Essentiality and Injectivity

Essentiality is an important notion closely related to injectivity. Depending on a class M of 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.