Алдынғы абелдік санаттар және нақты қосқандар туралы

Category
In mathematics, specifically in category theory, a pre abelian category is an additive category that has all kernels and cokernels. Spelled out in more detail, this means that a category C is pre abelian if:
C is preadditive, that is enriched over the monoidal category of abelian groups (equivalently, all hom sets in C are abelian groups and composition of morphisms is bilinear);
C has all finite products (equivalently, all finite coproducts); note that because C is also preadditive, finite products are the same as finite coproducts, making them biproducts;
given any morphism f: A → B in C, the equaliser of f and the zero morphism from A to B exists (this is by definition the kernel of f), as does the coequaliser (this is by definition the cokernel of f). Note that the zero morphism in item 3 can be identified as the identity element of the hom set Hom(A,B), which is an abelian group by item 1; or as the unique morphism A → 0 → B, where 0 is a zero object, guaranteed to exist by item 2.

Recall that all finite limits and colimits exist in a pre abelian category. In general category theory, a functor is called left exact if it preserves all finite limits and right exact if it preserves all finite colimits. (A functor is simply exact if it's both left exact and right exact.) In a pre abelian category, exact functors can be described in particularly simple terms. First, recall that an additive functor is a functor F: C → D between preadditive categories that acts as a group homomorphism on each hom set. Then it turns out that a functor between pre abelian categories is left exact if and only if it is additive and preserves all kernels, and it's right exact if and only if it's additive and preserves all cokernels. Note that an exact functor, because it preserves both kernels and cokernels, preserves all images and coimages. Exact functors are most useful in the study of abelian categories, where they can be applied to exact sequences.

On every pre abelian category there exists an exact structure that is maximal in the sense that it contains every other exact structure. The exact structure consists of precisely those kernel cokernel pairs where is a semi stable kernel and is a semi stable cokernel. Here, is a semi stable kernel if it is a kernel and for each morphism in the pushout diagram

the morphism is again a kernel. is a semi stable cokernel if it is a cokernel and for every morphism in the pullback diagram

the morphism is again a cokernel. A pre abelian category is quasi abelian if and only if all kernel cokernel pairs form an exact structure. An example for which this is not the case is the category of (Hausdorff) bornological spaces. The result is also valid for additive categories that are not pre abelian but Karoubian.

An abelian category is a pre abelian category such that every monomorphism and epimorphism is normal. A quasi abelian category is a pre abelian category in which kernels are stable under pushouts and cokernels are stable under pullbacks. A semi abelian category is a pre abelian category in which for each morphism the induced morphism is always a monomorphism and an epimorphism. The pre abelian categories most commonly studied are in fact abelian categories; for example, Ab is an abelian category. Pre abelian categories that are not abelian appear for instance in functional analysis.