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

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

An object $Q$ of $\mathfrak{C}$ is said to be $\mathcal{H}$-injective if for every morphism $f: A \to Q$ and every morphism $h: A \to B$ in $\mathcal{H}$ there exists a morphism $g: B \to Q$ extending (the domain of) $f$, i.e. $gh = f$. In other words, $Q$ is injective iff any $\mathcal{H}$-morphism into $Q$ extends (via composition on the left) to a morphism into $Q$.

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

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

If $\mathfrak{C}$ is an abelian category, an object A of $\mathfrak{C}$ is injective iff its hom functor HomC(–,A) is exact.

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

Enough injectives

Let $\mathfrak{C}$ be a category, H a class of morphisms of $\mathfrak{C}$ ; the category $\mathfrak{C}$ is said to have enough H-injectives if for every object X of $\mathfrak{C}$, there exist a H-morphism from X to an H-injective object.

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.

References

• J. Rosicky, Injectivity and accessible categories
• F. Cagliari and S. Montovani, T0-reflection and injective hulls of fibre spaces