TITLE:
Partial Groups, Simplicial K(G, 1)’s and Kan Complexes
AUTHORS:
Solomon Jekel
KEYWORDS:
Partial Group, Simplicial Set, Nerve, Kan Extension
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.13 No.11,
November
1,
2023
ABSTRACT: In our paper Simplicial K(G, 1)’s we constructed a sub-complex of the nerve of a group G determined by a partial group structure, and we proved, under a generalized associativity condition called regularity, that the sub-complex realizes as a K(G, 1). This type of sub-complex appears naturally in several topological and algebraic contexts. In this note we prove that regularity of a partial group implies that the Kan extension condition is satisfied on its nerve in dimensions greater than one, and in dimension one a weaker version of the extension condition holds.