I am reading Nigel Hitchin's notes to understand about gerbes.

It starts the article by saying the following :

Before giving a definition, it’s worthwhile to recognize when we, as mathematicians, might be in a situation where the language of gerbes could be relevant. We are basically in gerbe territory (for smooth manifolds) if a ny one of the following is being considered:

- a cohomology class in $H^3(X,\mathbb{Z})$
- a Cech cocycle $[g_{\alpha\beta\gamma}]\in H^2(X,C^\infty(S^1))$
In the last case, this means a 2-cocycle for the sheaf of germs of $C^\infty$ functions with values in the circle.

Here, sheaf $C^\infty(S^1)$ on $X$ defined as $C^\infty(S^1)(U)=\{\text{smooth }f:U\rightarrow S^1 \}$ for open $U\subseteq X$.

To understand gerbes, we need to consider the other creatures in a hierarchy to which gerbes belong, and here the lowest form of life consists of circle valued functions $f:X\rightarrow S^1$. Consider the following features of such a function:

- a cohomology class in $H^1(X,\mathbb{Z})$
- a Cech cocycle $[g_\alpha]\in H^0(X,C^\infty(S^1))$
The next step in hierarchy consists of unitary line bundle $L$, or its principal $S^1$ bundle of unitary frames. Here we have

- a cohomology class in $H^2(X,\mathbb{Z})$.
- a Cech cocycle $[g_{\alpha\beta}]\in H^1(X,C^{\infty}(S^1))$

I dont know what a unitary line bundle is. I only know what is a line bundle. I am guessing unitary line bundle is the one whose transition functions takes values in $S^1$ and not just in $\mathbb{C}^*$.

Given a line bundle $\pi:L\rightarrow X$, we have trivialization cover $\{U_\alpha\}=\mathcal{U}$ with trivializations $pi^{-1}(U_\alpha)\times U_\alpha\times \mathbb{C}$ and corresponding transition funcions $g_{\alpha\beta}:U_\alpha\cap U_\beta\rightarrow S^1$. So, we have $(g_{\alpha\beta})\in \mathcal{C}^1(\mathcal{U},C^{\infty}(S^1))$. We also have $g_{\alpha\beta}g_{\beta\gamma}=g_{\alpha\gamma}$ on $U_\alpha\cap U_\beta\cap U_\gamma$. This says that $\underline{g}=(g_{\alpha\beta})\in \mathcal{C}^1(\mathcal{U},C^{\infty}(S^1))$ is actaully a $1$ cocycle in the sense of Cech cocycles. This gives an element in first Cech cohomology $H^1(X,C^{\infty}(S^1))$.

It says further that

Now take an open covering of $X$ and a map $g_{\alpha\beta\gamma}:U_\alpha\cap U_\beta\cap U_\gamma\rightarrow S^1$ on each threefold intersection with $g_{\alpha\beta\gamma}=g_{\beta\alpha\gamma}^{-1}=g_{\alpha\gamma\beta}^{-1}=g_{\gamma\beta\alpha}^{-1}$ and satisfying the cocycle condition $$g_{\beta\gamma\delta}g_{\alpha\gamma\delta}^{-1}g_{\alpha\beta\delta}g_{\alpha\beta\gamma}^{-1}=1 \text{ on } U_\alpha\cap U_\beta\cap U_\gamma\cap U_\delta.$$ We shall say that this data defines a gerbe. By this we mean that it suffices to define a gerbe in the same way that a collection of transition functions defines a line bundle or a collection of coordinate charts defines a manifold.

I understand that we are trying to go a step above to second Cech cohomology group to define a gerbe. So, we need to give an element of $\mathcal{C}^2(\mathcal{U},C^{\infty}(S^1))$ that is a $2$ cocycle. This only means that we are asked to give $\overline{g}=(g_{\alpha\beta\gamma})$ with $g_{\alpha\beta\gamma}\in C^{\infty}(S^1)(U_\alpha\cap U_\beta\cap U_\gamma)$ such that $$g_{\beta\gamma\delta}g_{\alpha\gamma\delta}^{-1}g_{\alpha\beta\delta}g_{\alpha\beta\gamma}^{-1}=1 \text{ on } U_\alpha\cap U_\beta\cap U_\gamma\cap U_\delta$$ which is just the multiplicative version of coycle condition.

Can some one help me to realize why to impose the condition $$g_{\alpha\beta\gamma}=g_{\beta\alpha\gamma}^{-1}=g_{\alpha\gamma\beta}^{-1}=g_{\gamma\beta\alpha}^{-1}$$ on threefold intersection. I understand that if we allow $\underline{g}$ to be a $2$ couboundary then this condition holds true but I am sure we are not allowing $\overline{g}$ to be a $2$ coboundary, otherwise cohomology class of $\overline{g}$ would just be $0$.

So, what is happening here when they impose condition $g_{\alpha\beta\gamma}=g_{\beta\alpha\gamma}^{-1}=g_{\alpha\gamma\beta}^{-1}=g_{\gamma\beta\alpha}^{-1}$?

Is this some generalization of a condition in case of $1$ coycles seen above? If so, can some one suggest me the way to see this?