Cohomology Theories of Partial Groups

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.