Definition
A category C is additive if
- it has a zero object
- every hom-set Hom(A, B) has an addition, endowing it with the structure of an Abelian group, and such that composition of morphisms is bilinear
- all finitary biproducts exist.
Note that a category is called preadditive if just the second holds, whereas it is called semiadditive if both the first and the third hold.
Also, since the empty biproduct is a zero object in the category, we may omit the first condition. If we do this, however, we need to presuppose that the category C has zero morphisms, or equivalently that C is enriched over the category of pointed sets.
Read more about this topic: Additive Category
Famous quotes containing the word definition:
“One definition of man is “an intelligence served by organs.””
—Ralph Waldo Emerson (1803–1882)
“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)
“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)