Knowra Abelian category Abelian category An additive category in which every morphism has a kernel and cokernel, every monomorphism and epimorphism is normal, and finite direct sums exist. These properties make exact sequences and homological algebra available in an abstract setting.
Additive category : A category with abelian groups of morphisms, bilinear composition, a zero object, and finite biproducts. Abelian categories are additive categories with additional kernel, cokernel, and normality conditions.
Exact sequence : A sequence of objects and morphisms in which each morphism's image equals the next morphism's kernel. Abelian categories define exactness internally through kernels and images.
Module category : The category whose objects are modules over a ring and whose morphisms are module homomorphisms. Module categories are central examples where kernels, cokernels, and exact sequences are familiar.
Pre-abelian category : An additive category in which every morphism has a kernel and a cokernel. It has the basic limits but need not make every monomorphism a kernel or every epimorphism a cokernel.
Kernel (category theory) : A universal morphism that captures all maps annihilated by a given morphism. Every morphism in an abelian category has a kernel, supplying the categorical notion of its null part.
Image (category theory) : The subobject through which a morphism factors as an epimorphism followed by a monomorphism. The equality of image and kernel at each position determines exactness.
Category of abelian groups : The category with abelian groups as objects and group homomorphisms as morphisms. It is the basic concrete model of an abelian category.
Exact category : An additive category equipped with a chosen class of sequences treated as exact. It generalizes exact-sequence methods while requiring less structure than an abelian category.
Cokernel : A universal morphism that identifies outputs differing by the image of a given morphism. Cokernels pair with kernels to define images and exactness categorically.
Short exact sequence : An exact sequence consisting of a monomorphism, a middle morphism, and an epimorphism between three objects. It is the basic extension pattern used throughout homological algebra.
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