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 definition, general and/or definition:
“The following general definition of an animal: a system of different organic molecules that have combined with one another, under the impulsion of a sensation similar to an obtuse and muffled sense of touch given to them by the creator of matter as a whole, until each one of them has found the most suitable position for its shape and comfort.”
—Denis Diderot (1713–1784)
“The first general store opened on the ‘Cold Saturday’ of the winter of 1833 ... Mrs. Mary Miller, daughter of the store’s promoter, recorded in a letter: ‘Chickens and birds fell dead from their roosts, cows ran bellowing through the streets’; but she failed to state what effect the freeze had on the gala occasion of the store opening.”
—Administration in the State of Sout, U.S. public relief program (1935-1943)
“No man, not even a doctor, ever gives any other definition of what a nurse should be than this—”devoted and obedient.” This definition would do just as well for a porter. It might even do for a horse. It would not do for a policeman.”
—Florence Nightingale (1820–1910)