# In Search of Homology for Quasigroups of Bol-Moufang Type

## TL;DR

This paper develops a (co)homology theory for Bol-Moufang type quasigroups, using their extensions to define boundary operations and compute homology groups, providing new algebraic invariants for these structures.

## Contribution

It introduces a novel (co)homology framework for Bol-Moufang quasigroups based on their extensions, with explicit computations and theoretical insights.

## Key findings

- Computed second homology groups for specific quasigroups
- Defined boundary operations using extensions
- Suggested links between homology groups and categorical structures

## Abstract

We initiate (co)homology theory for quasigroups of Bol-Moufang type based on analysis of their extensions by affine quasigroups of the same type. We use these extensions to define second and third boundary operations, $\partial_2(x,y)$ and $\partial_3(x,y,z)$, respectively. We use these definitions to compute the second homology groups for several examples from the work of Phillips and Vojtechovsky. We speculate about the relation between these homology groups and those obtained from a small category with coefficients in a functor.