Functors from association schemes

We construct a wide subcategory of the category of finite association schemes with a collection of desirable properties. Our subcategory has a first isomorphism theorem analogous to that of groups. Also, standard constructions taking schemes to groups (thin radicals and quotients by thin residues) or algebras (adjacency algebras) become functorial when restricted to our category. We use our category to give a more conceptual account for a result of Hanaki concerning products of characters of association schemes; i.e. we show that the virtual representations of an association scheme form a module over the representation ring of the thin quotient of the association scheme.