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Abstract We introduce a set of axioms for locally topologically ordered quantum spin systems in terms of nets of local ground state projections, and we show they are satisfied by Kitaev’s Toric Code and Levin-Wen type models. For a locally topologically ordered spin system on$$\mathbb {Z}^{k}$$, we define a local net of boundary algebras on$$\mathbb {Z}^{k-1}$$, which provides a mathematically precise algebraic description of the holographic dual of the bulk topological order. We construct a canonical quantum channel so that states on the boundary quasi-local algebra parameterize bulk-boundary states without reference to a boundary Hamiltonian. As a corollary, we obtain a new proof of a recent result of Ogata [Oga24] that the bulk cone von Neumann algebra in the Toric Code is of type$$\mathrm {II}$$, and we show that Levin-Wen models can have cone algebras of type$$\mathrm {III}$$. Finally, we argue that the braided tensor category of DHR bimodules for the net of boundary algebras characterizes the bulk topological order in (2+1)D, and can also be used to characterize the topological order of boundary states. 

A unitary fusion category is called $$\mathbb{Z}/2\mathbb{Z}$$-quadratic if it has a $$\mathbb{Z}/2\mathbb{Z}$$ group of invertible objects and one other orbit of simple objects under the action of this group. We give a complete classification of $$\mathbb{Z}/2\mathbb{Z}$$-quadratic unitary fusion categories. The main tools for this classification are skein theory, a generalization of Ostrik's results on formal codegrees to analyze the induction of the group elements to the center, and a computation similar to Larson's rank-finiteness bound for $$\mathbb{Z}/3\mathbb{Z}$$-near group pseudounitary fusion categories. This last computation is contained in an appendix coauthored with attendees from the 2014 AMS MRC on Mathematics of Quantum Phases of Matter and Quantum Information.