There is a notion of Gerbe over a Manifold and a notion of Gerbe over a stack. Given a manifold $M$, there is a way to associate a stack $\underline{M}$ with it and this gives an embedding of category of smooth manifolds into category of Categories fibered in groupoids which can be found in [Orbifolds as Stacks](https://arxiv.org/abs/0806.4160).


This gives a hope that notion of gerbe on a manifold $M$ has to be some special case of notion of gerbe on stack $\underline{M}$. This question is about the difficulty I faced when realize that.


I have written definition of Gerbe over manifold and Gerbe over Stack in comments (hoping the question looks smaller with out the definitions). 


The problem I am looking at is : 



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> Is a gerbe over a manifold a special case of a gerbe over a stack? In particular, if $\mathfrak{X}=\underline{M}$, does a gerbe over $\mathfrak{X}=\underline{M}$ give a gerbe on manifold $M$? If so, How? How is the main question here.
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Has any one looked at this before?


[My previous question](https://mathoverflow.net/questions/307123/understanding-definition-of-gerbe-over-a-stack) says the definition of gerbe over stack mean that



> 
> Given a manifold $U$, there exists an open cover
>  $\{U\_\alpha\rightarrow U\}$ such that the restriction functor $\mathcal{D}(U)\rightarrow \mathfrak{X}(U)$ has following properties
> 
> 
> * given an object $x\in \mathfrak{X}(U)$ there is an object $y\_\alpha\in \mathcal{D}(U\_\alpha)$ and an isomorphism
>  $F(y\_\alpha)\rightarrow x|\_{U\_\alpha}$ for each index $\alpha$.
> * given an arrow $\phi:a\rightarrow b$ in $\mathfrak{X}(U)$ there exists an arrow $\tau\_\alpha:y\_\alpha\rightarrow z\_\alpha$ in $\mathcal{D}(U\_\alpha)$ such that $F(\tau\_\alpha)=\phi\_\alpha$ for each index
>  $\alpha$.
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> 


I start with a stack $\mathcal{D}\rightarrow \mathfrak{X}=\underline{M}$
and construct a gerbe over manifold $M$. I need to declare what is $\mathbb{G}(U)$ for each open set $U$ of $M$. I define that to be $\mathcal{D}(U)$ i.e., $\mathbb{G}(U)=\mathcal{D}(U)$. 


**motivation for declaring like that is first condition says that object set of $\mathcal{D}(U\_\alpha)$ is non empty and second condition says that morphism set of $\mathcal{D}(U\_\alpha)$ is non empty if I can produce some objects and morphisms in $\mathcal{C}(U\_\alpha)=\underline{M}(U\_\alpha)$.**


I have to check that it is locally non empty and locally connected.


I was able to see that this makes $\mathcal{G}$ to be locally non empty stack. Proof I have written in comments. 


Let $y,y'\in \mathbb{G}(U)=\mathfrak{R}(U)$. Then, we have $F(y),F(y')\in \mathfrak{X}(U)=\underline{M}(U)$. Assuming there is an arrow $F(y)\rightarrow F(y')$ in $\mathfrak{X}(U)$, [second condition in my answer](https://mathoverflow.net/questions/307123/understanding-definition-of-gerbe-over-a-stack) says that there is an arrow $y\rightarrow y'$ (locally atleast) in $\mathcal{D}(U)=\mathbb{G}(U)$ that maps to the arrow $F(y)\rightarrow F(y')$. This says $\mathbb{G}(U)$ is locally connected.


Thus, we have a Gerbe on manifold $M$ given a gerbe over stack $\underline{M}$ taking something for granted.


We have $F(x),F(y)\in \mathfrak{X}(U)=\underline{M}(U)$ i.e., $F(x):U\rightarrow M$ and $F(y):U\rightarrow M$ (smooth maps). An arrow $F(x)\rightarrow F(y)$ is a smooth map $\psi:U\rightarrow U$ such that $F(y)\circ \psi=F(x)$. One can quickly realize that this is not possible always (for example when $F(x)$ is constant map $p\in M$ and $F(y)$ is constant map $q\in M$ with $p\neq q$). Thus, **this idea does not work**.


We redefine $\mathbb{G}(U)$ as a category whose objects are objects of $\mathcal{D}(U)$ for which $F(y)$ is a diffeomorphism onto $U$. We have $$\mathbb{G}(U)=\{y\in \mathcal{D}(U): F(y)\text{ is a diffeomorphism onto } U\}.$$
Clearly, the $y$ we have choosen (locally) as preimage of inclusion map $i:U\rightarrow M$ is in $\mathbb{G}(U)$. 
Now, for $F(y),F(y)'\in \mathcal{C}(U)$, define $\Phi=F(y')^{-1}\circ F(y)$. Then, $F(y')\circ \Phi=F(y)$ which means there is an arrow $F(y)\rightarrow F(y')$. This solves the problem. Given $y,y'\in \mathcal{D}(U)$ there is an arrow $\phi:F(y)\rightarrow F(y')$ in $\mathcal{C}(U)$. As we are in case of gerbe, there exists arrow $\tau\_{\alpha}:y|\_{U\_\alpha}\rightarrow y'|\_{U\_\alpha}$ such that $F(\tau\_{\alpha})=\phi|\_{U\_\alpha}$, which in particular say that given $y,y'\in \mathbb{G}(U)$, they are locally connected by arrows $y|\_{U\_\alpha}\rightarrow y'|\_{U\_\alpha}$ which says $\mathbb{G}$ defines a gerbe on manifold $M$.


Is my construction correct? Can it be made more natural? For me, it was most reasonable thing to do. Any comments are welcome.



Is a gerbe over a manifold is a special case of a gerbe over a stack?
Yes. The category of manifolds embeds fully faithfully into the 2-category of stacks on Mfld, essentially by Yoneda, and the site structures are likewise compatible, so when restricted to the special case of your stack being a manifold, the definitions are equivalent.


