I understood gerbes as generalization of line bundle here.
In this, I am trying to understand how to generalize notion of trivialization of line bundle to the notion of trivialization of gerbes. I am following this notes of Hitchin for this.
It says that
We can’t point to a gerbe as a space in the same way that we can for a line bundle, but there is a fundamental notion which helps to understand them – a trivialization of a gerbe.
Further, it says,
Recall that for line bundles, trivialization is a non-vanishing section $s$ of $L$. A unitary trivialization $s/||s||$ is also a section of the principal $S^1$ bundle. In terms of transition functions, this is a collection of functions $f_\alpha:U_\alpha\rightarrow S^1$ such that on $U_\alpha\cap U_\beta$ we have $f_\alpha=g_{\alpha\beta}f_\beta$.
Given another trivialization $f_\alpha':U_\alpha\rightarrow S^1$ we have $f_\alpha'=g_{\alpha\beta}f_\beta'$ and so we see that $$\frac{f_\alpha'}{f_\alpha}=\frac{f_\beta'}{f_\beta}.$$ Thus, $f_\alpha'/f_\alpha$ is restriction of a global function to $U_\alpha$ and we see that the difference of two local trivializations of a line bundle is the global function $X\rightarrow S^1$.
Here, $L$ is a unitary line bundle or its principal $S^1$ bundle of unitary frames. I do not know what is a unitary bundle. I assume that $L$ is just a line bundle $\pi:L\rightarrow X$.
I understand that for line bundles, trivialization is same as giving a non vanishing section of line bundle.
He generalises notion of trivialization to that of gerbes.
In case of line bundles, this is collection of maps $f_\alpha:U_\alpha\rightarrow S^1$.
In case of gerbes it has to be on $1$ fold intersection extra i.e., $f_{\alpha\beta}:U_\alpha\cap U_\beta\rightarrow S^1$.
The condition $f_\alpha=g_{\alpha\beta}f_\beta$ in case of line bundles should have its similar generalization in case of gerbes. I fail to understand how this generalisation is the condition $$g_{\alpha\beta\gamma}=f_{\alpha\beta}f_{\beta\gamma}f_{\gamma\alpha}$$
Any suggestion on understanding this better is useful.
