Abelian category

An abelian category is an additive category with all kernels and cokernels in which every monomorphism and epimorphism is normal.

A complete abelian category is cofinitary when its opposite category is a finitary abelian category. Thus a cone of epimorphisms over an inverse directed diagram induces an epimorphism from its vertex to the limit.

A cocomplete abelian category is finitary in the directed-union sense if every cocone under a diagram indexed by a nonempty directed poset, with all its legs monomorphisms, induces a monomorphism from the colimit to the cocone vertex. The category of abelian groups has this property: two colimit representatives mapped to the same vertex become equal at a common upper index because that cocone leg is injective. This use of finitary concerns directed unions, rather than the preservation-of-filtered-colimits meaning used for finitary monads.

In a complete finitary abelian category, the comparison $j$ with diagonal matrix entries the identity and off-diagonal entries zero is monic. Express its source as the directed colimit of finite biproducts. Their maps into the product have finite-coordinate retractions and are therefore split monomorphisms; the finitary axiom applies to this cocone. If the category is also cofinitary abelian category, duality makes $j$ epic and hence invertible. For countably many copies of any object, the shift-and-fold argument in countable biproducts in an additive category force triviality then forces the identity to be zero, so every object is a zero object.

Objects are abelian groups and morphisms are group homomorphisms. Addition on hom-sets is pointwise, kernels are subgroup kernels, and cokernels are quotients by images. Thus this is an abelian category. Products are Cartesian products; coproducts are direct sums of finite-support families. The underlying-set functor creates directed colimits, giving a concrete description of their elements by eventual equality of representatives.

An abelian category has enough projectives when every object admits an epimorphism from a projective object in a category. Repeating this on the kernel in a category constructs a projective resolution of every object. Finite direct sums of projective objects remain projective and give the degreewise split horseshoe construction.

An abelian category with enough projectives is hereditary when every subobject of a projective object is projective. For $0\to K\to P\to A\to 0$ with $P$ projective, dimension shifting gives $L_{2}F(A)\cong L_{1}F(K)$. Hence hereditaryness is equivalent to vanishing of $L_{2}F$ for every additive right exact functor into any abelian category, by projectivity detected by all first left derived functors. It is also equivalent to left exactness of every $L_{1}F$: the derived long exact sequence gives injectivity of $L_{1}F(K)\to L_{1}F(P)$ when $L_{2}F$ vanishes, and applying left exactness to the displayed sequence proves the converse.

In a commutative diagram of two kernel rows, a monic arrow between the final objects makes the square between their kernels and middle objects a pullback in a category. The kernel universal property factors any categorical cone uniquely. Conversely, if the square between middle and final objects is a pullback, the induced arrow between their kernels in a category is an isomorphism, since both kernels have the same universal property after pullback.

In an abelian category, a morphism is an epimorphism if and only if its categorical cokernel object is zero. For the reverse implication, any difference of maps annihilating the morphism factors through its cokernel and is therefore zero. The cokernel invariance under pushout then proves both preservation and reflection of epimorphisms by pushout.

In an abelian category, the pullback in a category of an epimorphism $p:B\to C$ along $k:C^{\prime }\to C$ is the kernel of the epimorphism $[p,-k]:B\oplus C^{\prime }\to C$. If $v:C^{\prime }\to D$ annihilates its projection, $[0,v]$ annihilates that kernel and therefore factors through $[p,-k]$. Restriction to $B$ forces the factor to vanish, giving $v=0$ and proving the projection epic. This uses the definition's normality of epimorphisms.

In an abelian category, suppose a commutative square $gf=kh$ is a pullback in a category and $g$ is an epimorphism. Then $(f,h):A\to B\oplus C$ is the categorical kernel of the epimorphism $[g,-k]$. Since an epimorphism is the cokernel of its kernel, any compatible pair $u:B\to X$, $v:C\to X$ induces a unique map $D\to X$ by factoring $[u,-v]$. Hence the square is also a pushout in a category.

