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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Proving Fernbahnhof theorem (FHHF) using only concepts of abelian categories

Statement: let $F$ be a right exact functor. Describe a map $FH \rightarrow HF$. (from Vakil's notes 1.6H) attempt of a solution: Let $K$ be the kernel of $FA^i \rightarrow FA^{i+1}$. Since F is ...
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Projective Module as a Direct Sum of Left Ideals

I wonder if the following statement is true: Every projective $R$-module is a direct sum of projective left ideals of $R$. Most examples of non-free projective modules I have seen are all left ...
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Derived functors definition

I´m searching for a reference that defines $n^{th}$derived functors in an analogous way to the definition given in Mitchell´s "Theory of Categories" for the $0^{th}$ derived functor of $T$ covariant ...
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When is $K_0(i)$ an injection?

Suppose that $\mathcal A$ and $\mathcal B$ are two abelian categories such that $\mathcal A$ is a full subcategory of $\mathcal B$. If $i: \mathcal A\rightarrow\mathcal B$ is the inclusion functor, ...
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Is there a convention for precedence of operators in an additive category?

The laws for an additive category are that there must be a zero object, binary products, that every Hom-set is an abelian group, and that the morphism addition distributes over composition. My ...
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Complete abelian categories with projectieve generators are fully abelian.

This is my first time on stackexchange so if you need more detail from me , please ask. I was reading the book "Abelian Categories : An Introduction to the Theory of Functors" by Peter Freyd , and I ...
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Are the hom sets in the category of varieties abelian groups?

This is supposedly (though I know of the proof bud haven't read it) for the Hom sets of noetherian schemes. Since every variety can be thought of as a noetherian scheme then it seems right... when ...
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Proof of the First Isomorphism Theorem for Abelian Categories

It seems the first isomorphism theorem holds in any abelian category (see here and here). I'm trying to prove this fact for an arbitrary abelian category (with axioms as the ones given in Vakil's ...
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About a (so very) strong version of flatness

Let $A, B$ rings. WHat are $(A, B)$-bimodule $M$ such that $(-)\otimes_A M: Mod_A \to Mod_B$ preserve all (small) limits? A second question is: the characterization of bimodule ${}_AM_B$ such ...
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Is taking cokernels coproduct-preserving?

Let $\mathcal{A}$ be an abelian category, $A\,A',B$ three objects of $\mathcal{A}$ and $s: A\to B$, $t: A' \to B$ morphisms. Is the cokernel of $(s\amalg t): A\coprod A'\to B$ the coproduct of the ...
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Directed Colimits exact in the category of abelian groups

Starting right from the defintions, what would be the shortest way to prove, that the category of abelian groups, $\mathcal{Ab}$, has exact directed limits (This means for every directed set $I$ is ...
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Does “maximal submodule <=> simple quotient module” generalize to abelian categories?

Does the statement "If $A$, $B$ are modules over a commutative ring $R$, then $B$ is a maximal submodule of $A$ if and only if $A/B$ is a simple module" generalize to the setting of abelian ...
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When precisely can we replace quotient objects with subobjects in the definition of simple objects?

In a category with zero, a simple object is one that has only two quotients - itself and zero. Firstly - a point of confusion. The definition above says that quotient object requires a congruence, ...
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Commuting squares in abelian categories

Here $A,B,C$ and $D$ are all objects in an Abelian category. $\require{AMScd} \begin{CD} A @> >> B @> >> C;\\ @VVV @VVV @VVV\\ D @> >>E @> >> F; \end{CD} $ The ...
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Existence of product in the category of pre-sheaves of abelian categories

Let $X$ be $Top(X)$ be the category of open sets of $X$ with inclusion maps as morphism. Let $\mathcal{C}$ be abelian category and $\mathcal{C}_x$ denote the category of contravariant functors from ...
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A generalization of abelian categories including Grp

The category of groups shares various properties with abelian categories. For example, the Five lemma and Nine lemma hold in Grp. Is there a weakened notion of abelian category which also includes Grp ...
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Characterization of injective objects in abelian categories

In this link it is proved that in an abelian category $\mathcal C$ we have that $f:A\rightarrow B$ is mono iff the sequence $0\rightarrow A\rightarrow B$ is exact, where the arrow from $A$ to $B$ is ...
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Is every right adjoint of a surjective functor fully faithful?

