# 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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### Spectral sequence $\bigoplus_{k-j=q}\mathrm{Ext}^p(\mathcal{H}^j,\mathcal{H}^k)\Rightarrow \mathrm{Hom}^{p+q}(P,P)$

Reading the proof in Bondal-Orlov reconstruction theorem (http://arxiv.org/pdf/alg-geom/9712029v1.pdf), I found the spectral sequence in the title ...
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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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### Injective Resolution in Abelian Categories

Let $\mathcal{C}$ be an Abelian category. There is a fact that if $\mathcal{C}$ has enough injective objects, then any object in $\mathcal{C}$ has an injective resolution. By the definition of ...
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### Are split exact sequences exact in the opposite direction?

In an abelian category, let $$0\longrightarrow A \overset{f}{\longrightarrow}B\overset{g}{\longrightarrow}C\longrightarrow0$$ be a split short exact sequence with $\ell f=1_A,gr=1_C$. Is the sequence ...
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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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### If $f= \mathrm{ker}\,g$, then $g = \mathrm{coker}\,f$?

I didn't understand a step in the proof of Proposition 5.92 from Rotman's Introduction to Homological Algebra (2nd Ed.) where he says: "there is a morphism $g: B\to C$ [in a given abelian category ...
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### Which categories of linear representations are semisimple?

Let $k$ be a field of characteristic $0$. For which smooth algebraic groups $G$ over $k$ does the abelian category of linear representations $\mathsf{Rep}_k(G)$ (not assumed to be finite-dimensional) ...
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