# Questions tagged [abelian-categories]

Use this tag for questions about Abelian categories, which are categories that possess most of properties of categories of modules over a ring, and are easy to work with using techniques of homological algebra.

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### extension of regular logic

The internal language of some categories (such as abelian categories) support only a fraction of first order logic. But every category can be fully faithfully embedded into a category of presheaves, ...
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### Category of torsion free abelian groups is additive but not abelian

I’m trying to prove that the category of torsion free abelian groups is additive but not abelian… To prove that it is not abelian I’m trying to prove that that the multiplication map φ x2 in Z is not ...
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### Showing that the category of epimorphisms with same codomain in an abelian category is cofiltrant

This is part (i) of exercise 1.7 in Kashiwara and Schapira's "Sheaves on Manifolds". Let $\mathcal C$ be an abelian category. For any object $Z$ of $\mathcal C$, let $\mathcal P(Z)$ be the ...
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### What is the difference between $\mathbb C_X$-modules and sheaves of $\mathbb C$-vector spaces?

In the literature I have encountered the notions "category $\mathbb C_X-Mod$ of $\mathbb C_X$-modules" and "category $Sh_{\mathbb C}(X)$ of sheaves of $\mathbb C$-vector spaces". ...
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### Plan of proof about derived functors in general abelian category

I have to write a report about the derived functors of the inverse limit $\lim$ functor defined from the category of inverse systems (of modules, or maybe in some cases of cochain complexes). Now, the ...
I am working with the following definition of abelian category. a) It has a $0$ object. b) Every morphism has a kernel and a cokernel. Every monomorphismis a kernel and every epimorphism is a ...