)//coker(t) \to \ast //coker(t) \simeq \mathbf{B} coker(t)$

   which is a fibration resolution (say in the global projective model structure on simplicial presheaves $L_{lwhe} Funct(CartSp^{op}, sSet)$, which is sufficient for computing finite homotopy limits). Then compute the ordinary fiber product of 2-groupoids of

   $$coker(t)//coker(t) \to \ast // coker(t) \leftarrow \mathbf{B}\hat G \,.$$

   The result is a “big” 2-groupoid which however is easily seen to receive an inclusion from $\mathbf{B}^2 ker(t)$ that is an equivalence.

   For the second statement: this is the content of the theorem in section 4.3.4.1 “Geometric realization of topological $\infty$-groupoids” in the pdf at differential cohomology in a cohesive topos (schreiber).

   So under the condition that the quotient $coker(t)$ and the kernel $ker(t)$ are nice enough to be represented by a manifold (but I guess that’s what you are interested in anyway?) it follows that we have a homotopy fiber sequence of geometric realizations:

   $\vert \mathbf{B}^2 ker(t) \vert \to \vert \mathbf{B}\hat G\vert \to \vert \mathbf{B}coker(t)\vert$,

   yes.

   Indeed, again under the niceness assumption that the simplicial topological spaces representing these group stacks are “well-pointed” one can show (which you seem to be assuming anyway) using results by Danny Stevenson, that these geometric realizations are indeed the classifying spaces for the corresponding topological/smooth 2-bundles.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJul 10th 2013
   • (edited Jul 10th 2013)
   • PermaLink
   Author: Urs
   Format: MarkdownItexNow for the classification: generally, a homotopy fiber sequence of the form $$ \array{ \mathbf{B}^2 A &\to& \mathbf{B}\hat G \\ && \downarrow \\ && \mathbf{B}G } $$ exhibits a $(\mathbf{B}A)$-[[infinity-gerbe|2-gerbe]] over $\mathbf{B}G$ and is classified by a map $$ \mathbf{B}G \to \mathbf{B}\mathbf{Aut}(\mathbf{B}^2 A) $$ into the [[delooping]] of the [[automorphism infinity-group|automorphism 3-group]] of the 2-group $\mathbf{B}A$. This map encodes the group 3-cocycle with coefficients in the non-trival $G$-module $A$ that you are looking for (but it is also more general). Specifically in crossed module language, the 3-group $\mathbf{Aut}(\mathbf{B}^2 A)$ is given by a groupal [[crossed complex]] of the form $$ A \stackrel{0}{\to} A \stackrel{Ad}{\to} Aut(A) $$ and so an $\infty$-groupal map to that from $G$ is in lowest degree an $A$-action and then in top degree the corresponding 3-cocycles.

   Now for the classification:

   generally, a homotopy fiber sequence of the form

   $$\array{ \mathbf{B}^2 A &\to& \mathbf{B}\hat G \\ && \downarrow \\ && \mathbf{B}G }$$

   exhibits a $(\mathbf{B}A)$-2-gerbe over $\mathbf{B}G$ and is classified by a map

   $$\mathbf{B}G \to \mathbf{B}\mathbf{Aut}(\mathbf{B}^2 A)$$

   into the delooping of the automorphism 3-group of the 2-group $\mathbf{B}A$.

   This map encodes the group 3-cocycle with coefficients in the non-trival $G$-module $A$ that you are looking for (but it is also more general).

   Specifically in crossed module language, the 3-group $\mathbf{Aut}(\mathbf{B}^2 A)$ is given by a groupal crossed complex of the form

   $$A \stackrel{0}{\to} A \stackrel{Ad}{\to} Aut(A)$$

   and so an $\infty$-groupal map to that from $G$ is in lowest degree an $A$-action and then in top degree the corresponding 3-cocycles.

5. • CommentRowNumber5.
   • CommentAuthorRyan Thorngren
   • CommentTimeJul 10th 2013
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   Author: Ryan Thorngren
   Format: MarkdownItexThanks again, Urs! Is the criterion that in the fiber sequence of smooth infinity-groupoids each object be presented by simplicial smooth manifolds mean by finite dimensional manifolds? What does present mean here? These classifying spaces are certainly not themselves homologically finite dimensional if the cokernel and kernel are finite.