An abelian category has finite limits, and every morphism factors through its image as an epimorphism followed by a monomorphism. Every epimorphism is the cokernel of its kernel, hence a regular epimorphism, and epimorphisms in an abelian category are stable under pullback. Therefore every abelian category is a regular category.

For $f:A\to B$ in an abelian category,The canonical morphism $\mathrm{coim}f\to \mathrm{im}f$ is an isomorphism, giving the canonical epimorphism--isomorphism--monomorphism factorization of $f$.

In an abelian category, the coimage of a morphism is its domain modulo its kernel in a category, expressed as a cokernel in a category. The canonical arrow from the coimage to the image is an isomorphism, giving the image factorization in an abelian category.

A sequence is exact at an object when the image of the incoming morphism equals the kernel of the outgoing morphism. It is exact when this holds everywhere.

A short exact sequence in an abelian category has one nonzero morphism entering the middle object and one leaving it: $0\to A\stackrel{i}{\to }B\stackrel{p}{\to }C\to 0$. Equivalently, $i$ is the kernel in a category of the epimorphism $p$. It splits if $p$ has a section; in that case the additive category splitting argument identifies $B$ with the biproduct $A\oplus C$. Categories of modules give familiar examples.

Two projective presentations of the same object in an abelian category, with kernels $K,K^{\prime }$ and middle objects $P,P^{\prime }$, satisfy $K\oplus P^{\prime }\cong K^{\prime }\oplus P$. Form their pullback in a category. Its two induced short exact sequences in an abelian category split because each quotient is a projective object in a category. The two resulting biproduct descriptions of the pullback give the isomorphism.

Pulling back the final arrow of a short exact sequence in an abelian category preserves its kernel object and yields another short exact sequence in an abelian category. The new kernel inclusion is specified by its old kernel component and zero component into the new quotient. The pullback stability of epimorphisms in an abelian category ensures the new final arrow remains epic.

In a commutative diagram of two exact five-term sequences, if the first and fourth vertical maps are epimorphisms and the second and fifth are monomorphisms, then an isomorphism in the middle follows; in particular, four surrounding isomorphisms force the fifth.

For commuting five-term exact sequences in an abelian category, the first vertical map being epic and the second invertible give an invertible induced map on the images entering the middle objects, because these images are the cokernels of the first horizontal maps. The fourth vertical map being invertible and the fifth monic similarly give an invertible map on the images leaving the middle objects, because these are the kernels of the last horizontal maps. The image factorization in an abelian category packages each middle object between these two images in a short exact sequence. The short five lemma then makes the middle vertical map invertible. The proof is categorical and does not require an embedding into a category of modules.

A morphism between two short exact sequences in an abelian category induces an exact sequence

A complex in an abelian category is a sequence $\cdots \to A_{n+1}\stackrel{d_{n+1}}{\to }{A}_n\stackrel{d_n}{\to }{A}_{n-1}\to \cdots$ with $d_n{d}_{n+1}=0$.

Let $\mathbf{Z}$ have the integers as objects, with $\mathbf{Z}(n,p)=\mathbb{Z}$ for $p=n$ or $p=n-1$ and zero otherwise. Composition is bilinear, the generator of $\mathbf{Z}(n,n)$ is the identity, and the composite of two degree-lowering generators is zero. Additive functors $\mathbf{Z}\to \mathcal{A}$ are exactly chain complexes in the additive category $\mathcal{A}$.

The homology object is $H_n=\ker d_n/\mathrm{im}{d}_{n+1}$, formed categorically as the cokernel of the image-to-kernel monomorphism.

Let $q:C_n\to \mathrm{coker}{d}_{n+1}$. Since $d_n{d}_{n+1}=0$, there is an induced ${\bar{d}}_n:\mathrm{coker}{d}_{n+1}\to C_{n-1}$. In an abelian category,The left side is the usual homology construction and the right side is its construction in the opposite category, so the definition of homology is self-dual.