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### In a closed monoidal abelian category, are the compact projectives a monoidal subcategory?

Question: In a closed monoidal abelian category such that the unit object is compact projective, must the tensor product of compact projective objects be compact projective?
Recall that an object ...

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171 views

### Can we obtain a derived category from an additive category? Like a category of Banach modules?

Let $A$ be a Banach algebra, let $A$-mod be the category of left Banach modules (as defined in Helemskii's "Banach and locally convex algebras"), $A$-mod is an additive category, but not abelian ...

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123 views

### Intuition for hereditary torsion theories

I'm looking for intuition and references for the definition of a hereditary torsion theory and two facts found here. First, the definition and facts:
Definition. A torsion theory $(\mathcal ...

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121 views

### Exactness of pure functors

I can't prove this lemma in "Notes on motivic cohomology, Beilinson, Macpherson, Schechtman":
Lemma. A pure functor is exact.
Definitions: A mixed category $\mathcal{M}$ is a ...

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**1**answer

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### A canonical algebra of type $(2,2,r)$ is derived equivalent to a path algebra of type $\tilde{D}_{r+2}$ (references)

According to several articles I could find, a canonical algebra of type $(2,2,r)$ is derived equivalent to a path algebra of type $\tilde{D}_{r+2}$, where $r \geq 2$.
I don't know how to obtain this ...

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205 views

### Tensor product of pullbacks of abelian categories

Does the tensor product of abelian categories commute with pullbacks? In more detail:
Let $k$ be a field. We consider $k$-linear small abelian categories ...

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316 views

### When is/isn't the monoidal unit compact projective?

I am interested in developing intuition for when the monoidal unit in a closed monoidal abelian category is or isn't compact projective. As such, my question is not looking for a yes/no answer, but ...

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**1**answer

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### Splitting lemma for semigroups or monoids

I know there is a splitting lemma for groups, but is there a similar lemma for semigroups or monoids? And do you have the proof of that?

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### Holomorphic vector bundles over a Riemann surface does not satisfy $\mathbf{AB2}$ but satisfies $\mathbf{AB1}$

in Grothendieck's Tohoku paper (page 127), Grothendieck asserts that the category of holomorphic vector bundles $\mathbf{Bund}(X)$ over a fixed Riemann surface $X$ does not satisfy $\mathbf{AB2}$, but ...

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**1**answer

114 views

### What is the universal property of being the maximal common subobject of two objects in a semisimple category?

Imagine a semisimple abelian category $\mathcal{C}$, for example representations of a finite group.
Take two (nonsimple) objects $X, Y$ that are subobjects of a common object $Z$, and decompose them ...

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**1**answer

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### Projectives and Injectives in Functor Categories

Would it be possible to enlighten me (or even better give a reference) about enough projectives (injectives) in functor categories?
Here is a precise question. Let $C$ be a small category, whose ...

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**1**answer

318 views

### Can one characterize the category of finite-dimensional vector spaces? [duplicate]

Let $K$ be a field. Does the category of finitely generated $K$-modules have a nice characterization, for example as the unique abelian category satisfying a certain simple condition? For example, we ...

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### Quotient categories and essential extension

Let $A$ be a right Noetherian positively graded ring. Let $Gr(A)$ be the category of right graded $A$-modules, and $Tors(A)$ be the full subcategory of $Gr(A)$ of torsion modules. Let $QGr(A)$ be the ...

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254 views

### About an embedding of abelian categories into categories of modules

Let $k$ be a field. Let $C$ be an abelian $k$-linear category with a symmetric tensor product $\otimes$ and internal homomorphisms, such that $\mathrm{End}(1)=k$. Let $M$ be another $k$-linear abelian ...

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221 views

### Extension to a right exact functor

Setup. Let $k$ be a field, $\mathcal{C}$ be a $k$-linear abelian category, $\mathcal{D}$ an arbitrary $k$-linear category. Let $\mathcal{C}'$ be a full subcategory of $\mathcal{C}$ with the following ...

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346 views

### Deligne-Mumford Stacks and exactness of products

Let me start by saying that this is probably a very ingenuous question but I still need to digest most definitions before being able to formulate a more precise question.
In this paper: ...

