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:

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

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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