Following [Thurston](http://library.msri.org/books/gt3m/), an  **orbifold**  is a topological space which looks locally like a finite quotient of $\mathbb R^n$ by a finite group of $O(n)$: this is expressed using charts as for differentiable manifolds, the finite groups being part of the structure.


A fundamental example is the quotient of a manifold by a group acting properly discontinuously - but not necessarily freely. The notion of orbifold seems well-established among mathematicians, and I am wondering if there is also a well-established notion of morphism between these objects.



> 
> Is there a good notion of morphism between orbifolds?
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A good notion should reasonably include the following maps:


* Coverings between orbifolds (in particular, from manifold to orbifolds),
* Fiberings between orbifolds of different dimension, such as a Seifert 3-manifold fibering over its base 2-orbifold,
* The (at least smooth) maps $[0,1] \to O$ that are used to define the orbifold fundamental group of $O$



Is there a good notion of morphism between orbifolds?
I believe this is worked out very nicely in "Geometrization of Three-Dimensional Orbifolds via Ricci Flow" by Bruce Kleiner, John Lott (<http://arxiv.org/abs/1101.3733>). 


An *atlas* for an $n$-orbifold $\mathcal O$ consists of a Hausdorff paracompact topological space $|\mathcal O|$ together with an open covering $\{U\_\alpha\}$, local models $\{(\hat U\_\alpha,G\_\alpha)\}$ ($U\_\alpha$ connected open subset of $\mathbb R^n$) and homeomorphisms $\varphi\_\alpha:U\_\alpha\to \hat U\_\alpha/G\_\alpha$ satisfying a compatibility condition. An *orbifold* is then defined by an equivalence class of such atlas. (See page 6 of Kleiner-Lott.)


A *smooth map* $f:\mathcal O\_1\to\mathcal O\_2$ between orbifolds is given by a continuous map $|f|:|\mathcal O\_1|\to |\mathcal O\_2|$ with the property that for each $p\in |\mathcal O\_1|$, there are local models $(\hat U\_i,G\_i)$ ($i=1$, $2$) and a smooth map $\hat f:\hat U\_1\to \hat U\_2$ equivariant with respect to a homomorphism $\rho:G\_1\to G\_2$ such that $\pi\_2\circ \hat f = |f|\circ \pi\_1$ where $\pi\_i:\hat U\_i \to U\_i$ is the projection ($\rho$ is not required to be injective or surjective). (See page 7 of Kleiner-Lott.)


I think this satisfies your requirements and is in the spirit of Thurston. 


**Edit**: Perhaps I should mention Remark 2.8 in the Kleiner-Lott paper
(also in regard to other answers to this post), which recalls 
that an orbifold can also be seen as a smooth proper étale grupoid (and 
Morita-equivalent grupoids correspond to equivalent orbifolds). A grupoid 
morphism gives rise to an orbifold map, but these correspond to a stricter
class of maps called *good maps*. The advantage of these maps
is that one can pull back orbi-vector bundles.


