# Tag Info

### Name for abelian category in which every short exact sequence splits

The abelian categories in which all short exact sequences split I would call "split abelian categories", reserving the term "semisimple abelian category" for a more restrictive condition. Roughly, ...
• 15.3k
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### What is a triangle?

To answer your first precise questions: Yes, every distinguished triangle in $D(A)$ comes from a short exact sequence. For every distinguished triangle $X \to Y \to \mathrm{Cone}(f) \stackrel{+1}\to$...
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### What's there to do in category theory?

There is a majestic paper by Mac Lane MacLane, Saunders. "Possible programs for categorists." Category Theory, Homology Theory and their Applications I. Springer, Berlin, Heidelberg, 1969. 123-131. ...

### Abelian category equivalent to a non-abelian category

Here is a manifestly invariant definition of an abelian category $\mathcal{C}$. It is a category with finite limits and colimits such that: (It is pointed) the map from the initial to the final ...
• 13.3k
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### Abelian category equivalent to a non-abelian category

What you were told is wrong, for we have the following: Proposition. If two categories are equivalent and one of them is abelian, then so is the other. A proof (and some related results) can be ...
• 6,690
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### Is the category of left exact functors abelian?

The following pair of examples follows the idea of Jeremy Rickard suggested in a comment on Math Stack Exchange under the link. Inverting the arrows, it suffices to construct an example of abelian ...
• 15.3k
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### Categorical presentation of direct sums of vector spaces, versus tensor products

One way to think about what the monoidal structure on vector spaces is doing is that it is telling us that vector spaces do not really form a category, or not "just" a category: they form a ...
• 116k
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### Functorial kernel in derived category

Let $\mathcal{C}$ be a stable $\infty$-category. Then $\mathcal{C}$ has a homotopy category $h \mathcal{C}$, which is triangulated. The collection of morphisms $f: X \rightarrow Y$ of $\mathcal{C}$ ...
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### What's there to do in category theory?

Through extended TQFT and the cobordism hypothesis, many questions in topological quantum field theory have been turned into explicit questions about higher category theory. A TQFT is formalized as a ...
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### What was the error in the proof of Roos' theorem?

Taking a brief look at the Roos note we see that detailed proofs of the statements aren't provided. There is no argument in the note one could say is wrong. The paper with the corrected statement very ...
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### Example of an abelian category with enough projectives and injectives which are not dual

The category of countable abelian groups is an essentially small abelian category, and has enough projectives and injectives (the countable free abelian groups and the countable divisible groups ...
• 34.4k
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### Are there (enough) injectives in condensed abelian groups?

Indeed, there are no nonzero injective condensed abelian groups. Let $I$ be an injective condensed abelian group. We can find some surjection $$\bigoplus_{j\in J} \mathbb Z[S_j]\to I$$ for some index ...
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### What's there to do in category theory?

I apologize in advance for this very long answer. I am pretty sure that many people could write a better version of it. Unfortunately, they are not doing it. So, here we are. The very beginning of ...
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• 9,569
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### Abelian categories satisfying AB5*

The snarky response would be "the opposite category of any of the categories you could name on the spot". The less-snarky response is to observe that some of these are quite natural. For ...
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### Can $\mathcal O_X$ be recognized abstract-nonsensically?

I think you are a little confused about what your characterization of $R$ does, and this causes problems as you generalize to sheaves. There is no characterization of $R$ as an element of the ...
• 139k