I am reading Ieke Moerdijk's article "Orbifolds as Groupoids : an Introduction".


In that notes author defines a notion of generalized map between Lie groupoids. 



> 
> Let $\mathcal{G}$ and $\mathcal{H}$ be Lie groupoids. By a generalized map from $\mathcal{G}$ to $\mathcal{H}$ we mean a homomorphism of Lie groupoids $\mathcal{G}'\rightarrow \mathcal{H}$ where $\mathcal{G}'$ is a morita equivalent groupoid to $\mathcal{G}$.
> 
> 
> 


I do not understand the necessity/motivation behind introducing this notion of generalized map. 


What can one expect to do by considering maps of this generality?


Even the notion of Morita equivalent is little extra that what one would expect.



> 
> Two Lie groupoids $\mathcal{G}$ and $\mathcal{G}'$ are said to be Morita equivalent if there exists a third groupoid $\mathcal{H}$ and equivalences (a morphsim of Lie groupoids said to be an equivalence if [somthing happens](https://mathoverflow.net/questions/301054/morita-equivalence-of-lie-groupoids))
>  $$\mathcal{G}\xleftarrow{\phi} \mathcal{H}\xrightarrow{\phi'} \mathcal{G}'.$$
> 
> 
> 


I am not able to see the necessity of considering this general notion of equiavelnce. 


Any comments are welcome.


Edit : This is just to bump the question so that it gets some attention and an answer/comment.


Edit: Orbifold definition that I am familiar with is the following.



> 
> Definition : Let $X$ be a locally compact Hausdorff space. An orbifold structure on $X$ is given by an orbifold groupoid $\mathcal{G}$ and a homeomorphism $f:|\mathcal{G}|\rightarrow X$. If $\mathcal{H}\rightarrow\mathcal{G}$ is a equivalnece, then $|\phi|:|\mathcal{H}|\rightarrow |\mathcal{G}|$ is a homeomorphism and the composition $f\circ|\phi|:|\mathcal{H}|\rightarrow |\mathcal{G}|\rightarrow X$ is viewed as defining an equivalent orbifold structure on $X$. An orbifold $\underline{X}$ is a space $X$ equipped with an equivalnece class of orbifold structures. 
> 
> 
> 



Necessity/Motivation for generalised homomorpisms
Let me attempt a very simple-minded answer.


Say your objects of interest are orbifolds. You have an orbifold V and you want to describe it through a groupoid, usually an action groupoid $G\ltimes M \rightrightarrows M$. This means that you have an action of $G$ on $M$ and that $V\simeq G\backslash M$, i.e. $V$ is the orbit space of the action groupoid. 


Then you may think that a map between two such orbifolds $V\_1$ and $V\_2$ is a groupoid morphism between the corresponding action groupoids. 


But this does not work because there are many ways in which you can construct an action groupoid with orbit space $V$. There are many different ways in which you can see an orbifold as quotient of a manifold. And you cannnot expect to lift any orbifold map to any possible such presentation.


Morita equivalence is the equivalence relation that guarantees you that two different groupoids have the same orbit space. Thus that they, in a way, correspond to different ways of presenting the same orbifold. 


Generalized morphism are the kind of correspondence that you can establish between one presentation of the first orbifold and one presentation of the other orbifold. 


This is the reason why you need the generality of Morita equivalence and generalized morphisms in the context of orbifolds. 


Nardin comment is a very terse way to say this: an orbifold (more generally a stack) is described by a Morita equivalence class of groupoids. 


From this point of view it is very relevant to understand which groupoid invariants are, in fact, Morita invariant. Those will define orbifold (stack) invariants.


