An Ehresmann-Schein-Nambooripad theorem for locally Ehresmann \(P\)-Ehresmann semigroups (Q2300150)

The classic Ehresmann-Schein-Nambooripad (ESN) theorem identifies the category of inverse semigroups with the category of inductive groupoids and their respective functors. It has been extended in all sorts of directions, one of which is into the realm of restriction and Ehresmann semigroups (a good survey is by \textit{C. Hollings} [Eur. J. Pure Appl. Math. 5, No. 4, 414--450 (2012; Zbl 1389.20078)]). The reviewer [J. Pure Appl. Algebra 216, No. 3, 618--632 (2012; Zbl 1257.20058)] introduced the classes of \(P\)-restriction and \(P\)-Ehresmann semigroups, as common generalizations of restriction and Ehresmann semigroups with regular \(^*\)-semigroups. The author found an ESN theorem for a class of \(P\)-restriction semigroups [Bull. Malays. Math. Sci. Soc. (2) 42, No. 2, 535--568 (2019; Zbl 1474.20113)]. Here he does the same for the correspondingly wider class of \(P\)-Ehresmann semigroups, though again only for those such \(S\) in which \(eSe\) is Ehresmann for each `projection' \(e\) of \(S\). The notion of an inductive groupoid is now replaced by an ``\(lepe\)-generalized category'', along with suitably defined functors. While the approach is slightly different from that of the author's cited paper, it is shown how the results of the latter may be deduced from those of the current one.