Representations of inverse monoids by partial automorphisms (Q1590063)

A category \(\mathbf C\) is \((E,M)\)-structured if it has subcategories \(E\) and \(M\) such that (1) \(\text{Iso}({\mathbf C})\subseteq E\cap M\) and (2) each morphism \(f\) may be factored as \(f=em\) where \(e\in E\) and \(m\in M\) and if \(e_1m_1=f=e_2m_2\), then \(e_1=e_2u\) and \(um_1=m_2\) for a unique isomorphism \(u\). An \((E,M)\)-structured category \(\mathbf C\) is said to be properly \((E,M)\)-structured if \(E\subseteq\text{Epi}({\mathbf C})\) and \(M\subseteq\text{Mono}({\mathbf C})\). The author shows that any inverse semigroup of endomorphisms of an object in a properly \((E,M)\)-structured category may be embedded in an inverse monoid of partial automorphisms between retracts of that object.