Cohomology Theories of Partial Groups

TL;DR

This paper develops two cohomology theories for partial groups, generalizing classical group cohomology, and demonstrates their equivalence and application to extension problems.

Contribution

It introduces two parallel cohomology frameworks for partial groups and proves their equivalence, enabling new insights into partial group extensions.

Findings

The algebraic and simplicial cohomology theories coincide.

Cohomology can be used to classify and compute partial group extensions.

Explicit examples of partial group extensions are provided.

Abstract

We initiate a systematic study of cohomology theories for partial groups, algebraic structures introduced by Chermak that generalize groups by allowing only partially defined products. Inspired by classical group cohomology, we develop two parallel approaches - an algebraic theory based on Chermak's framework and a simplicial-set-based theory using local coefficient systems - and show that they coincide. As an application, we illustrate how the extension theory of partial groups, as developed by Broto and Gonzalez, can be interpreted and computed using our cohomology theory, including explicit examples such as extensions of free partial groups, and compare these results with classical group extensions.