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In literature, a strict 2-group is defined as a group object in the category of categories or groupoids. It has many well known equivalent descriptions viz:

1. A strict monoidal category in which all objects and morphisms are invertible.

2 A crossed module.

3 A strict 2-category with one object in which all 1-morphisms and 2-morphisms are invertible.

4 A category object in Grp

I am here interested in especially 2 .

Now from Brown-Spencer result, we know that that the category of crossed module is equivalent to the category of strict 2- groups.

Now there is a notion of cocycle description of non-abelian gerbe over a manifold $M$ given by the following data: Let $\lbrace U_{\alpha}\rbrace$ ($\alpha \in I$) be an open cover of $M$.Let $W$ be a Strict 2-group and let $(G, H,\tau,\alpha)$ be its corresponding crossed module of groups where $\tau:H \rightarrow G$ and $\alpha:G \rightarrow \mathrm{Aut}(H)$. Let $g_{ij}: U_i \cap U_j \rightarrow G$ and $h_{ijk}:U_i\cap U_j \cap U_k\rightarrow H$ (where $i,j,k \in I$) be a collection of functions such that $g_{ij}.g_{jk}=\tau(h_{ijk}).g_{ik}$ on $U_i\cap U_j \cap U_k$.

Recently in higher gauge theory, it had been found that there are some relations between the above cocycle description of non-abelian gerbes over a manifold and local description of principal 2-bundles over a manifold or loop space of a manifold (where the structure group is the strict 2-group corresponding to the crossed module defined in gerbe.)

Now there is a notion of weak 2-group which is defined by weakening the description (1) i.e a weak 2- group is a weak monoidal category $(M,\times,1)$ in which all morphisms are invertible and for each object $x \in M$ there exist an object $y$ such that $x \times y$ is isomorphic to 1 and $y \times x$ is isomorphic to 1. Now if we have a coherent choice of inverses satisfying some coherent diagrams then we call a weak 2-group a coherent 2-group.

My questions are the following:

A. Is there any known equivalent description of weak or coherent 2-groups analogous to crossed module description of strict 2-Groups?

B Is there any weaker notion of a cocycle description of non-abelian gerbe over a manifold where the crossed module is replaced by a suitable description of weak or coherent 2-group?

C Is there any relation between the local description of a principal 2-bundle (where the structure group is a weak or coherent 2-Group) over a manifold and the notion of gerbe (suitably weakened) over a manifold?

For the definition of principal 2-bundle , I have followed the following references:

  1. https://arxiv.org/pdf/0803.3692.pdf (by Wockel)

  2. http://math.ucr.edu/home/baez/2conn.pdf ( Baez and Schrieber)

For the definition of cocycle description of non-abelian gerbe over a manifold I have followed:

http://math.ucr.edu/home/baez/2conn.pdf (Baez and Schrieber)

For the definition of weak 2-groups and strict 2-Groups, I have followed

https://arxiv.org/pdf/math/0307200.pdf (Baez and Lauda).

Also in the paper 'Higher gauge theory: 2 connection on 2 Bundles' https://arxiv.org/abs/hep-th/0412325 (Baez and Scrieber) in the section 'Discussion and open problems' a question had been proposed that

"How does the discussion in this paper generalize when the standard fiber F of the 2-bundle is allowed to be something which is not a strict 2-group?"

It would be really helpful to know about the current status of the above question.

Any clarification regarding my question is highly appreciated.

Thanks in advance.

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  • $\begingroup$ You can remove “ In literature, a strict 2-group is defined as a group object in the category of categories or groupoids. It has many well known equivalent descriptions viz: 1. A strict monoidal category in which all objects and morphisms are invertible. 2 A crossed module. 3 A strict 2-category with one object in which all 1-morphisms and 2-morphisms are invertible. 4 A category object in Grp I am here interested in especially 2. Now from Brown-Spencer result, we know that that the category of crossed module is equivalent to the category of strict 2- groups.” $\endgroup$ – Praphulla Koushik Feb 25 at 18:45
  • $\begingroup$ That does not add any extra clarity for the question.. removing that part reduce the length making it easier to read $\endgroup$ – Praphulla Koushik Feb 25 at 18:45
  • $\begingroup$ @PraphullaKoushik Thanks for your kind suggestion. But I feel it is necessary. The only definition of weak 2 group I know is as a weak monoidal category whereas in the proof(that I know) of Brown-Spencer result they have treated Strict 2 group as a group object in Cat . So to get the answer to (A) we need to mention that strict 2 group is same as a Strict Monoidal category with all objects and morphisms invertible. The things that I feel unnecessary to this question is (3) and (4). Still I feel its better to mention all the descriptions that I know of Strict 2 group. $\endgroup$ – Adittya Chaudhuri Feb 26 at 6:00
  • $\begingroup$ :) as you wish. $\endgroup$ – Praphulla Koushik Feb 26 at 6:04
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    $\begingroup$ I'm also interested in the analogue of a weak crossed module. $\endgroup$ – Student Sep 25 at 9:28

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