# Weak 2-groups and non-abelian gerbe over a manifold

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.