Abelian categories are categories that possess most of properties of categories of modules over a ring, e.g. abelian group structure on morphisms, existence of kernels and cokernels of morphisms, existence of direct products and directs sums, etc.

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Additive functors and zero homomorphisms

The question: Let $F:\mbox{Ab} \to \mbox{Ab}$ be an additive functor; if $f$ is a zero homomorphism, then so is $F(f)$; if $A$ is the zero group, then so is $F(A)$. This boils down to showing that, ...
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Abelian categories, axioms AB5 and AB5* and incompatability

This is a homework exercise, so please don't post full solutions to the question below. Grothendieck (I believe) introduced several axioms an abelian category A voluntarily could satisfy. In ...
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Example of relative Ext functor

Greetings, I've been reading Maclane's "Homology" and ran into the following question: Let $(R,S)$ be a resolvent pair of ring, i.e $R$ is an $S$-algebra and we have a functor $\Psi \colon ...
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How to define Homology Functor in an arbitrary Abelian Category?

In the Category of Modules over a Ring, the i-th Homology of a Chain Complex is defined as the Quotient Ker d / Im d where d as usual denotes the differentials, indexes skipped for simplicity. How ...
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Limits in the category of exact sequences

Let $\mathbf C$ be an abelian category admitting projective limits. Let's consider the category whose objects are those of the form $$ 0\to A\to B\to C\to 0 $$ and whose morphisms are triples of ...
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Complement of a submodule

Let $N \subseteq M$ be a subobject in an abelian category (say, modules). A complement of $N$ in $M$ is then defined to be a subobject $Q \subseteq M$ which is maximal with respect to the condition $Q ...