A Tale of Two Categories: Inductive groupoids and Cross-connections

Abstract

A groupoid is a small category in which all morphisms are isomorphisms. An inductive groupoid is a specialised groupoid whose object set is a regular biordered set and the morphisms admit a partial order. A normal category is a specialised small category whose object set is a strict preorder and the morphisms admit a factorisation property. A pair of 'related' normal categories constitutes a cross-connection. Both inductive groupoids and cross-connections were identified by Nambooripad as categorical models of regular semigroups. We explore the inter-relationship between these seemingly different categorical structures and prove a direct category equivalence between the category of inductive groupoids and the category of cross-connections.

The mathematical work of K.S.S. Nambooripad

“…Similarly the normal category R G was constructed from three constituent categories and the cross-connection was defined between L G and R G . A direct category equivalence between the category of inductive groupoids and crossconnections was also proved [11,Theorem 5.1].…”

“…By transitivity, Nambooripad's results (Theorems 2.3 and 4.4) imply that the category of inductive groupoids is equivalent to the category of cross-connections: but these constructions look evidently disconnected. Their interrelationship was explored by Azeef and Volkov [10,11] and a direct category equivalence was constructed. In addition to giving the relationship between the ideal structure and the idempotent structure of regular semigroups, their results also give a road map for the transfer of problems of inductive groupoid origin to the cross-connection framework and vice versa.…”

Structure theory of regular semigroups

“…The subject of Section 4.2 is an alternative description of the structure of regular semigroups due to Nambooripad which generalizes Grillet's approach (Section 3.2) for all regular semigroups. Additionally, we discuss a recent work by Muhammed and Volkov [44,45] on the relationship between Nambooripad's two approaches. We close this section by outlining a generalization of Nambooripad's description of regular semigroups via ordered groupoids over biordered sets for a class of semigroups with distinguished sets of idempotents.…”

“…It should be emphasized that the equivalence of categories established is not the composition of the equivalences given in the proofs of Theorems 4.1 and 4.2. In [45] the same authors go further by providing an equivalence between the category of regular inductive groupoids and the category of cross-connections in such a way that they avoid using semigroups and restrict themselves to a purely categorical framework. It is worth mentioning that cross-connections seem to encode much more information than regular inductive groupoids.…”

Cross-connection structure of concordant semigroups

“…In [30], a category equivalence between the category of regular semigroups and the category of crossconnected normal categories was also proved. Recently, the first author and Volkov [3,4] showed the direct equivalence of the above discussed approaches to arbitrary regular semigroups: the ESN approach and the cross-connection approach.…”