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41 votes

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, ...
Leonid Positselski's user avatar
38 votes
Accepted

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 $...
Dan Petersen's user avatar
  • 39.4k
37 votes

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. ...
35 votes

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 ...
Dylan Wilson's user avatar
  • 13.3k
26 votes
Accepted

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 ...
Fred Rohrer's user avatar
  • 6,690
24 votes
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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 ...
Leonid Positselski's user avatar
24 votes
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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 ...
Qiaochu Yuan's user avatar
23 votes
Accepted

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 $\mathcal{C}$ ...
Jacob Lurie's user avatar
  • 17.6k
22 votes

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 ...
22 votes
Accepted

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 ...
adversary's user avatar
  • 236
22 votes
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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 ...
Jeremy Rickard's user avatar
20 votes
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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 ...
Peter Scholze's user avatar
19 votes

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 ...
18 votes
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Abelian category with enough injectives but not functorially

Since the dual of an abelian category is also an abelian category, the question is equivalent to the same question for projective resolutions. I will show that the category $\mathbf{Ab}^{\...
R. van Dobben de Bruyn's user avatar
18 votes

Abelian categories that are not monoidal

Let $\mathcal{A}$ be an additive category with a monoidal structure such that the maps $$ \otimes \colon \mathcal{A}(A,B)\times\mathcal{A}(C,D) \to \mathcal{A}(A\otimes C,B\otimes D) $$ are ...
Neil Strickland's user avatar
18 votes

Abelian categories that are not monoidal

In the paper Hovey, Mark, Additive closed symmetric monoidal structures on R-modules, J. Pure Appl. Algebra 215, No. 5, 789-805 (2011). ZBL1223.18005. Hovey shows the following theorem (Theorem 3.3) ...
AT0's user avatar
  • 1,447
17 votes
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When is the category of finitely presented modules abelian?

Wojowu's idea is right: Lemma. Let $R$ be a ring, let $\mathbf{Mod}_R$ be the category of (left) $R$-modules, and let $\mathbf{Mod}_R^{\text{fp}}$ be the subcategory of finitely presented modules. ...
R. van Dobben de Bruyn's user avatar
16 votes
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In an abelian category with no nontrivial Serre subcategories, does every short exact sequence split?

The category of finite abelian $p$-groups (where $p$ is your favourite prime) is an abelian category with no proper nonzero Serre subcategories, but not every short exact sequence splits.
Jeremy Rickard's user avatar
15 votes
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Let $F:\mathscr{A}\to\mathscr{B}$ be an equivalence of Abelian categories. Must $F$ be additive?

I am posting the comment above as an answer. An equivalence of categories preserves identity morphisms, finite product, and finite coproducts. Thus, it also preserves diagonal morphisms and ...
15 votes
Accepted

Can a category be enriched over abelian groups in more than one way?

You can easily find examples among categories with one element: a category with one element is a (multiplicative) monoid, and $Ab$-enrichment over it is a choice of an addition which turns it into a ...
Wojowu's user avatar
  • 27.6k
14 votes

Is every "nice" abelian category with enough projectives an additive presheaf category?

The category $[C^{op}, \text{Ab}]$ of $\text{Ab}$-valued presheaves on any (small, for simplicity) $\text{Ab}$-enriched category is about as nice as it gets - locally finitely presentable, ...
Qiaochu Yuan's user avatar
14 votes

What is a triangle?

You really seem to be looking for intuition for the triangulated structures on derived categories of Abelian categories, so here goes: (Co-)chain complexes are like (Abelianised) pointed homotopy ...
Adrian Clough's user avatar
14 votes

How exotic can an infinite biproduct in an additive category be?

I think your memory is right on Question 1: an Eilenberg swindle implies that if an infinite bipower $\bigoplus_X A$ of an object $A$ exists in an additive category, then $A=0$. The point is that ...
Mike Shulman's user avatar
  • 65.6k
14 votes
Accepted

Is the category of chain complexes a reflexive or coreflexive subcategory of the category of functors?

Yes, it's reflective and coreflective, under mild assumptions on the codomain category $\mathcal A.$ The adjoints are given, by definition, by Kan extension along the quotient from the abelian group-...
Kevin Carlson's user avatar
13 votes
Accepted

Concrete examples of Freyd-Mitchell embedding

For some abelian categories it is also very easy to describe such a ring quite explicitly if the category you start with is similar enough to a module category. Let's say you consider $\mathsf{Ch}(A\...
Johannes Hahn's user avatar
13 votes
Accepted

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 ...
Achim Krause's user avatar
  • 9,139
12 votes

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 ...
Will Sawin's user avatar
  • 139k
12 votes
Accepted

Is a categorical coproduct of epimorphisms (monomorphisms) always an epimorphism (a monomorphism)?

Question 1: Yes. The $I$-coproduct-functor $\bigsqcup_I\colon\prod_{i\in I}\mathbf{C}\to\mathbf{C}$ is left-adjoint (its right adjoint is the diagonal functor $\Delta_{\mathbf{C}}^I\colon \mathbf{C}\...
Oskar's user avatar
  • 614
12 votes

Do the isomorphism classes of indecomposable objects in $R{\text{-mod}}$ form a set?

In Conjecture $1_{\infty}$ of Simson, Daniel, On large indecomposable modules, endo-wild representation type and right pure semisimple rings., Algebra Discrete Math. 2003, No. 2, 93-118 (2003). ...
Jeremy Rickard's user avatar
12 votes

What are abelian categories enriched over themselves?

To make sense of enrichment over a category $V$, you want $V$ to have a monoidal structure. Indeed, you want to be able to compose morphisms so you need a way to go from "something in $\hom(a,b)$ ...
Maxime Ramzi's user avatar
  • 14.2k

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