# Injective object

In mathematics, especially in the field of category theory, the concept of injective object is a generalization of the concept of injective module. This concept is important in homotopy theory and in theory of model categories. The dual notion is that of a projective object.

## General Definition

In a locally small category, it is equivalent to require that the hom functor $Hom_{\mathfrak {C}}(-,Q)$ carries ${\mathcal {H}}$ -morphisms to epimorphisms (surjections).

The classical choice for ${\mathcal {H}}$ is the class of monomorphisms, in this case, the expression injective object is used.

## Abelian case

The abelian case was the original framework for the notion of injectivity.

## Injective hull

A H-morphism g in ${\mathfrak {C}}$ is called H-essential if for any morphism f, the composite fg is in H only if f is in H.

If f is a H-essential H-morphism with a domain X and an H-injective codomain G, G is called an H-injective hull of X. This H-injective hull is then unique up to a noncanonical isomorphism.