On lax epimorphisms and the associated factorization (Q2154260)

Lax epimorphisms or co-fully-faithful morphisms in a 2-category are 1-cells \(f:A\to B\) which induce fully faithful contravariant Hom-functors \(\mathrm{Hom}(f,C)\) for all \(C\). The authors give a concrete description of an \((\mathcal E,\mathcal M)\)-factorization system in \(\mathsf{Cat}\), where \(\mathcal E\) is the class of all lax epimorphisms. After proving that lax epimorphisms are closed under 2-colimits, they further show that under suitable conditions if a 2-category has all 2-colimits then it admits an orthogonal factorization system \((\mathit{LaxEpi},\mathit{LaxStrongMono})\). Finally they study several characterizations of the class of lax epimorphisms in enriched 2-categories \(\mathcal V\text{-}\mathsf{Cat}\) under suitable properties on the enriching symmetric monoidal category \(\mathcal V\).