Association schemoids and their categories

On strong homotopy for quasi-schemoids

“…Let C be a small category and K(C) = (C, S) the discrete quasi-schemoid associated with C; that is, the partition S is defined by S = {{f }} f ∈mor(C) . The correspondence K induces a pair of adjoints K : Cat / / qASmd : U o o in which U is the forgetful functor and the right adjoint to K. It is remarkable that the functor K is a fully faithful embedding; see [8,Remark 3.1,Diagram (6.1)]. Furthermore, the correspondences S( ) and  mentioned above give rise to functors.…”

“…where Gpd denotes the category of groupoids and ı : Gr → Gpd is the natural fully faithful embedding; see [8,Section 3,Diagram (6.1)]. Observe that the composite U • S( ) is not the usual embedding from Gpd to Cat.…”

“…Matsuo and the author [8] have proposed the notion of quasi-schemoids generalizing that of association schemes from a small categorical point of view. Roughly speaking, the new object is indeed a small category with suitable coloring for morphisms.…”

“…Thus one might expect a relevant notion of a homotopy group for a quasi-schemoid, as in [7], and an application of categorical matrix Toda brackets due to Hardie, Kamps and Marcum [5] to our category qASmd. As for homological algebra on schemoids, in order to develop categorical representation theory, we may consider the Bose-Mesner algebra introduced in [8, Section 2] and an appropriate functor category with a quasi-schemoid and an abelian category as source and target, respectively; see [8,Sections 5 and 6] for first steps in this direction. These topics are addressed in subsequent work.…”