   Thanks again, Urs!

   Is the criterion that in the fiber sequence of smooth infinity-groupoids each object be presented by simplicial smooth manifolds mean by finite dimensional manifolds? What does present mean here? These classifying spaces are certainly not themselves homologically finite dimensional if the cokernel and kernel are finite.

6. • CommentRowNumber6.
   • CommentAuthorjim_stasheff
   • CommentTimeJul 11th 2013
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   Author: jim_stasheff
   Format: Text"under the condition that the quotient coker(t) and the kernel ker(t) are nice enough to be represented by a manifold" is there any need for that? cf. a very nice exposition by Freed and Hopkins in a recent Bull AMS

   "under the condition that the quotient coker(t) and the kernel ker(t) are nice enough to be represented by a manifold"
   
   is there any need for that? cf. a very nice exposition by Freed and Hopkins in a recent Bull AMS
7. • CommentRowNumber7.
   • CommentAuthorDavidRoberts
   • CommentTimeJul 11th 2013
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   Author: DavidRoberts
   Format: MarkdownItexI wasn't assuming that the 2-group satisfied any properties, and I mean that there is a _3-bundle_ (=2-gerbe) over the classifying space of $coker(t)$. If we have a central extension of ordinary groups $A \to \hat{G} \to G$, then there is an $A$-bundle gerbe over $BG$, namely the lifting bundle gerbe ('lifting 2-bundle') of the universal bundle using this central extension. This is the one level down version of what Urs wrote. When I said 'bundle gerbe' in my first paragraph, I meant what you might call a _basic_ bundle gerbe over the group $coker(t)$, not the bundle _2-gerbe_ over the classifying space of $coker(t)$. These things are closely related, but clearly not the same

   I wasn’t assuming that the 2-group satisfied any properties, and I mean that there is a 3-bundle (=2-gerbe) over the classifying space of $coker(t)$. If we have a central extension of ordinary groups $A \to \hat{G} \to G$, then there is an $A$-bundle gerbe over $BG$, namely the lifting bundle gerbe (’lifting 2-bundle’) of the universal bundle using this central extension. This is the one level down version of what Urs wrote.

   When I said ’bundle gerbe’ in my first paragraph, I meant what you might call a basic bundle gerbe over the group $coker(t)$, not the bundle 2-gerbe over the classifying space of $coker(t)$. These things are closely related, but clearly not the same

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeJul 11th 2013
   • (edited Jul 11th 2013)
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   Author: Urs
   Format: MarkdownItex> Is the criterion that in the fiber sequence of smooth infinity-groupoids each object be presented by simplicial smooth manifolds mean by finite dimensional manifolds? No, infinite dimensional is fine. In fact simplicial paracompact topological spaces is fine. The only thing crucial is paracompactness, for that makes the [[nerve theorem]] apply, and this is how one sees that the geometric realization of $\infty$-stacks defined abstractly by a left adjoint to the constant $\infty$-stack functor coincides on these objects with the traditional [[geometric realization of simplicial topological spaces]]. > What does present mean here? So there is a caonical functor $$ Top^{\Delta^{op}} \to L_{lwhe} Funct(CartSp_{top}^{op}, sSet) \simeq Sh_\infty(CartSp_{top}) $$ or else $$ SmthMfd^{\Delta^{op}} \to L_{lwhe} Funct(CartSp_{smooth}^{op}, sSet) \simeq Sh_\infty(CartSp_{smooth}) $$ which takes a [[simplical topological space]] $X_\bullet$, regards it as the [[simplicial presheaf]] which sends a test space $U$ to the simplicial set $[k] \mapsto Hom_{Top}(U,X_k)$, and then views this as an object in the local [[model structure on simplicial presheafs]] and thereby as an object in the [[simplicial localization]] $L_{lwhe}(-)$ of that category at the local weak homotopy equivalences, which is our $\infty$-topos of $\infty$-stacks. So it's essentially a (re-)presentation as in the Yoneda lemma, only that it's a degreewise Yoneda lemma on simplicial presheaf followed by $\infty$-stackification.

