Let $\mathcal{ B}$ be a conformal net. We give the notion of a proper action of a finite hypergroup acting by vacuum preserving unital completely positive (so-called stochastic) maps, which generalizes the proper actions of finite groups. Taking fixed points under such an action gives a finite index subnet $\mathcal{ B}^K$ of $\mathcal{ B}$, which generalizes the $G$-orbifold. Conversely, we show that if $\mathcal{ A}\subset \mathcal{ B}$ is a finite inclusion of conformal nets, then $\mathcal{ A}$ is a generalized orbifold $\mathcal{ A}=\mathcal{ B}^K$ of the conformal net $\mathcal{ B}$ by a unique finite hypergroup $K$. There is a Galois correspondence between intermediate nets $\mathcal{ B}^K\subset \mathcal{ A} \subset \mathcal{ B}$ and subhypergroups $L\subset K$ given by $\mathcal{ A}=\mathcal{ B}^L$. In this case, the fixed point of $\mathcal{ B}^K\subset \mathcal{ A}$ is the generalized orbifold by the hypergroup of double cosets $L\backslash K/ L$. If $\mathcal{ A}\subset \mathcal{ B}$ is an finite index inclusion of completely rational nets, we show that the inclusion $\mathcal{ A}(I)\subset \mathcal{ B}(I)$ is conjugate to a Longo--Rehren inclusion. This implies that if $\mathcal{ B}$ is a holomorphic net, and $K$ acts properly on $\mathcal{ B}$, then there is a unitary fusion category $\mathcal{ F}$ which is a categorification of $K$ and $\mathrm{ Rep}(\mathcal{ B}^K)$ is braided equivalent to the Drinfel'd center $Z(\mathcal{ F})$. More generally, if $\mathcal{ B}$ is completely rational conformal net and $K$ acts properly on $\mathcal{ B}$, then there is a unitary fusion category $\mathcal{ F}$ extending $\mathrm{ Rep}(\mathcal{ B})$, such that $K$ is given by the double cosets of the fusion ring of $\mathcal{ F}$ by the Verlinde fusion ring of $\mathcal{ B}$ and $\mathrm{ Rep}(\mathcal{ B}^K)$ is braided equivalent to the M\"uger centralizer of $\mathrm{ Rep}(\mathcal{ B})$ in $Z(\mathcal{ F})$.
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