Relative cohomology of algebraic theories

TL;DR

This paper develops a framework for relative cohomology in algebraic theories using abelian categories, providing new tools for analyzing models of algebraic systems and their cohomological properties.

Contribution

It introduces a construction of relative abelian categories for algebraic models and applies this to define an analogue of Hochschild-Mitchell cohomology.

Findings

Established a method to construct relative abelian categories for algebraic models.

Defined a Hochschild-Mitchell type cohomology for the Yoneda embedding.

Provided examples illustrating the application of the theory.

Abstract

We construct relative abelian categories in the sense of MacLane for models of algebraic systems in (co)complete abelian categories. As an example, we consider an analogue of Hochschild-Mitchell cohomology for the functor of Yoneda embedding.