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### Cisinski-Deglise model structure on chain complexes

Let $\newcommand{\A}{\mathscr{A}}\A$ be a grothendieck abelian category. In their paper "Local and stable homological algebra in grothendieck abelian categories", Cisinski and Déglise define the ...

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### how to make the category of chain complexes into an $(\infty,1)$-category

Related to this question, I would like to know, if there is an explicit presentation
of the $(\infty,1)$-category of a model category of abelian chain complexes, but this time in terms of simplicial ...

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557 views

### Properties of quotient categories.

I asked this on math.stackexchange.com, but didn't get any answer.
Let $\mathcal{A}$ be an abelian category and $\mathcal{C}$ a localizing subcategory in the sense of Gabriel. (A Serre/thick/dense ...

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### Why are the left exact functors from an abelian category to abelian groups cocomplete and have a injective generator?

Let $\mathcal{C}$ be an abelian category, $\mathcal{Ab}$ the category of abelian groups and $Lex(\mathcal{C}, \mathcal{B})$ the category of left exact functors between abelian categories.
What is the ...

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348 views

### Applications and examples of quotient categories of abelian categories

I want to get motivated to learn more about quotient categories of abelian categories by a Serre subcategory or even by a localizing category as they are described in Pierre Gabriel's thesis "Des ...

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321 views

### Why is $Lex(\mathcal{A},\mathcal{Ab})$ abelian? Does $Lex(\mathcal{A},\mathcal{Ab})\rightarrow Func(\mathcal{A},\mathcal{Ab})$ admit a left-adjoint?

What is the best way to show, that $Lex(\mathcal{A},\mathcal{Ab})$ is abelian, where $\mathcal{A}$ is an abelian category and $\mathcal{Ab}$ is the category of abelian groups from scratch?
There is ...

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**1**answer

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### Examples of applications of the Freyd-Mitchell embedding theorem.

The Freyd-Mitchell embedding theorem states the following:
Let $\mathcal{A}$ be a small abelian category. There exists a unital ring $R$ and a full, faithful and exact functor ...

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### Recovering an abelian category out of its derived category

I'm trying to learn more about derived category stuff and my curiosity has made me to ask these questions. Sorry if I'm being sloppy, I'm a new learner.
In Wikipedia it has been stated that since ...

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### Any abelian category as filtered colimit of categories of projective modules

Recently I have heard somewhere that any (edit: small) abelian category can be expressed as the colimit of categories of projective modules over some rings. The remark was that this is "basically just ...

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### Why does tensor product in Ab(V) require colimits in V?

In Tom Leinster's book on operads, he gives Ab(V), the category of abelian groups in a symmetric monoidal category V, as an example of a multicategory that doesn't arise from a monoidal category, ...

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### Cocomplete but not complete abelian category

This is a duplicate of the following question to which I did not receive any answer: http://math.stackexchange.com/questions/238247/complete-but-not-cocomplete-category
Let $\mathfrak C$ be an ...

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### Additive functors preserving quasi-isomorphism

Let $F: \mathcal{A} \rightarrow \mathcal{B}$ be an additive functor between abelian categories (with enough injectives and projectives) and $K^\cdot, L^\cdot$ objects of $\textrm{Ch}(\mathcal{A})$. ...

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### A finite diagram in an abelian category which may not be locally small

This question was posted in StackExchange, but there has been no answer so far.
This question is motivated by this.
I will use the notations of my answer to this.
We say a category $\mathcal C$ is ...

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### Does mapping cylinder category have enough injectives?

Let $A, B$ be two abelian categories, and $\tau : A \to B$ a left exact functor.
We define a category $C$ as follows:
objects: triples $(M, N, \varphi)$ where $M\in A, N\in B$ and $\varphi: N\to ...

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### Cov. right-exact additive functors that don't commute with direct sums?

Background
Recently, I have been writing up some notes on derived functors and I came across the Eilenberg-Watts theorems [1], which essentially explain why it is hard to find derived functors ...

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### Subcategories of abelian categories generated by finitely many objects

Hello!
I am trying to understand the structure of the smallest abelian subcategory of an abelian category that contains one object $X$ and all endomorphisms of that object (or rather containing a ...

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### Higher “Cartan-Eilenberg” Resolutions

I am a number theory graduate student learning a bit of homological algebra, and I am curious about higher complexes in abelian categories. I apologize if my post is slightly vague as I am not an ...

