Homology, cohomology and extensions of ordered groupoids

Abstract

This thesis contains contributions to the homology, cohomology and extensions of ordered groupoids. We study the simplicial homology of ordered groupoids. We also discuss the (co)homology of the set of identities of ordered groupoids and relate the cohomology of the set of identities of an ordered groupoid to the cohomology of the ordered groupoid. We discuss the β -relation on ordered groupoids; the analogue of the minimum group congruence for inverse semigroups and show that for β-transitive ordered groupoids, the homology of the ordered groupoid is isomorphic to that of its levelled groupoid. In the applications of the discussion on the cohomology of ordered groupoids, we relate the second cohomology group of ordered groupoids to the set of extensions of ordered groupoids with abelian kernel. In particular we show that for an ordered groupoid QI obtained from the ordered groupoid Q by attaching the symbol I ∉ Q and a QI-module A0 obtained as an extension of the Q{module A, Hn(QI ,A0) is in one-to-one correspondence with the set of extensions of A by Q. Finally, we follow the approach of Huebschmann but using appropriate constructions for ordered groupoids and verify that our constructions do have the properties required in the arguments of Huebschmann to show that the set of n-fold extensions of an abelian ordered groupoid A by an ordered groupoid Q is isomorphic to Hn+1(QI, A0).