Association schemoids and their categories

We propose the notion of association schemoids generalizing that of association schemes from small categorical points of view. In particular, a generalization of the Bose-Mesner algebra of an association scheme appears as a subalgebra in the category algebra of the underlying category of a schemoid. In this paper, the equivalence between the categories of grouopids and that of thin association schemoids is established. Moreover linear extensions of schemoids are considered. A general theory of the Baues-Wirsching cohomology deduces a classification theorem for such extensions of a schemoid. We also introduce two relevant categories of schemoids into which the categories of schemes due to Hanaki and due to French are embedded, respectively.