On Mitchell's embedding theorem for a quasi-schemoid

TL;DR

This paper extends Mitchell's embedding theorem to tame schemoids, enabling model category structures on chain complexes and exploring Morita equivalence, including applications to Hamming schemes and simplicial complexes.

Contribution

It establishes Mitchell's embedding theorem for tame schemoids, introduces Morita equivalence for schemoids, and constructs new schemoids from simplicial complexes.

Findings

Mitchell's embedding theorem is valid for tame schemoids.

A model category structure on chain complexes over schemoid functor categories is developed.

Hamming schemes are Morita equivalent to certain group-based association schemes.

Abstract

A quasi-schemoid is a small category whose morphisms are colored with appropriate combinatorial data. In this note, Mitchell's embedding theorem for a tame schemoid is established. The result allows us to give a cofibrantly generated model category structure to the category of chain complexes over a functor category with a schemoid as the domain. Moreover, a notion of Morita equivalence for schemoids is introduced and discussed. In particular, we show that every Hamming scheme of binary codes is Morita equivalent to the association scheme arising from the cyclic group of order two. In an appendix, we construct a new schemoid from an abstract simplicial complex, whose Bose-Mesner algebra is closely related to the Stanley-Reisner ring of the given complex.