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).