Let $X$ be a topological space. Let $\mathcal{F}$ be a fibered category over $X$; seen as an assignment of a category $\mathcal{F}(U)$ for each open $U\subseteq X$.

A fibered catgeory $\mathcal{F}$ over $X$ satisfying some properties, is called a stack over $X$. Further, if $\mathcal{F}(U)$ is a groupoid for each open $U\subseteq X$, we call $\mathcal{F}$ to be a stack of groupoids over the topological space $X$.

A stack $\mathcal{F}$ over a topological space $X$ is said to be a gerbe over $X$, if the following conditions are satisfied:

- there exists an open cover $\{U_\alpha\}$ of $X$ such that $Obj(\mathcal{F}(U_\alpha))\neq \emptyset$ for every $\alpha$.
- Fix an open set $U\subseteq X$ and objects $a,b$ of $\mathcal{F}(U)$. Then there exists an open cover $\{V_\alpha\}$ of $U$ such that $\text{Hom}_{\mathcal{F}(V_\alpha)}\left(a|_{V_\alpha},b|_{V_\alpha}\right)\neq \emptyset$ for every $\alpha$.

Given a gerbe $\mathcal{G}$ over $X$, one can choose an open cover $\{U_\alpha\}$ of $X$ and objects $a_\alpha$ of $\mathcal{G}(U_\alpha)$. This gives sheaves of groups $\underline{\text{Aut}}(a_\alpha)$ for each $\alpha$. Appropriate usage of second condition in the definition of gerbe defines an outer isomorphism of sheaves on $U_{\alpha\beta}$; namely

$$\lambda_{\alpha\beta}:\underline{\text{Aut}}(a_\beta)|_{U_{\alpha\beta}}\rightarrow \underline{\text{Aut}}(a_\alpha)|_{U_{\alpha\beta}}.$$

Given a gerbe $\mathcal{G}$ on $X$; the collection of sheaves of groupoids $\underline{\text{Aut}}(a_\alpha)$ and outer automorphisms $\{\lambda_{\alpha\beta}\}$ is called the band of the gerbe of $X$.

I understand the definition and some immediate remarks about bands. I could not get the main idea behind the association of a band for a gerbe. Above definition is from the notes Introduction to the language of gerbe and stacks by Ieke Moerdijk. I have had a look at the notes on 1-gerbes and 2-gerbes by Lawrence Breen.

Question :

- How should one think about the band of a gerbe?
- It was mentioned in another question that, band of a gerbe was a misguided attempt. Still there are many notes that talks about band of a gerbe. So, is the present day definition different from that of the definition of Giraud? What was the motivation for the change? Is there a better definition or usage or understanding of the notion of a band of a gerbe since then?