We investigate the Squier complexes of presentations of groups and inverse monoids using the theory semiregular, regular, and pseudoregular groupoids. Our main interest is the class of regular groupoids, and the new class of pseudoregular groupoids.
Our study of group presentations uses monoidal, regular groupoids. These are equivalent to crossed modules, and we recover the free crossed module usually associated to a group presentation, and a free presentation of the relation module with kernel the fundamental group of the Squier complex, the module of identities among the relations. We carry out a similar study of inverse monoid presentations using pseudoregular groupoids. The relation module is defined via an intermediate construction – the derivation module of a homomorphism, – and a key ingredient is the factorisation of the presentation map from a free inverse monoid as the composition of an idempotent pure map and an idempotent separating map. We can then use the properties of idempotent separating maps, and properties of the derivation module as a left adjoint, to derive a free presentation of the relation module. The construction of its kernel – the module of identities – uses further key facts about pseudoregular groupoids
