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:
“The tremendous outflow of intellectuals that formed such a prominent part of the general exodus from Soviet Russia in the first years of the Bolshevist Revolution seems today like the wanderings of some mythical tribe whose bird-signs and moon-signs I now retrieve from the desert dust.”
—Vladimir Nabokov (1899–1977)
“The man who knows governments most completely is he who troubles himself least about a definition which shall give their essence. Enjoying an intimate acquaintance with all their particularities in turn, he would naturally regard an abstract conception in which these were unified as a thing more misleading than enlightening.”
—William James (1842–1910)