# 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.

## Definition

Q is H-injective if, given AB in H, any AQ extends to BQ.

Let ${\displaystyle {\mathfrak {C}}}$ be a category and let ${\displaystyle {\mathcal {H}}}$ be a class of morphisms of ${\displaystyle {\mathfrak {C}}}$.

An object ${\displaystyle Q}$ of ${\displaystyle {\mathfrak {C}}}$ is said to be ${\displaystyle {\mathcal {H}}}$-injective if for every morphism ${\displaystyle f:A\to Q}$ and every morphism ${\displaystyle h:A\to B}$ in ${\displaystyle {\mathcal {H}}}$ there exists a morphism ${\displaystyle g:B\to Q}$ extending (the domain of) ${\displaystyle f}$, i.e. ${\displaystyle g\circ h=f}$.

The morphism ${\displaystyle g}$ in the above definition is not required to be uniquely determined by ${\displaystyle h}$ and ${\displaystyle f}$.

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

The classical choice for ${\displaystyle {\mathcal {H}}}$ 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 (and still the most important one). If ${\displaystyle {\mathfrak {C}}}$ is an abelian category, an object A of ${\displaystyle {\mathfrak {C}}}$ is injective iff its hom functor HomC(–,A) is exact.

Let

${\displaystyle 0\to A\to B\to C\to 0}$

be an exact sequence in ${\displaystyle {\mathfrak {C}}}$ such that A is injective. Then the sequence splits and B is injective if and only if C is injective.[1]

## Enough injectives

Let ${\displaystyle {\mathfrak {C}}}$ be a category, H a class of morphisms of ${\displaystyle {\mathfrak {C}}}$ ; the category ${\displaystyle {\mathfrak {C}}}$ is said to have enough H-injectives if for every object X of ${\displaystyle {\mathfrak {C}}}$, there exist a H-morphism from X to an H-injective object. Again, H is often the class of monomorphisms, and the classical definition of having enough injectives is that for every every object X of ${\displaystyle {\mathfrak {C}}}$, there exist a monomorphism from X to an injective object.

## Injective hull

A H-morphism g in ${\displaystyle {\mathfrak {C}}}$ is called H-essential if for any morphism f, the composite fg is in H only if f is in H. If H is the class of monomorphisms, g is called an essential monomorphism.

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.