   Is the criterion that in the fiber sequence of smooth infinity-groupoids each object be presented by simplicial smooth manifolds mean by finite dimensional manifolds?

   No, infinite dimensional is fine. In fact simplicial paracompact topological spaces is fine. The only thing crucial is paracompactness, for that makes the nerve theorem apply, and this is how one sees that the geometric realization of $\infty$-stacks defined abstractly by a left adjoint to the constant $\infty$-stack functor coincides on these objects with the traditional geometric realization of simplicial topological spaces.

   What does present mean here?

   So there is a caonical functor

   $$Top^{\Delta^{op}} \to L_{lwhe} Funct(CartSp_{top}^{op}, sSet) \simeq Sh_\infty(CartSp_{top})$$

   or else

   $$SmthMfd^{\Delta^{op}} \to L_{lwhe} Funct(CartSp_{smooth}^{op}, sSet) \simeq Sh_\infty(CartSp_{smooth})$$

   which takes a simplical topological space $X_\bullet$, regards it as the simplicial presheaf which sends a test space $U$ to the simplicial set $[k] \mapsto Hom_{Top}(U,X_k)$, and then views this as an object in the local model structure on simplicial presheafs and thereby as an object in the simplicial localization $L_{lwhe}(-)$ of that category at the local weak homotopy equivalences, which is our $\infty$-topos of $\infty$-stacks.

   So it’s essentially a (re-)presentation as in the Yoneda lemma, only that it’s a degreewise Yoneda lemma on simplicial presheaf followed by $\infty$-stackification.

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeJul 11th 2013
   • (edited Jul 11th 2013)
   • PermaLink
   Author: Urs
   Format: MarkdownItexre #6 : > is there any need for that? Need for regularity assumptions of stacks arise depending on what you want them to do. For instance it is not true that for every group $\infty$-stack $G$ the geometric realization of its delooping is a classifying topological space for $G$-principal $\infty$-bundles. That requires that $G$ is nice enough. Things like that.

   re #6 :

   is there any need for that?

   Need for regularity assumptions of stacks arise depending on what you want them to do. For instance it is not true that for every group $\infty$-stack $G$ the geometric realization of its delooping is a classifying topological space for $G$-principal $\infty$-bundles. That requires that $G$ is nice enough. Things like that.

10. • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeJul 11th 2013
    • (edited Jul 11th 2013)
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    Author: Urs
    Format: MarkdownItexre #7 > I mean that there is a 3-bundle (=2-gerbe) I will keep objecting to that way of using the word "$n$-gerbe" as synonymous with "$(n+1)$-bundle". * A _gerbe_ in the sense of Giraud is just a connected stack of groupoids. That need not even be a [[fiber infinity-bundle|higher fiber bundle]]. * A _$G$-gerbe_ in the sense of Giraud is a $\mathbf{B}G$-[[fiber infinity-bundle|fiber 2-bundle]], so a rather special fiber 2-bundle. Even if regarded as an [[associated infinity-bundle|associated 2-bundle]] to a principal 2-bundle these are $\mathbf{Aut}(\mathbf{B}G)$-principal 2-bundles only. * An _$A$-bundle gerbe_ is a $\mathbf{B}A$-[[principal 2-bundle]].

    re #7

    I mean that there is a 3-bundle (=2-gerbe)

    I will keep objecting to that way of using the word “$n$-gerbe” as synonymous with “$(n+1)$-bundle”.

    • A gerbe in the sense of Giraud is just a connected stack of groupoids. That need not even be a higher fiber bundle.

    • A $G$-gerbe in the sense of Giraud is a $\mathbf{B}G$-fiber 2-bundle, so a rather special fiber 2-bundle. Even if regarded as an associated 2-bundle to a principal 2-bundle these are $\mathbf{Aut}(\mathbf{B}G)$-principal 2-bundles only.

    • An $A$-bundle gerbe is a $\mathbf{B}A$-principal 2-bundle.

