Well-behaved Epireflections for Kan Extensions

Abstract

Let $K:\mathbb{B}\rightarrow \mathbb{A}$ be a functor such that the image of the objects in $\mathbb{B}$ is a cogenerating set of objects for $\mathbb{A}$. Then, the right Kan extensions adjunction $\mathbf{Set}^K\dashv Ran_K$ induces necessarily an epireflection with stable units and a monotone-light factorization. This result follows from the one stating that an adjunction produces an epireflection in a canonical way, provided there exists a prefactorization system which factorizes all of its unit morphisms through epimorphisms. The stable units property, for the so obtained epireflections, is thereafter equivalently restated in such a manner it is easily recognizable in the examples. Furthermore, having stable units, there is a strong but quite simple sufficient condition for the existence of an associated monotone-light factorization, which has proved to be fruitful.