# Questions tagged [abelian-categories]

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### Small abelian categories and module categories - preservation of injective and projective objects

A soft question on small abelian categories: https://en.wikipedia.org/wiki/Grothendieck%27s_T%C3%B4hoku_paper Wikipedia: "The article "Sur quelques points d'algèbre homologique" by ...
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### Bounds on homological dimension of functor categories

Let $A$ be a Grothendieck abelian category. I will say that $A$ is of global dimension less or equal to $n$ if $Ext^{k}_{A}(a, b) = 0$ for $k > n$ and all $a, b \in A$. This is equivalent to saying ...
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### How can one characterize categories of exact functors?

Does there exist any intrinsic characterization of additive categories equivalent to $\operatorname{Ex}(A,Ab)$, that is, of exact functors from a small abelian category $A$ into abelian groups? Any ...
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### Filling square to push-out in abelian category

Let $\mathcal{C}$ be an abelian category. In $\mathcal{C}$ we consider the diagram \begin{array}{ccc} A&&\\\ \downarrow&&\\\ C&\rightarrow&D \end{array} with arrows being ...
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### Methods for finding complex for subobjects of homology

Let $\mathcal{C}$ be an abelian category and $$C_\bullet:C_n \rightarrow C_{n-1}\rightarrow \ldots \rightarrow C_1\rightarrow C_0$$ a complex in $\mathcal{C}$. Suppose we have for each $i$ a ...
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### Functorial kernel in derived category

By the work of Verdier, we know that cones in a triangulated category $\mathcal{T}$ are functorial if and only if $\mathcal{T}$ is semisimple abelian. However, in these notes, it is said that In the ...
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### Is the category of Yetter-Drinfeld modules abelian?

Is $YD(H)$ the category of Yetter--Drinfeld modules over a Hopf algebra (defined over a field $k$) necessarily abelian? If not then what is the simplest example of a Hopf algebra $H$ for which $YD(H)$ ...
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### Categorical presentation of direct sums of vector spaces, versus tensor products

My apologies in advance if this question is to vague, but here goes.... In the category of vector spaces, products are given by direct sums. In general category theory, the existence of products is a ...
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### Example of pseudocoherent complex which is not locally quasi-isomorphic to a strict pseudocoherent one

I ask this question here even if I posted it also on math.stackexchange (recieving no answer so far) because I have read some analogous question but for perfect complexes on this site, even though ...
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### Which abelian categories possess an exact faithful functor into abelian groups that respects products?

Let $A$ be an abelian category closed with respect to small coproducts (that is, and AB3 category). Which assumptions are sufficient to ensure the existence of an exact faithful functor from $A^{op}$ ...
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### From Topoi to Grothendieck categories

This question is mostly about a reference request. Let $\mathcal{E}$ be a Grothendieck topos. I am looking for a reference of the following two facts. I am aware that $(2) \Rightarrow (1)$ by Gabriel-...
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### Why do some tricks in homological algebra work over the category of C*-algebras?

The category of $C^*$-algebras is not abelian (a "proof" that it is pre-abelian can be found here, but it does not seem correct; I can't find any authoritative sources). However, it's ...
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### Semisimple Abelian categories with infinite sums

A semisimple category is an abelian category in which every object is a finite direct sum of simple objects. A) Why does one impose the finiteness condition here? B) If one condsiders infinite direct ...
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### When uniquely divisible objects can be embedded into ind-torsion ones?

Let $A$ be an AB3 abelian category. We will say that an object $M$ of $A$ is uniquely divisible if for any integer $n\neq 0$ the endomorphism $nid_M$ is invertible. We will say that $M'$ is ind-...
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### Any exact faithful functor is represented by a unique projective generator

In the book 'Tensor Categories' by Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych and Victor Ostrik on page 10 it says: 'Conversely, it is well known (and easy to show) that any exact faithful functor ...
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### Is the center of an abelian rigid monoidal category, abelian?

Is the Drinfeld-Majid center of an abelian rigid monoidal category, abelian? [stated in 1J of On the center of fusion categories" by Bruguières and Virelizier (link at Virelizier's page)] In ...
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### Embedding abelian categories into abelian sheaves

The Yoneda functor from an abelian category into sheaves of abelian groups is shown to be exact in The Stacks Project, Lemma 19.9.2. I like this proof because it is constructive and it doesn't use ...
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### Projective (or injective) object in a subcategory

Let $\mathcal{A}$ be an abelian category and $\mathcal{B}$ be a full subcategory of $\mathcal{A}$. Suppose that $\mathcal{B}$ is abelian and that the inclusion of $\mathcal{B}$ in $\mathcal{A}$ is ...
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### Freyd-Mitchell for $k$-linear categories

I don't know much about the proof of the Freyd–Mitchell embedding theorem and I could not find an answer to my question looking naïvely online, but at the same time I feel like this is the kind of ...
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### Condition for an additive functor to be an equivalence

Consider an additive functor $F : \mathcal{A} \longrightarrow \mathcal{B}$ between abelian categories and suppose that $F$ is a dense functor, that is, for every object $B$ in $\mathcal{B}$ there is ...
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### Abelian category from the category of Hopf algebras

The kernel of a Hopf algebra map $\phi:H_1 \to H_2$ is in general not a Hopf sub-algebra of $H_1$. Is there some replacement or alteration of the notion of a kernel in the Hopf algebra setting. Same ...
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### Baer sum and endomorphisms

I work in an Abelian category. If I take the Baer sum $M' + M''$ of two extensions $M'$ and $M''$ of $M_2$ by $M_1$, i.e., $$0 \to M_1 \to M' \to M_2 \to 0$$ is exact, and the same for $M''$, then ...
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### Are there (enough) injectives in condensed abelian groups?

The question is very simple : does $Cond(\mathbf{Ab})$, the category of condensed abelian groups (as defined in Scholze's Lectures in Condensed Mathematics), have enough injectives ? Does it, in fact,...
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### Characterization of geometric morphisms without referring explicitly to the left adjoint?

Recall that a functor $f_\ast : \mathcal E \to \mathcal F$ between toposes is called a geometric morphism if it has a left exact left adjoint $f^\ast$. Is there an intrinsic characterization of such ...
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### Concrete examples of Freyd-Mitchell embedding

I originally posted this on math.SE (https://math.stackexchange.com/questions/3438528/concrete-examples-of-freyd-mitchell-embedding) but since it's been a few days I figured I would crosspost it here. ...
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Let $R$ be a ring and $M$ be a $R$-module. Let $rad(M)$ be the radical of $M$, that is, the intersection of all maximal submodules of $M$. Moreover, let $soc(M)$ be the socle of $M$, that is, the sum ...
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### Grothendieck groupoid associated to bicategory

Given a finite abelian category $\mathcal{C}$, we can associate to $\mathcal{C}$ its Grothendieck group $\mathsf{Gr}(\mathcal{C})$, which is the free abelian group generated by isomorphism classes of ...
$\newcommand{\im}{\operatorname{Im}}$Consider the following (subpart of) a double complex, using the same notation as in George Bergman's pre-print or in these lecture notes: \require{AMScd}\begin{...
Looking for example at $R$-modules for some commutative $R$, we have the direct sum and the tensor product acting analogously to addition and multiplication. After studying a little bit about co-...