General Definition
Let be a category and let be a class of morphisms of .
An object of is said to be -injective if for every arrow and every morphism in there exists a morphism extending (the domain of), i.e . In other words, is injective iff any -morphism extends (via composition on the left) to any morphism into .
The morphism in the above definition is not required to be uniquely determined by .
In a locally small category, it is equivalent to require that the hom functor carries -morphisms to epimorphisms (surjections).
The classical choice for is the class of monomorphisms, in this case, the expression injective object is used.
Read more about this topic: Injective Object
Famous quotes containing the words general and/or definition:
“In democratic ages men rarely sacrifice themselves for another, but they show a general compassion for all the human race. One never sees them inflict pointless suffering, and they are glad to relieve the sorrows of others when they can do so without much trouble to themselves. They are not disinterested, but they are gentle.”
—Alexis de Tocqueville (1805–1859)
“Mothers often are too easily intimidated by their children’s negative reactions...When the child cries or is unhappy, the mother reads this as meaning that she is a failure. This is why it is so important for a mother to know...that the process of growing up involves by definition things that her child is not going to like. Her job is not to create a bed of roses, but to help him learn how to pick his way through the thorns.”
—Elaine Heffner (20th century)