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1. • CommentRowNumber1.
   • CommentAuthorZhen Huan
   • CommentTimeAug 4th 2020
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   Author: Zhen Huan
   Format: TextAs far as I know, we don't have a classification of ALL 2-group representation in nontrivial case. The only relevant work I know is the paper "Central Extension of Smooth 2-groups and a finite-dimensional String 2-group" by Schommer-Pries. In it the central extensions of smooth 2-groups are classified in term of Segal-Mitchison topological group cohomology. Is there any results in this direction?

   As far as I know, we don't have a classification of ALL 2-group representation in nontrivial case. The only relevant work I know is the paper "Central Extension of Smooth 2-groups and a finite-dimensional String 2-group" by Schommer-Pries. In it the central extensions of smooth 2-groups are classified in term of Segal-Mitchison topological group cohomology.
   Is there any results in this direction?
2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeAug 4th 2020
   • (edited Aug 4th 2020)
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   Author: Urs
   Format: MarkdownItex(It seems you don't mean to say "representations" but "central extensions"? There is a close relation, but the general topic of ("linear") representations of higher groups is deep and murky, while the central extensions of higher groups are much better understood.) Generally, extensions of an $\infty$-group $G$ by a braided $\infty$-group $A$ (both can be the coherent 2-groups that you are after, but may also be more general) are equivalently $\mathbf{B}A$-principal bundles over $\mathbf{B}G$. (this is Section 4.3 in NSS12 [here](https://arxiv.org/pdf/1207.0248.pdf#page=38), or Section 3.6.14 in dcct [here](https://arxiv.org/pdf/1310.7930v1.pdf#page=341)) By the general classification of principal $\infty$-bundles, (Theorem 3.17 in NSS12 [here](https://arxiv.org/pdf/1207.0248.pdf#page=25)) this means that these extensions are classified by maps $$ \mathbf{B}G \longrightarrow \mathbf{B}^2 A $$ hence by $$ \pi_0 \mathbf{H}(\mathbf{B}G, \mathbf{B}^2 A) $$ hence by $$ H^2_{Grp}(G, A) $$ the "cohesive $\infty$-group degree-2 cohomology of $G$ with coeffcients in $A$". Again, you may restrict this to coherent 2-groups if desired. This holds for $G$ being group $\infty$-stacks, but to compute classifications you'll want to restrict to controlled cases where, say, $G$ is a Lie group. In the case that $G$ is a Lie group, the above reduces to Segal-Brylinski smooth group cohomology (Theorem 4.4.36 in dcct [here](https://arxiv.org/pdf/1310.7930v1.pdf#page=488)), Combining all this, one finds in particular the String 2-group extensions (Section 5.1.4 in dcct [here](https://arxiv.org/pdf/1310.7930v1.pdf#page=583) )

   (It seems you don’t mean to say “representations” but “central extensions”? There is a close relation, but the general topic of (“linear”) representations of higher groups is deep and murky, while the central extensions of higher groups are much better understood.)

   Generally, extensions of an $\infty$-group $G$ by a braided $\infty$-group $A$ (both can be the coherent 2-groups that you are after, but may also be more general) are equivalently $\mathbf{B}A$-principal bundles over $\mathbf{B}G$.

   (this is Section 4.3 in NSS12 here, or Section 3.6.14 in dcct here)

   By the general classification of principal $\infty$-bundles, (Theorem 3.17 in NSS12 here) this means that these extensions are classified by maps

   $$\mathbf{B}G \longrightarrow \mathbf{B}^2 A$$

   hence by

   $$\pi_0 \mathbf{H}(\mathbf{B}G, \mathbf{B}^2 A)$$

   hence by

   $$H^2_{Grp}(G, A)$$

   the “cohesive $\infty$-group degree-2 cohomology of $G$ with coeffcients in $A$”. Again, you may restrict this to coherent 2-groups if desired.

   This holds for $G$ being group $\infty$-stacks, but to compute classifications you’ll want to restrict to controlled cases where, say, $G$ is a Lie group.

   In the case that $G$ is a Lie group, the above reduces to Segal-Brylinski smooth group cohomology (Theorem 4.4.36 in dcct here),

   Combining all this, one finds in particular the String 2-group extensions (Section 5.1.4 in dcct here )

3. • CommentRowNumber3.
   • CommentAuthorZhen Huan
   • CommentTimeAug 4th 2020
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   Author: Zhen Huan
   Format: TextThanks a lot. I realize my question led to misunderstanding. What I'm curious about is the classification of the representations of a specific 2-group E which is a central extenstion of G by A. I will read the paper first.

   Thanks a lot. I realize my question led to misunderstanding. What I'm curious about is the classification of the representations of a specific 2-group E which is a central extenstion of G by A. I will read the paper first.
4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeAug 5th 2020
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   Author: Urs
   Format: MarkdownItexThe only systematic way, that I am aware of, to get the interesting 2-representations of 2-groups remains that in: Appendix A "2-Vector spaces and the canonical 2-representation" of [arXiv:0806.1079](https://arxiv.org/abs/0806.1079) As formulated there this invokes strict 2-group models of the given 2-group. This should be just a technical convenience, not a restriction. For the string 2-groups the strict 2-group model is established in [arXiv:math/0504123](https://arxiv.org/abs/math/0504123).

   The only systematic way, that I am aware of, to get the interesting 2-representations of 2-groups remains that in:

   Appendix A “2-Vector spaces and the canonical 2-representation” of arXiv:0806.1079

   As formulated there this invokes strict 2-group models of the given 2-group. This should be just a technical convenience, not a restriction. For the string 2-groups the strict 2-group model is established in arXiv:math/0504123.

5. • CommentRowNumber5.
   • CommentAuthorZhen Huan
   • CommentTimeAug 5th 2020
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   Author: Zhen Huan
   Format: TextThanks!

   Thanks!