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### Is the bounded derived category of coherent sheaves of a variety a small category?

The question is in the title.
I am trying to apply the Mitchell (Freyd-Mitchell?) embedding theorem, which states that for every small abelian category $A$, there exists a ring $R$ such that A ...

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350 views

### Morita equivalence of acyclic categories

(Crossposted from math.SE.)
Call a category acyclic if only the identity morphisms are invertible and the endomorphism monoid of every object is trivial. Let $C, D$ be two finite acyclic categories. ...

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### Terminology - subcategories of Abelian categories

Hello,
I have terminological question. Consider the following properties of a full subcategory $B \subset A$, where $A$ is an abelian category, and we assume $B$ to be closed under finite direct ...

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### Module category equivalent to graded module category?

Main Question
Let $R$ be a graded ring, graded by the nonnegative integers. Denote by $\mathrm{gr}R-\mathrm{Mod}$ the category of $\mathbb{Z}$-graded left $R$-modules with morphisms that preserve ...

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774 views

### The composition of derived functors - commutation fails hazardly?

Hello,
When we have left exact functors $F: A \to B , G: B \to C$ (between abelian categories), we would like sometimes to state that $D(GF)=D(G)D(F)$ (functors between bounded below derived ...

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### Indecomposable projectives correspond to irreducibles - reference

Hello,
We have the following assertion:
In an abelian category that has enough projectives and in which every object has finite length, indecomposable projectives correspond bijectively to ...

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530 views

### Incarnations of a theorem of Eilenberg

Let $R$ be any ring, let $\text{Mod}_R$ be the category of right $R$-modules and let $\text{Ab}$ be the category of abelian groups. There is a classical theorem of Eilenberg (I think) which says that ...

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### Categories with canonical factorizations into products satisfying two particular properties

An old splitting theorem for (Hausdorff) locally compact abelian (LCA) groups says that any LCA group $L$ is isomorphic to a direct product of $\mathbb{R}^n$ and $L_1$, where $L_1$ contains a ...

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### DG vs. abelian quotients

The following, if true, should probably be "standard," but I don't know where to look. I'd rather be slightly imprecise about hypotheses in the hope that there's a good general answer. Feel free to ...

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### Is the colimit of a filtered diagram of module categories in AbCat induced by a cofiltered diagram of rings equivalent to the category of modules over the limit?

Given a cofiltered diagram of commutative rings $F:D\to \mathrm{CRing}$, we obtain a filtered diagram $\mathrm{Mod}(F):D^{op}\to \mathrm{ExAbCat}$ (where $\mathrm{ExAbCat}$ is the category of Abelian ...

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### Tame abelian tensor categories

In the article "Tannaka duality for geometric stacks" (arxiv, see nlab for a summary) Jacob Lurie introduced the notion of a tame abelian tensor category. An abelian tensor category is called tame if ...

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### Is there an additive functor between abelian categories which isn't exact in the middle?

Suppose $F: C\to D$ is an additive functor between abelian categories and that
$$0\to X\xrightarrow f Y\xrightarrow g Z\to 0$$
is and exact sequence in $C$. Does it follow that ...

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### The category of abelian group objects

Let $C$ be a category, say with finite products. What can be said about the category $Ab(C)$ of abelian group objects of $C$? Is it always an abelian category? If not, what assumptions on $C$ have to ...

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### What are examples of cogenerators in R-mod?

Fill in the blank, please :)
Let $\mathcal C$ be a complete and cocomplete abelian category. A generator in $\mathcal C$ is an object $X \in \mathcal C$ such that every object $Y \in \mathcal C$ ...

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### Mitchell's embedding theorem

Mitchell's embedding theorem http://en.wikipedia.org/wiki/Mitchell%27s_embedding_theorem tells us that every small abelian category ${\cal A}$ has a full, faithful and exact embedding $V : {\cal A} ...

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### F(0) = 0? F: additive functor

If I define an additive functor to be a functor on abelian categories such that the action of F on Hom(A,B) is a group homomorphism, do I necessarily have that F(zero object) = zero object?

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### Tot and colimits

This must be a well-known exercise with spectral sequences, but I don't know a reference for it. I'm trying to figure out when does $Tot$ commute with colimits.
More precisely, let $X$ be a double ...