Homological Algebra/Definition of abelian category

-enriched category An -enriched category is a category such that: , is an abelian group. , is bilinear. zero object A zero object is an object in an -enriched category that is both initial and terminal. We usually denote it by . biproduct Given an -enriched category , a biproduct of is a tuple such that: We usually denote by . additive category An additive category is an -enriched category such that: There is a zero product in . Every has a biproduct. (co-)kernel Given in an -enriched category. A (co-)kernel of is a (co-)equalizer of and . abelian category An abelian category is an additive category where: Every morphism has a kernel and cokernel. Every monomorphism is a kernel and every epimorphism is a cokernel. The category of all left -modules of a ring is an abelian category. Exercises Given in an -enriched category with zero object. Prove that iff factors through . Given a biproduct of and . Prove that is a coproduct of and and is a product of and . In an -enriched category with zero object, a kernel of can be equivalently be characterized as a pullback of along .