A field guide to categories with Aₙ fusion rules

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We collate information about the fusion categories with Aₙ fusion rules. This note includes the classification of these categories, a realisation via the Temperley-Lieb categories, the auto-equivalence groups (both braided and tensor), identifications of the subcategories of invertible objects, and explicit descriptions of the Drinfeld centres. The first section describes the classification of these categories (as monoidal, dagger, pivotal, and braided categories). The second section describes the properties of these categories.