Talk:Essential extension

Essential extensions in general categories
Let C be a category. A morphism f: X → Y in C is essential if a morphism g: Y → Z is a monomorphism if and only if g ° f is a monomorphism. Taking g to be the identity morphism of Y shows that an essential morphism f must be a monomorphism.

If X has an injective envelope Y, then Y is the largest essential extension of X. But the largest essential extension may not be injective.

Hazelmaye (talk) 15:23, 18 June 2021 (UTC)