Roughly, an orbifold is the quotient of a manifold by a finite group. For example, take a sheet of paper and add a small crease perpendicular to one side at the halfway point. Then, line up the two halves of the side. This may be thought of as the plane $\mathbb R^2$ modulo the group $\mathbb Z^2$ . Now, let us give the definition.

Define a category $\mathcal X$ : The objects are pairs $(G,X)$ , where $G$ is a finite group acting effectively on a connected smooth manifold $X$ . A morphism $\Phi$ between two objects $(G^\prime, X^\prime)$ and $(G,X)$ is a family of open embeddings $\phi:X^\prime\rightarrow X$ which satisfy

• for each embedding $\phi\in\Phi$ , there is an injective homomorphism $\lambda_\phi:G^\prime\rightarrow G$ such that $\phi$ is $\lambda_\phi$ equivariant
• For $g\in G$ , we have
  
  and if $(g\phi)(X) \cap \phi(X) \ne \emptyset$ , then $g\in \lambda_\phi(G^\prime)$ .
• $\Phi = \{g\phi, g\in G$ }, for any $\phi \in \Phi$

Now, we define orbifolds. Given a paracompact Hausdorff space $X$ and a nice open covering $\mathcal U$ which forms a basis for the topology on $X$ , an orbifold structure $\mathcal V$ on $X$ consists of

1. For $U\in \mathcal U$ , $\mathcal V(U) = (G_U,\tilde U)\stackrel \tau \rightarrow U$ is a ramified cover $\tilde U \rightarrow U$ which identifies $\tilde U\slash G_U \cong U$
2. For $U\subset V \in \mathcal U$ , there exists a morphism $\phi_{VU}(G_U, \tilde U)\rightarrow (G_V, \tilde V)$ covering the inclusion
3. If $U\subset V\subset W \in \mathcal U$ , $\phi_{WU} = \phi_{WV}\circ\phi_{VU}$

References:

[1] Kawasaki T., The Signature theorem for V-manifolds. Topology 17 (1978), 75-83. MR0474432 (57:14072)