Essentiality and Injectivity

Abstract

Essentiality is an important notion closely related to injectivity.Depending
on a classMof morphisms of a category A, three different types of essentiality are
considered in literature. Each has its own benefits in regards with the behaviour of
M-injectivity. In this paper we intend to study these different notions of essentiality
and to investigate their relations to injectivity and among themselves. We will see,
among other things, that although these essential extensions are not necessarily
equivalent, they behave almost equivalently with regard to injectivity.