11. • CommentRowNumber11.
    • CommentAuthorDavidRoberts
    • CommentTimeJul 11th 2013
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    Author: DavidRoberts
    Format: MarkdownItexYes, I should have been consistent, as I use bundle 2-gerbe later when I could/should have used 3-bundle. I think the indexing issues may have been a point of confusion in reading my answer at MO.

    Yes, I should have been consistent, as I use bundle 2-gerbe later when I could/should have used 3-bundle. I think the indexing issues may have been a point of confusion in reading my answer at MO.

12. • CommentRowNumber12.
    • CommentAuthorRyan Thorngren
    • CommentTimeJul 19th 2013
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    Author: Ryan Thorngren
    Format: MarkdownItexRegarding David's picture: what is the relationship between this principal 3-bundle with fiber $ker t$ on $B(coker t)$ and the surface holonomy in $ker t$ one obtains from the big 2-bundle? It seems like there should be some $ker t$ 2-bundle giving this holonomy, yet we end up with a 3-bundle.

    Regarding David’s picture: what is the relationship between this principal 3-bundle with fiber $ker t$ on $B(coker t)$ and the surface holonomy in $ker t$ one obtains from the big 2-bundle? It seems like there should be some $ker t$ 2-bundle giving this holonomy, yet we end up with a 3-bundle.

13. • CommentRowNumber13.
    • CommentAuthorDavidRoberts
    • CommentTimeJul 19th 2013
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    Author: DavidRoberts
    Format: MarkdownItex@Ryan - which big 2-bundle are you referring to?

    @Ryan - which big 2-bundle are you referring to?

14. • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeJul 19th 2013
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    Author: Urs
    Format: MarkdownItexHi Ryan, sorry for the slow reply, I needed to concentrate on a bunch of other things (as one can see here, I suppose...). So since that 3-bundle classifies the extension, equivalently the extension is the universal trivialization of that 3-bundle. This means that the 2-connection is the data that trivializes that 3-connection. Sorry if this sounds mysterious, I don't have more time right now. But this is discussed in a good bit of detail at _[[twisted differential string structure]]_ and in the pdf-s linked to there.

    Hi Ryan,

    sorry for the slow reply, I needed to concentrate on a bunch of other things (as one can see here, I suppose…).

    So since that 3-bundle classifies the extension, equivalently the extension is the universal trivialization of that 3-bundle. This means that the 2-connection is the data that trivializes that 3-connection.

    Sorry if this sounds mysterious, I don’t have more time right now. But this is discussed in a good bit of detail at twisted differential string structure and in the pdf-s linked to there.

15. • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeJul 19th 2013
    • PermaLink
    Author: Urs
    Format: MarkdownItexI have another minute, let me give you more specific pointers to this phenomenon: the definition and construction of String-2-connections as universal trivializations of that 3-connection which you are asking about we gave in section 6.9 of * _Cech cocycles for differential characteristic classes_ ([arXiv:1011.4735](http://arxiv.org/abs/1011.4735)) You can see it in action (with both meanings of the term :-) in def. 3.2 and the following discussion of * _Twisted differential String and Fivebrane structures_ ([arXiv:0910.4001](http://arxiv.org/abs/0910.4001)) and in section 4.2 of * _The E8 moduli 3-stack of the C-field in M-theory_ ([arXiv:1202.2455](http://arxiv.org/abs/1202.2455)) . In both cases there is a 3-connection which is the universal [[Chern-Simons circle 3-bundle with connection]] and depending on context one asks for universally trivializing (or "twisted trivializing", hence identifying with a pre-specified class possibly different from 0) 1. the underlying topological class in degree 4-cohomology; 1. the underlying class and the de Rham class of the 4-form curvature; 1. the underlying class and the curvature itself. These different kinds of trivializations of the 3-connection give three different flavors of String 2-group principal 2-connections. Etc.

