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:
“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)
“I’m beginning to think that the proper definition of “Man” is “an animal that writes letters.””
—Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)
“Beauty, like all other qualities presented to human experience, is relative; and the definition of it becomes unmeaning and useless in proportion to its abstractness. To define beauty not in the most abstract, but in the most concrete terms possible, not to find a universal formula for it, but the formula which expresses most adequately this or that special manifestation of it, is the aim of the true student of aesthetics.”
—Walter Pater (1839–1894)