Given a set $\Sigma$ of morphisms in a category C, we construct a functor F which sends elements of $\Sigma$ to split monomorphisms. Moreover, we prove that F is weakly universal with that property when considered in the world of locally posetal 2-categories. Besides, we also use locally posetal 2-categories in order to construct weak left adjoints to those functors for which any object in the codomain admits a weak reflection. We then apply these two results in order to restate the Injective Subcategory Problem for $\Sigma$ into the existence of some kind of weak right adjoint for F.
Keywords: split monomorphism, locally posetal 2-category, category of fractions, weak reflection, weak adjoint, injective subcategory problem
2010 MSC: 55P60, 18A40, 18G05 (primary); 18A20, 18D05, 18B35, 18A10 (secondary)
Theory and Applications of Categories, Vol. 34, 2019, No. 46, pp 1526-1551.