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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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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Intuition behind Snake Lemma

I've been struggling with this for some time. I can prove the Snake Lemma, but I don't really “understand” it. By that I mean if no one told me Snake Lemma existed, I would not even ...
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What are exact sequences, metaphysically speaking?

Why is it natural or useful to organize objects (of some appropriate category) into exact sequences? Exact sequences are ubiquitous - and I've encountered them enough to know that they can provide a ...
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Proving the snake lemma without a diagram chase

Suppose we have two short exact sequences in an abelian category $$0 \to A \mathrel{\overset{f}{\to}} B \mathrel{\overset{g}{\to}} C \to 0 $$ $$0 \to A' \mathrel{\overset{f'}{\to}} B' \...
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When is the derived category abelian?

I read in the book Methods of homological algebra of Gelfand and Manin that the derived category of an abelian category $A$ is never abelian. Now to me this seems to be wrong, because if $A=0$ then $D(...
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How to define Homology Functor in an arbitrary Abelian Category?

In the Category of Modules over a Ring, the i-th Homology of a Chain Complex is defined as the Quotient Ker d / Im d where d as usual denotes the differentials, indexes skipped for simplicity. How ...
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Abelian categories and axiom (AB5)

Let $\mathcal{A}$ be an abelian category. We say that $\mathcal{A}$ satisfies (AB5) if $\mathcal{A}$ is cocomplete and filtered colimits are exact. In Weibel's Introduction to homological algebra, ...
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Hom is a left-exact functor

If $0 \to A \to B\to C$ is a left exact sequence of $R$-module, then for any $R$-module $M$, $0 \to Hom_R(M,A)\to Hom_R(M,B)\to Hom_R(M,C)$ is left exact. I proved the above, and highlighted what I'...
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Meaning of “efface” in “effaceable functor” and “injective effacement”

I'm reading Grothendieck's Tōhoku paper, and I was curious about the reasoning behind the terms "effaceable functor" and "injective effacement". I know that in English, to efface something means ...
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Equivalent conditions for a preabelian category to be abelian

Let's fix some terminology first. A category $\mathcal{C}$ is preabelian if: 1) $Hom_{\mathcal{C}}(A,B)$ is an abelian group for every $A,B$ such that composition is biadditive, 2) $\mathcal{C}$ has ...
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Why do universal $\delta$-functors annihilate injectives?

Let $\mathcal{A}$ and $\mathcal{B}$ be abelian categories. Suppose $\mathcal{A}$ has enough injectives, and consider a universal (cohomological) $\delta$-functor $T^\bullet$ from $\mathcal{A}$ to $\...
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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 : X\...
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583 views

On equivalent definitions of Ext

Let $A$ be an abelian category and $X$, $Y$ two objects of $A$. Let's define Ext in this way: Ext$^i_A(X,Y)$=Hom$_{D(A)}(X[0],Y[i])$ Where $X[0]$ is the complex with all zeros except in degree 0 ...
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Limits in the category of exact sequences

Let $\mathbf C$ be an abelian category admitting projective limits. Let's consider the category whose objects are those of the form $$ 0\to A\to B\to C\to 0 $$ and whose morphisms are triples of ...
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An explicit imbedding of $(R\mathbf{-Mod})^{op}$ into $S\mathbf{-Mod}$

Given a ring $R$ consider $(R\mathbf{-Mod})^{op}$, the opposite category of the category of left $R$-modules. Since it is the dual to an abelian category and the axioms of abelian categories are self-...
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The projective model structure on chain complexes

Let $\mathcal{A}$ be an abelian category with enough projective objects and let $\mathcal{M}$ be the category of chain complexes in $\mathcal{A}$ concentrated in non-negative degrees. Quillen [1967, ...
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Motivation for Definition of Derived Category

On the $n$Lab entry about derived categories, I read the derived category of an abelian category $\mathsf A$ is the localization of $\mathsf{Ch}_\bullet (\mathsf A)$ at the quasi-isomorphisms. My ...
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Applications of $Ext^n$ in algebraic geometry

I have been doing a project about $\operatorname{Ext}^n$ functors for my commutative algebra class. I used the approach via extensions of degree n. Basically I have shown the long exact sequence ...
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Abelian category induced by commutative ring

If $R$ is any ring, then ${}_R \mathsf{Mod}$ is an abelian category. We cannot detect commutativity of $R$ from ${}_R \mathsf{Mod}$, since for example $R$ and the matrix ring $M_n(R)$ are always ...
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Has category theory solved major math problems?

All: I am new to category theory. Just wonder if category theory has solved any major math problems for other mathematics fields? or what are the major applications of the category theory ? ...
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Derived functors of torsion functor

Let $A$ be a domain. For every $A$-module $M$ consider its torsion submodule $M^{tor}$ made up of elements of $M$ which are annihilated by a non zero-element of $A$. If $f \colon M \to N$ is a ...
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The construction of the localization of a category

I was reading the construction of the localization of a category in the book "Methods of homological algebra" of Manin and Gelfand. Let me remind you the definition of the localization of a category: ...
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The Freyd-Mitchell Embedding Theorem and projective (injective) objects

Given a small abelian category $\mathcal{A}$, the Freyd-Mitchell Embedding Theorem gives me a fully faithful exact functor $F:\mathcal{A}\rightarrow R$-$\mathsf{Mod}$, for some unital ring $R$, so ...
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Does $\cdots \to G_1\overset f\to G_2 \overset g\to G_3\to \cdots$ exact imply $0\to \ker(g) \to G_2 \to \operatorname{coker}(f)\to 0$ exact?