    I have another minute, let me give you more specific pointers to this phenomenon:

    the definition and construction of String-2-connections as universal trivializations of that 3-connection which you are asking about we gave in section 6.9 of

    • Cech cocycles for differential characteristic classes (arXiv:1011.4735)

    You can see it in action (with both meanings of the term :-) in def. 3.2 and the following discussion of

    • Twisted differential String and Fivebrane structures (arXiv:0910.4001)

    and in section 4.2 of

    • The E8 moduli 3-stack of the C-field in M-theory (arXiv:1202.2455) .

    In both cases there is a 3-connection which is the universal Chern-Simons circle 3-bundle with connection and depending on context one asks for universally trivializing (or “twisted trivializing”, hence identifying with a pre-specified class possibly different from 0)

    1. the underlying topological class in degree 4-cohomology;

    2. the underlying class and the de Rham class of the 4-form curvature;

    3. the underlying class and the curvature itself.

    These different kinds of trivializations of the 3-connection give three different flavors of String 2-group principal 2-connections. Etc.

16. • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeJul 20th 2013
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    Author: Urs
    Format: MarkdownItexHi Ryan, I have spotted * Sergei Gukov, Anton Kapustin, _Topological Quantum Field Theory, Nonlocal Operators, and Gapped Phases of Gauge Theories_ ([arXiv:1307.4793](http://arxiv.org/abs/1307.4793)) and the pointer to your work on p. 8 there. Is this the context of your questions here? That looks very interesting. That would give me a bit of perspective of what you are after. For instance, would it be correct to guess that you are interested in the case that $ket(t)$ is a discrete group?

    Hi Ryan,

    I have spotted

    • Sergei Gukov, Anton Kapustin, Topological Quantum Field Theory, Nonlocal Operators, and Gapped Phases of Gauge Theories (arXiv:1307.4793)

    and the pointer to your work on p. 8 there. Is this the context of your questions here? That looks very interesting.

    That would give me a bit of perspective of what you are after. For instance, would it be correct to guess that you are interested in the case that $ket(t)$ is a discrete group?

17. • CommentRowNumber17.
    • CommentAuthorRyan Thorngren
    • CommentTimeJul 21st 2013
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    Author: Ryan Thorngren
    Format: MarkdownItexThanks again for your answers, Urs. And yes, that is the context I'm considering these in. Good looking out! Basically we're trying to analyze these higher gauge theories with finite gauge group in dimensions through 4.

    Thanks again for your answers, Urs.

    And yes, that is the context I’m considering these in. Good looking out! Basically we’re trying to analyze these higher gauge theories with finite gauge group in dimensions through 4.

18. • CommentRowNumber18.
    • CommentAuthorRyan Thorngren
    • CommentTimeJul 23rd 2013
    • PermaLink
    Author: Ryan Thorngren
    Format: MarkdownItexCan this trivialization be understood as meaning that the (3-form) curvature of this connection represents the class of the bundle?

    Can this trivialization be understood as meaning that the (3-form) curvature of this connection represents the class of the bundle?

19. • CommentRowNumber19.
    • CommentAuthorRyan Thorngren
    • CommentTimeJul 23rd 2013
    • PermaLink
    Author: Ryan Thorngren
    Format: MarkdownItexI guess my confusion is that in characterizing the string 2-connection that there is a characteristic class whose differential refinement is supposed to map the string 2-connection to the connection on the circle 3-bundle. I'm not sure what the analogue in this situation is. Can we think about this extension as killing some homotopy class? Considering the group cohomology picture, I came up with the constraint that for any 3-simplex in our space with a 2-connection for the big 2-group with 01, 12, 23 1-holonomies $x_1, x_2, x_3$ in the cokernel and face holonomies $y_0, y_1, y_2, y_3$, which we can take to be in the kernel, that if $\beta$ is a cocycle representing the extension class, $\alpha(x_1)(y_0) - y_1 + y_2 - y_3 = \beta(x_1,x_2,x_3)$. I think that this means what I conjectured above, ie. that the curvature 3-form of this 2-connection represents the extension class, but I am not sure how to see it in this topological picture.

    I guess my confusion is that in characterizing the string 2-connection that there is a characteristic class whose differential refinement is supposed to map the string 2-connection to the connection on the circle 3-bundle. I’m not sure what the analogue in this situation is. Can we think about this extension as killing some homotopy class?

