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That's because of the required property of equivariance of the local lifting of maps between orbifolds. This problem disapears in Diffeology. Indeed, there exists smooth maps between orbifolds (as diffeologies) that have no local equivariant liftings at all. This is shown in the example 25 of our paper on orbifolds [IKZ]. The function $f \colon \mathbf{C} \to \mathbf{C}$ defines by projectioon a well-defined smooth map between $\mathcal{Q} =\mathbf{C}/\mathbf{Z}_m$ to $\mathcal{Q}_n$ that has no equivarian local liftings on the neighborhood of $0$, any small you take the neighborhood:

That's because of the required property of equivariance of the local lifting of maps between orbifolds. This problem disapears in Diffeology. Indeed, there exist smooth maps between orbifolds (as diffeologies) that have no local equivariant liftings at all. This is shown in the example 25 of our paper on orbifolds [IKZ]. The function $f \colon \mathbf{C} \to \mathbf{C}$ defines by projectioon a well-defined smooth map between $\mathcal{Q} =\mathbf{C}/\mathbf{Z}_m$ to $\mathcal{Q}_n$ that has no equivarian local liftings on the neighborhood of $0$, any small you take the neighborhood:

What is interesting is that, a contrario, if $f$ is a diffeomorphism, then it has always a local equivariant lifting, in any local representation. This is the Lemma 21 of [YKZ].

What is interesting is that, a contrario, a local diffeomorphism between orbifold has always a local equivariant lifting, in any local representation. This is the Lemma 21 of [YKZ].

That's because of the required property of equivariance of the local lifting of maps between orbifolds. This problem disapears in Diffeology. Indeed, there exists smooth maps between orbifolds (as diffeologies) that have no one local equivariant liftings. This is shown in the example 25 of our paper on orbifolds [IKZ]. The function $f \colon \mathbf{C} \to \mathbf{C}$ defines by projectioon a well-defined smooth map between $\mathcal{Q} =\mathbf{C}/\mathbf{Z}_m$ to $\mathcal{Q}_n$ that has no equivarian local liftings on the neighborhood of $0$, any small you take the neighborhood:

That's because of the required property of equivariance of the local lifting of maps between orbifolds. This problem disapears in Diffeology. Indeed, there exists smooth maps between orbifolds (as diffeologies) that have no local equivariant liftings at all. This is shown in the example 25 of our paper on orbifolds [IKZ]. The function $f \colon \mathbf{C} \to \mathbf{C}$ defines by projectioon a well-defined smooth map between $\mathcal{Q} =\mathbf{C}/\mathbf{Z}_m$ to $\mathcal{Q}_n$ that has no equivarian local liftings on the neighborhood of $0$, any small you take the neighborhood: