A topos associated with a colored category

TL;DR

This paper introduces a topos-theoretic framework for colored categories, enabling cohomology theories that connect to association schemes and their group cohomology, revealing new algebraic insights.

Contribution

It establishes that functor categories over colored categories form topoi and develops a cohomology theory linking colored categories with association schemes.

Findings

Functor categories over colored categories are topoi.

Cohomology of colored categories generalizes group cohomology.

Relation between cohomology of colored categories and association schemes' representations.

Abstract

We show that a functor category whose domain is a colored category is a topos.The topos structure enables us to introduce cohomology of colored categories including quasi-schemoids. If the given colored category arises from an association scheme, then the cohomology coincides with the group cohomology of the factor scheme by the thin residue. Moreover, it is shown that the cohomology of a colored category relates to the standard representation of an association scheme via the Leray spectral sequence.