    Considering the group cohomology picture, I came up with the constraint that for any 3-simplex in our space with a 2-connection for the big 2-group with 01, 12, 23 1-holonomies $x_1, x_2, x_3$ in the cokernel and face holonomies $y_0, y_1, y_2, y_3$, which we can take to be in the kernel, that if $\beta$ is a cocycle representing the extension class,

    $\alpha(x_1)(y_0) - y_1 + y_2 - y_3 = \beta(x_1,x_2,x_3)$.

    I think that this means what I conjectured above, ie. that the curvature 3-form of this 2-connection represents the extension class, but I am not sure how to see it in this topological picture.

20. • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeJul 28th 2013
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    Author: Urs
    Format: MarkdownItexYes, the standard String extension is the step in the [[Whitehead tower]] of $O$ after $Spin$ which co-kills $\pi_3$. Generally, the $String(G)$-extension is the universal trivialization of the canonical 4-class $[c]$ of $B G$, for $G$ a simply connected compact simple Lie group. I understand that you are not mainly interested in the String 2-group, which is a _central_ higher extension of its cokernel 1-group, but maybe it is good to talk about that first, because it is a simpler example than the general example that you are after. The general case is rouhgly like that of String, but more intricate. And in this case at least I can answer your question about how the 3-form curvature of a 2-connection relates to the obstruction class of the extension. So the String 2-group is defined to fit into the homotopy pullback diaram $$ \array{ \mathbf{B}String(G) &\to& \ast \\ \downarrow &\swArrow& \downarrow \\ \mathbf{B} G &\stackrel{\mathbf{c}}{\to}& \mathbf{B}^3 U(1) } $$ so in words (and the following is in fact precise if you read it in homotopy type theory): "a point in $\mathbf{B}String$ is a point in $\mathbf{B}Spin$ together with a trivialization of its image in $\mathbf{B}^3 U(1)$ under $\mathbf{}c$". The point of this is that the same pattern remains when we pass to connections. Then String 2-connections sit in the homotopy pullback $$ \array{ \mathbf{B}String(G)_{conn} &\to& \ast \\ \downarrow &\swArrow& \downarrow^{\mathrlap{0}} \\ \mathbf{B} G_{conn} &\stackrel{\mathbf{c}_{conn}}{\to}& \mathbf{B}^3 U(1)_{conn} } \,. $$ (These is the most restrictive notion of String 2-connections, but it is the one that relates directly to the 2-connections that you are looking at. More genrally one can replace here the right vertical map by the inclusion of something more general than the zero 3-connection.) Now the differential cohomology class $\mathbf{c}_{conn}$ is a universal 3-connection with some curvature 4-form. This curvature 4-form is the de Rham image of the 4-class $[c] \in \pi_0 \mathbf{H}(\mathbf{B}G, \mathbf{B}^3 U(1)) \simeq H^4(B G, \mathbb{Z})$. And: the above fiber product says that 1. this curvature 4-form vanishes on $\mathbf{B}String_{conn}$; 1. a "point in" $\mathbf{B}String_{conn}$ is a trivialization of this trivial 3-connection. This is the 2-connection whose curvature 3-form is the rivialization of the trivial 4-form, hence is a closed 3-form. So the 3-form curvature of a String 2-connection (in the above restricted sense) is the Chern-Simons "secondary invariant" which reflects the vanishing of the curvature 4-form which is the de Rham image of the 4-class which in turn in the class that classifies the String-extension. That's how it works for the higher central extension of the form of String 2-groups. In the more general case the situation is a more complicated version of this story. As I mentioned before, in the general case the role of the map $\mathbf{c} : \mathbf{B}G \to \mathbf{B}^3 U(1)$ is played by a map $\mathbf{B}G \to \mathbf{B}\mathbf{Aut}(\mathbf{B}ker(t))$. From here on one could walk hrough the above story and see how it generalizes.

    Yes, the standard String extension is the step in the Whitehead tower of $O$ after $Spin$ which co-kills $\pi_3$.

