The pullback closure operator and generalisations of perfectness

A categorical closure operator induced via pullback by a pointed endofunctor is introduced. Various notions of a perfect morphism relative to a pointed endofunctor and the induced closure are then considered. The main result explores how these notions are interrelated, linking also with earlier notions of perfectness.The author acknowledges financial support from the University of Cape Town, from the Foundation for Research Development through the Categorical Topology Research Group at the University of Cape Town, and from the University of L'Aquila.