Let $F\colon C\rightarrow D$ a functor, which is surjective and $G$ a right adjoint of $F$. Is $G$ always fully faithful? What if $C,D$ are abelian and $F$ is additive? I know that it is always ...
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Mistake in Popescu's book “Abelian Categories with Applications to Rings and Modules”

Corollary 5.5 a) in chapter 1 on page 13 in Popescu's book "Abelian Categories with Applications to Rings and Modules" says: Let $F\colon C\rightarrow C^\prime$ be a functor and $G$ be a full and ...
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Showing the intrinsic addition of an Abelian Category is asociative

Let $\mathcal A$ be an abelian category. Let $(P\oplus Q,\eta_1,\eta_2)$ be the coproduct of $P$ and $Q$, then if $\delta_1=[id_P,0]$ and $\delta_2=[0,id_Q]$, then $(P\oplus Q,\delta_1,\delta_2)$ is ...
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Proof of exactness at the first two non-zero objects in the ker-coker sequence (snake lemma).

I am reading MacLane's chapter on Abelian Categories and I am proving the fact, needed for the snake lemma, that the sequence $0\to \text{Ke}f\to \text{Ke}g\to\text{Ke}h$ is exact at $\text{Ke}f$ and ...
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Does a linear quotient map have sections

Suppose $V$ is a vector space with vector subspace $N$. Then there is a natural projection $$ \pi_N: V \to V/N $$ from the vector space $V$ to the quotient space $VN$ of $V$ modulo $N$. Does ...
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(AB1) failure in $\mathcal{K(A)}$, without triangulated categories

I am trying to prove that there exists an abelian category $\mathcal A$ such that its homotopy category $\mathcal{K(A)}$ is (additive but) not abelian, without passing through triangulated categories ...
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Is every additive monofunctor between abelian categories left exact?

Is there an additive functor between abelian categories, which preserves monomorphisms, but is not left exact?
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Simple question on exact sequences.

I am just learning about abelian categories and I would like to hear some advice on how to think about the concepts of kernel, cokernel, image etc correctly; I am a bit confused. I am trying to prove ...
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Properties of quotient categories.

Let $\mathcal{A}$ be an abelian category and $\mathcal{C}$ a localizing subcategory in the sense of Gabriel. (A Serre subcategory or "thick" subcategory, such that the quotient functor $T\colon ...
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Questions about epimorphisms and projectives in functor categories

Suppose $I$ is a small category, $R$ is a ring and $_R\mathrm{Mod}$ is the category of left $R$-modules. How do I show that the category $[I,~_R\mathrm{Mod}]$ of all functors from $I$ to ...
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Definition of Exact functors [duplicate]

As we know by definition, a functor for example $T\colon R\textrm{-}\mathsf{Mod}\to\mathsf{Ab}$ (from the category of $R$-modules to the category of abelian groups) is "exact" precisely when for any ...
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Abelian categories with direct sums

Does any abelian category admits direct sums? If not, categories admiting direct sums have a special name? I'm asking this since I am writing a proof that requires direct sums but I only know that ...
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135 views

What's stronger: projective or locally free? flat or locally free?

maybe that's an idiot question, however I did not found anything related in the classical references. It's know that a finitely generated projective $A$-module $M$ is locally free ,since each ...
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202 views

Applications of Mitchell's embedding theorem

I don't understand what is the advantage of viewing a particular category as a category of modules over some ring. Can anybody tell me some application of Mitchell's embedding theorem so that I can ...
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216 views

The definition of the quotient category in abelian category.

