# Tagged Questions

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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### The homotopy category of complexes

I have some trouble in proving Exercise A3.51 of Eisenbud's book "Commutative Algebra with a view toward Algebraic Geometry", pag. 688. The solution is sketched at pag. 754 at the end of the book. The ...
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### Abelian subcategory generated by a full subcategory.

If $\mathcal{C}$ is a full subcategory of an abelian category $\mathcal{C}'$ to what extent does the abelian subcategory generated by $\mathcal{C}$ depend on the ambient category $\mathcal{C}'$? ...
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### What is higher kernel explicitly?

Let $\mathcal{A}$ be an abelian category (for simplicity you can think that $\mathcal{A}$ is the category of modules over ring $R$). Let $[1]$ be the category with two objects and one arrow between ...
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### Why are functors exact if they preserve all short exact sequences?

If a functor $F\colon \mathcal C → \mathcal D$ of abelian categories preserves short exact sequences, why is it exact? I know the argument is supposed to be that you can split up long exact sequences ...
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### Reference request: Derived category of category with sufficiently many injectives

I'm studying derived categories and have encountered problem with references I have. Namely, proof of the following theorem: Theorem: Let $\mathcal A$ be Abelian category and $\mathcal I$ full ...
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### Filtered colimits are exact in abelian categories

It is well known that filtered colimits commute with finite limits in $\mathsf{Set}$, and hence in every algebraic category - $R\mathsf{Mod}$ in particular. Unless I'm wrong, from the Mitchell ...
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### Examples of thick subcategory

I'd like to know several examples of thick/Serre subcategory of an abelian category, I have no one in mind now. Help me please!
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### Kernel of biproduct projection is the other biproduct injection

For some reason I'm unable to figure out what should be a trivial step in a proof.. Let $A\oplus B$ be a biproduct with injections $i_1,i_2$ and projections $p_1,p_2$. I aim to prove $i_1=\ker p_2$. ...
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### Why is the additive category of Hilbert spaces not abelian

As an answer to this post Additive category that is not abelian it was said that the additive category of Hilbert spaces is not abelian. Why is that? Also what category of Hilbert spaces is this? ...
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### A quasi-isomorphism between the total complex of a Cartan-Eilenberg resolution and the complex per se.

Problem (Weibel's Introduction to Homological Algebra, Exercise 5.7.1) Suppose $A$ is a (not necessarily bounded below) chain complex over an abelian category $\mathcal A$ where axiom (AB4) holds, ...
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### Does “modular category” make sense without saying “abelian” or “linear”?

I know the term "modular category" only from representations of quantum groups, TQFTs and fusion (finitely semisimple linear) categories. There, a modular category is a ribbon fusion category where a ...
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### Freyd: “is a subobject of” is not transitive

On page 20 of Abelian Categories, Freyd writes Note that the relation "is a subobject of" is not transitive. On page 91 of Awodey's Category Theory (there are several typos in this page; the ...
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### Why is the coproduct in $\mathsf{Grp}$ so different from the coproduct in $\mathsf{Ab}$?

Why is the coproduct in $\mathsf{Grp}$ so different from the coproduct in $\mathsf{Ab}$? What about $\mathsf{Grp}$ makes for a seemingly far-more-complicated coproduct? If your answer revolves around "...
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