Association schemoids and their categories

TL;DR

This paper introduces association schemoids as a categorical generalization of association schemes, explores their algebraic structures, equivalences with groupoids, and classifies their extensions using cohomology.

Contribution

It generalizes association schemes to a categorical framework, establishes equivalences with groupoids, and develops a cohomological classification of schemoid extensions.

Findings

Equivalence between categories of groupoids and thin association schemoids.

Development of a cohomology-based classification of schemoid extensions.

Embedding of scheme categories by Hanaki and French into schemoid categories.

Abstract

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.