I want to understand the definition of morphisms in this category. My question is how can I construct directed sets and direct systems, and therefore understanding the colimite. Definition: Given a ...
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Additive category and zero map

Let $A$ be an additive category. Namely $A$ has a zero object, $A$ has finite products and coproducts, and Every Hom-set is an Abelian group such that composition of morphisms is bilinear. ...
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Decomposing Semisimple Perverse Sheaves

Assume $\mathbf{G}$ is an algebraic group over an algebraic closure $\overline{\mathbb{F}_p}$ for some prime $p>0$. Let $\mathscr{M}\mathbf{G}$ be the category of all ...
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Abstract nonsense proof of snake lemma

During my studies, I always wanted to see a "purely category-theoretical" proof of the Snake Lemma, i.e. a proof that constructs all morphisms (including the snake) and proves exactness via universal ...
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Long Exact Sequence on Homology in an Abelian Category

Let $\mathcal A$ be an abelian category and let $0 \xrightarrow{} X \xrightarrow f Y \xrightarrow g Z \xrightarrow{} 0$ be an exact sequence of chain complexes in $\mathcal A$. I am using the ...
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Are bicategories and lax 2-categories the same?

My question is that whether the definition of bicategories is the same as the definition of lax 2-categories. I heard that they are both week versions of 2-categories. Are they the same? If not, how ...
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Isomorphism between (source of) kernels of parallel arrow of a pullback square, by adjunction

Let $\mathcal A$ be an abelian category, $\alpha_1 \colon A_1 \to B$, $\alpha_2 \colon A_2 \to B$ two morphisms and $A_1 \leftarrow A_1\times_B A_2 \to A_2$ their pullback (with morphisms, say, $p_1, ...
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MacLane's Abelian Categories Chapter

I have just started reading MacLane's chapter Abelian Categories in Categories for the Working Mathematician. I am stuck in the second page, where he discusses the equivalence relation on a set of ...
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1answer
208 views

split exact complexes and biproducts

I have an split exact complex in an abelian category, that is, a chain complex which is exact and maps $s_n : C_n \to C_{n+1}$ st. $dsd = d$. I would like to prove that this implies that $C_n \cong ...
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Why $Kom(\mathcal{A})$ may not be triangulated, while $D(\mathcal{A})$ may not be abelian?

Let $\mathcal{A}$ be an abelian category, let $Kom(\mathcal{A})$ be the category of complex with a shift functor $T$, and Let $D(\mathcal{A})$ be the derived category of $\mathcal{A}$. Why: (1) ...
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intersections in abelian category

Let $\mathcal{A}$ be an abelian category. We fix an object $A$ and we consider the category $mono(A)$ whose objects are the monomorphisms $u:B\rightarrow A$ and where a morphism from $u:B\rightarrow ...
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Is $\mathbb{AB}$ an additive (or even abelian) category?

Notation. Let $U_0 \in U_1 \in U_2$ be Grothendieck universes, each containing $\mathbb N$. Let $\mathbf{Cat}_{U_0}$ be the ($U_1$-small) 2-category of all $U_0$-small categories, ...
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Difficult category theory: kernels

In a category with a zero object ($0$ = initial and terminal) and zero morphisms are unique $A \to 0 \to B$ for every $A,B$, define the kernel of a map as the equalizer of $(f,0)$ and cokernel dually. ...
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If an abelian category has a generator then it is well-powered

The title of my question is Proposition 3.35 in Freyd's Abelian categories. The proof says If $G$ is a generator and $A$ is any object, then a subobject $A' \longrightarrow A$ is distinguished by ...
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Equivalent characterizations of faithfully exact functors of abelian categories

Let $F\colon \mathcal{A} \rightarrow \mathcal{B}$ be a functor of abelian categories. We will define some properties of $F$ before we state a question. Let $X \rightarrow Y \rightarrow Z$ be a ...
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What do we call a functor which is exact and reflects exact sequences in abelian categories? [duplicate]

Possible Duplicate: what is a faithfully exact functor? Let $F\colon \mathcal{A} \rightarrow \mathcal{B}$ be a functor of abelian categories. We say $F$ reflects exact sequences if $F$ ...
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Free abelian groups and abelian categories

Why is the category of free abelian groups not an abelian category?
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If a functor between categories of modules preserves injectivity and surjectivity, must it be exact?

Let $A$ and $B$ be commutative rings. Let $F$ be a functor from the category of $A$ modules to the category of $B$ modules. Suppose that $F$ preserves injectivity and surjectivity: whenever $f : ...
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Semisimple objects in abelian categories

Let $\mathcal A$ be any Grothendieck abelian category and $0 \neq M \in \cal A$ an object. It is true that $M$ admits a simple subquotient? It is certainly true for $\mathcal A=R-Mod$ since $M$ ...