Injective object

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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[edit]

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[edit]

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[edit]

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[edit]

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.



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