Additive categoryIn mathematics, specifically in , an additive category is a C admitting all finitary biproducts. There are two equivalent definitions of an additive category: One as a category equipped with additional structure, and another as a category equipped with no extra structure but whose objects and morphisms satisfy certain equations. A category C is preadditive if all its hom-sets are abelian groups and composition of morphisms is bilinear; in other words, C is over the of abelian groups.
SubcategoryIn mathematics, specifically , a subcategory of a C is a category S whose are objects in C and whose morphisms are morphisms in C with the same identities and composition of morphisms. Intuitively, a subcategory of C is a category obtained from C by "removing" some of its objects and arrows. Let C be a category. A subcategory S of C is given by a subcollection of objects of C, denoted ob(S), a subcollection of morphisms of C, denoted hom(S).
Equaliser (mathematics)In mathematics, an equaliser is a set of arguments where two or more functions have equal values. An equaliser is the solution set of an equation. In certain contexts, a difference kernel is the equaliser of exactly two functions. Let X and Y be sets. Let f and g be functions, both from X to Y. Then the equaliser of f and g is the set of elements x of X such that f(x) equals g(x) in Y. Symbolically: The equaliser may be denoted Eq(f, g) or a variation on that theme (such as with lowercase letters "eq").
Pointed spaceIn mathematics, a pointed space or based space is a topological space with a distinguished point, the basepoint. The distinguished point is just simply one particular point, picked out from the space, and given a name, such as that remains unchanged during subsequent discussion, and is kept track of during all operations. Maps of pointed spaces (based maps) are continuous maps preserving basepoints, i.e.
