A note on a paper of Joubert (Q688989)

This note concerns a theorem of \textit{G. Joubert} [Cah. Topologie Géom. Différ. 8, 1-117 (1966; Zbl 0192.102)]. Recall that a \(\vee\)- prehomomorphism \(\vartheta : S \to T\) between inverse semigroups is a map such that \(\vartheta(xy) \leq \vartheta(x)\vartheta(y)\) for all \(x, y \in S\) and that if \(G\) is a group then \(K(G)\) is the inverse semigroup of all cosets of subgroups of \(G\) where the product \(A \otimes B\) of two such cosets is the smallest coset containing \(AB\). Joubert's Theorem is as follows: Given an inverse semigroup \(S\) there is a group \(G\) and an idempotent-determined \(\vee\)-prehomomorphism \(\varphi: S \to K(G)\) such that \(G = \bigcup\{\varphi(s); s\in S\}\). This result is equivalent to the existence of an \(E\)-unitary cover of \(S\) [\textit{D. B. McAlister} and \textit{N. Reilly}, Pac. J. Math. 68, 161-174 (1977; Zbl 0368.20043)]. In this paper the author uses contemporary terminology and notation to present a simplified version of Joubert's original proof which is couched in the language and notation of Ehresmann's school of category theory.