Relative epis need not be surjective (Q1062130)

In concrete categories (\({\mathcal A},U)\) over Set, i.e., \({\mathcal A}\to^{U}Set\) is faithful, one would like to characterize those \({\mathcal A}\)-morphisms, which are (i.e., whose underlying Set-map is) surjective. Any such morphism is epi, but the converse need not be true. The second author has shown [Algebra Univers. 17, 65-72 (1983; Zbl 0524.18004)] that in categories of ordered universal algebras, subject to certain completeness and continuity conditions, together with appropriately continuous homomorphisms the surjective morphisms are precisely the relative epis. (A morphism f of a category \({\mathcal C}\) is relative epi, if for every \({\mathcal C}\)-morphism g with cod g\(=cod f\) there exist \({\mathcal C}\)-morphisms r and s with \(\bullet\) \(\to^{r}\bullet \to^{f}\bullet =\bullet \to^{s}\bullet \to^{g}\bullet\), such that s is epi.) The authors construct a variety \({\mathcal V}\) of (unordered) universal algebras and a relative epimorphism in \({\mathcal V}\) which is not surjective. This also shows that the universal epis (i.e., epis whose pullbacks are again epis [see \textit{D. Pumplün,} Math. Ann. 198, 131-146 (1972; Zbl 0229.18002)]) and the persistent epis (i.e., epis whose pullbacks along all morphisms with the same codomain exist and are again epis [see \textit{G. M. Kelly}, J. Aust. Math. Soc. 9, 124-142 (1969; Zbl 0169.326)]) need not be surjective, since in the presence of pullbacks these classes of epis coincide with the relative epis.