    Generally, the $String(G)$-extension is the universal trivialization of the canonical 4-class $[c]$ of $B G$, for $G$ a simply connected compact simple Lie group.

    I understand that you are not mainly interested in the String 2-group, which is a central higher extension of its cokernel 1-group, but maybe it is good to talk about that first, because it is a simpler example than the general example that you are after. The general case is rouhgly like that of String, but more intricate.

    And in this case at least I can answer your question about how the 3-form curvature of a 2-connection relates to the obstruction class of the extension.

    So the String 2-group is defined to fit into the homotopy pullback diaram

    $$\array{ \mathbf{B}String(G) &\to& \ast \\ \downarrow &\swArrow& \downarrow \\ \mathbf{B} G &\stackrel{\mathbf{c}}{\to}& \mathbf{B}^3 U(1) }$$

    so in words (and the following is in fact precise if you read it in homotopy type theory): “a point in $\mathbf{B}String$ is a point in $\mathbf{B}Spin$ together with a trivialization of its image in $\mathbf{B}^3 U(1)$ under $\mathbf{}c$”.

    The point of this is that the same pattern remains when we pass to connections. Then String 2-connections sit in the homotopy pullback

    $$\array{ \mathbf{B}String(G)_{conn} &\to& \ast \\ \downarrow &\swArrow& \downarrow^{\mathrlap{0}} \\ \mathbf{B} G_{conn} &\stackrel{\mathbf{c}_{conn}}{\to}& \mathbf{B}^3 U(1)_{conn} } \,.$$

    (These is the most restrictive notion of String 2-connections, but it is the one that relates directly to the 2-connections that you are looking at. More genrally one can replace here the right vertical map by the inclusion of something more general than the zero 3-connection.)

    Now the differential cohomology class $\mathbf{c}_{conn}$ is a universal 3-connection with some curvature 4-form. This curvature 4-form is the de Rham image of the 4-class $[c] \in \pi_0 \mathbf{H}(\mathbf{B}G, \mathbf{B}^3 U(1)) \simeq H^4(B G, \mathbb{Z})$. And: the above fiber product says that

    1. this curvature 4-form vanishes on $\mathbf{B}String_{conn}$;

    2. a “point in” $\mathbf{B}String_{conn}$ is a trivialization of this trivial 3-connection. This is the 2-connection whose curvature 3-form is the rivialization of the trivial 4-form, hence is a closed 3-form.

    So the 3-form curvature of a String 2-connection (in the above restricted sense) is the Chern-Simons “secondary invariant” which reflects the vanishing of the curvature 4-form which is the de Rham image of the 4-class which in turn in the class that classifies the String-extension.

    That’s how it works for the higher central extension of the form of String 2-groups. In the more general case the situation is a more complicated version of this story.

    As I mentioned before, in the general case the role of the map $\mathbf{c} : \mathbf{B}G \to \mathbf{B}^3 U(1)$ is played by a map $\mathbf{B}G \to \mathbf{B}\mathbf{Aut}(\mathbf{B}ker(t))$. From here on one could walk hrough the above story and see how it generalizes.

21. • CommentRowNumber21.
    • CommentAuthorRyan Thorngren
    • CommentTimeJul 31st 2013
    • PermaLink
    Author: Ryan Thorngren
    Format: MarkdownItexThanks, Urs. After thinking about it for some time this is starting to make some sense to me. I am really beginning to appreciate the general theory here, and especially via your work on physical examples. I think the physical prospects are very exciting.

    Thanks, Urs. After thinking about it for some time this is starting to make some sense to me. I am really beginning to appreciate the general theory here, and especially via your work on physical examples. I think the physical prospects are very exciting.

22. • CommentRowNumber22.
    • CommentAuthorRyan Thorngren
    • CommentTimeAug 21st 2013
    • (edited Aug 21st 2013)
    • PermaLink
    Author: Ryan Thorngren
    Format: MarkdownItexOne more question: How does one compute the automorphism n-group as a crossed complex like Urs did above for $B^2A$?

    One more question:

    How does one compute the automorphism n-group as a crossed complex like Urs did above for $B^2A$?