Given a (part of a) long exact sequence of abelian groups (or modules over some commutative ring) $$ \cdots \to G_1\overset f\to G_2 \overset g\to G_3 \to \cdots $$ we have the short exact sequence $$ ...
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Freyd: “is a subobject of” is not transitive

On page 20 of Abelian Categories, Freyd writes Note that the relation "is a subobject of" is not transitive. On page 91 of Awodey's Category Theory (there are several typos in this page; the ...
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Arbitrary products of quasi-coherent sheaves?

I have a short question: Does the category of quasi-coherent sheaves on a scheme have arbitrary products? I know that it does if the scheme is affine and I know that they will not be isomorphic to ...
7
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Example of relative Ext functor

Greetings, I've been reading Maclane's "Homology" and ran into the following question: Let $(R,S)$ be a resolvent pair of ring, i.e $R$ is an $S$-algebra and we have a functor $\Psi \colon \...
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101 views

Applications of diagram lemmas

I'm currently reading Theo Bühler's survey on exact categories about which he says This article is written for the reader who wants to learn about exact categories and knows why. Very few ...
6
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280 views

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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Is it possible for $R \oplus M$ and $R \oplus N$ to be isomorphic to each other if $M$ and $N$ are not isomorphic?

Suppose $M$ and $N$ are non-isomorphic $R$-modules (where $R$ is a commutative ring with a unit element).Can we conclude that $R \oplus M \not\simeq R \oplus N$ ? If not in this most general setup, ...
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How to construct a tensor product of two preadditive categories in pure categorical fashion?

Let $\mathsf C$ and $\mathsf D$ be two preadditive categories (by an preadditive category I mean a category together with compatible abelian group structure on every hom-set). I thought of ...
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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 $u:B\rightarrow A$...
6
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Converse of the Nine-Lemma (aka $ 3\times 3$ lemma)

I have been asked to either prove or disprove a sort of converse to the well know "Nine Lemma" (Also sometimes called the $3 \times 3$ Lemma I believe) The basic concept of the Nine Lemma is that if ...
6
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Homotopy of double chain complexes

Consider complexes $(A,d_1), (A',d_1)$, $(C,d_2), (C',d_2)$ and morphisms $f_1,f_2: (A,d_1)\to (A',d_1)$ and $g_1,g_2: (C,d_2)\to (C',d_2)$ of degrees $0$. Consider the functor $(-\otimes-)$, then $$...
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Tensor products and morphisms

Let $C$ be semisimple category with simple objects $X_1, \dots, X_r$. Suppose we have a fusion relation $X_i\otimes X_j =\bigoplus_{l=1}^r N_{ij}^l X_l$. Let $m\in \mathbb{N}$ and let $g:mX_j \to ...
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Prove the FHHF theorem using as much abstract non-sense as possible

This is my second attempt to solve exercise 1.6H from Vakil: Assume $F$ is a covariant right-exact functor and show that we get a map $HF^i \rightarrow H^iF$. Attempt to a solution: Apply $F$ to the ...
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Abelian categories, axioms AB5 and AB5* and incompatability

This is a homework exercise, so please don't post full solutions to the question below. Grothendieck (I believe) introduced several axioms an abelian category A voluntarily could satisfy. In ...
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Why can we use flabby sheaves to define cohomology?

In my algebraic geometry class, we defined sheaf cohomology using flabby sheaves, and the functor on the category of sheaves on a space $X$: $$ D: \mathcal F \mapsto D\mathcal F $$ where $$ D\...
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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 ...
5
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showing exact functors preserve exact sequences (abelian categories, additive functors, and kernels)

I'm working through Vakil's algebraic geometry text and I've been stuck on Exercise 1.6.E (page 52 on http://math.stanford.edu/~vakil/216blog/FOAGjun1113public.pdf.) Suppose that $F$ is an exact ...
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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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Is $\coprod \subseteq \prod$ true in any (complete cocomplete) Abelian category?

Consider $A_i, \; i \in I$ a collection of objects in an Abelian category with arbitrary products and coproducts $\mathcal{C}$. Is there always a functorial monomorphism $\coprod_{i}A_i \...
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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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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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Counter-example for abelian category that is not concrete

I am trying to figure out a counter-example for abelian category that is not concrete category. Does the category of representations over a quiver work? Thanks,
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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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Category of vector bundles over a space is additive?

Consider the category of vector bundles over a fixed base space. Then this category is not abelian, since the kernel of a morphism of bundles is in general not a vector bundle. But is it additive? ...
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Showing epimorphism without using the Freyd-Mitchell Embedding Theorem

In an Abelian category $\mathscr{C}$ consider a commutative diagram as follows: $$\require{AMScd}\begin{CD} 0@>>>\ker f@>\theta>>W @>{f}>> Y\\ @. @. @V{\phi}VV @|{id} \\ @. ...
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Long exact sequence into short exact sequences

This question is the categorical version of this question about splitting up long exact sequences of modules into short exact sequence of modules. I want to understand the general mechanism for ...
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Mod-$R$, Mod-$S$ and Mod-$R \otimes S$

Let $R,S,T$ be commutative rings and assume that $R,S$ are $T$-algebras. In an answer to this question, Pierre-Yves Gaillard gives an example of an $R \otimes_T S$-module that cannot be written as ...