Equivariant Tannaka-Krein reconstruction and quantum automorphism groups of discrete structures

We define quantum automorphism groups of a wide range of discrete structures. The central tool for their construction is a generalisation of the Tannaka-Krein reconstruction theorem. For any direct sum of matrix algebras M, and any concrete unitary 2-category of finite type Hilbert-M-bimodules C, under reasonable conditions, we construct an algebraic quantum group G which acts on M by α, such that the category of α-equivariant corepresentations of G on finite type Hilbert-M-bimodules is equivalent to C. Moreover, we explicitly describe how to get such categories from connected locally finite discrete structures. As an example, we define the quantum automorphism group of a quantum Cayley graph.