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.

learn more… | top users | synonyms

0
votes
0answers
34 views

Question about generators and Hom functor

In a given category $\mathcal{C}$ I want to prove the following statement: If $U$ es a generator in the category $\mathcal{C}$ if and only if the left-exact functor $Hom_{\mathcal{C}}(U,-): \mathcal{...
0
votes
0answers
33 views

image of direct sum

Let $\mathcal A$ be an abelian category. Show that $$\mathrm{Im}(\bigoplus_{1\leqslant i\leqslant n}\ \varphi_i)\simeq \bigoplus_{1\leqslant i\leqslant n}\mathrm{Im}\varphi_i$$ There are two ...
4
votes
1answer
68 views

Why is the category of finitely generated modules over a non-noetherian ring not abelian?

I am learning about abelian categories for a talk I have to give next week. One of the first questions I had upon learning this definition is "does there exist an additive category that is not abelian?...
4
votes
1answer
80 views

What's wrong with my understanding of the Freyd-Mitchell Embedding Theorem?

It's truly bizarre that there exists no full modern exposition of this theorem, as noted elsewhere. Anyway, I thought I'd poke through and see if I could get the gist of how it works as somebody who ...
3
votes
1answer
60 views

Completeness of 2-category of Monoidal Categories

Is the 2-category of monoidal categories complete? If not, can any conditions be imposed to satisfy completeness?
3
votes
0answers
38 views

Does Projectiveness always imply flatness?

I know that a project module is always flat, deduced form the properties and abundance of free modules. I'm trying to figure out how essential role the free modules play in this result. So I'd like to ...
3
votes
1answer
38 views

Can tensor abelian categories always be embedded into the category of modules?

Let $(\mathcal A, +,\otimes,I)$ a small symmetric monoidal abelian category. I know that $\mathcal A$ can be embedded into the category of $R$-module for a certain ring $R$. But can we make such ...
1
vote
0answers
30 views

Interaction of a functor with internal hom

An additive functor between abelian categories $F: \mathscr{C} \to \mathscr{D}$ induces a functor on categories of chain complexes $F: \mathscr{C}^\bullet \to \mathscr{D}^\bullet$. The internal hom ...
0
votes
0answers
37 views

Does a binormal category always admit an additive structure?

Let $\mathsf{C}$ be a category. We call $\mathsf{C}$ binormal if it has a null object, has all equilizers and coequilizers, all monomorphisms are kernels and all epimorphisms are cokernels (whereby a ...
1
vote
0answers
29 views

Any straightforward proof of “in an abelian category, a pullback yields a monomorphism at cokernel level”?

Here is the question I encountered: $$\require{AMScd} \begin{CD} s @>{f^\prime}>> a @>{\varphi^\prime}>> \bar a\\ @V{g^\prime}VV @V{g}VV @V{\bar g}VV\\ b @>{f}>> c @>{...
1
vote
1answer
57 views

Splitness of short exact sequences

Consider the following commutative diagram in an abelian category: $$\require{AMScd}\begin{CD} @.0@.0@.0\\ @.@VVV@VVV@VVV\\ 0@>>>A@>>>B@>>>C@>>>0\\ @.@VVV@VVV@VVV\\...
1
vote
1answer
29 views

Abelian subcategory which is not a Serre subcategory

Is there an example of an abelian subcategory $\mathcal{B}$ of an abelian category $\mathcal{A}$ such that : The inclusion functor $\mathcal{B}\to \mathcal{A}$ is exact. $\mathcal{B}$ is not closed ...
0
votes
0answers
28 views

$\mathbf{Ch}(\mathcal{A})$ has enough projectives

This is the Exercise 2.2.2 from Weibel's book. Suppose an Abelian category $\mathcal{A}$ has enough projectives, then so does the category $\mathbf{Ch}(\mathcal{A})$ of chain complexes over $\mathcal{...
10
votes
0answers
105 views

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 ...
3
votes
1answer
46 views

Bounded derived category and hereditary categories

Let $\mathcal{A}$ be an abelian category with enough projectives (injectives). I tried to prove that if every element $M$ of $\mathcal{D}^{b}(\mathcal{A})$ satisfies $$ M \cong \bigoplus_{i} H^{i}(M)[-...
2
votes
1answer
50 views

Ext functors and extensions.

Given two short exact sequences, $$ 0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0 $$ $$ 0 \to B \xrightarrow{\alpha} Y \xrightarrow{\beta} X\to 0$$ I want to show that if the short exact ...
4
votes
1answer
50 views

Is the inverse limit of exact sequences exact?

We are in the category of $R$-modules. Let us consider an inverse system $\{M_n^\bullet\}_{n\geq 1}$ where each $M_n^\bullet$ is an exact sequence. $$ \dots \longrightarrow M_n^{i-1}\longrightarrow ...
1
vote
1answer
45 views

Equivalence of definition of product in a category

I was reading Mitchel book on categories and the following observation without proof is given: A family of morphisms given by $\lbrace p_{i}:A \to A_{i} \rbrace$ is the product of $A_{i}$ if and only ...
5
votes
1answer
75 views

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} \\ @. ...
1
vote
1answer
36 views

$\operatorname{Ext}^{n}$ as the class of Yoneda extensions of degree $n$.

Given an abelian category $\mathcal{A}$, we can define $\operatorname{Ext}^{n}(A,B)$ as the class of extensions of degree $n$ of $A$ by $B$. How can one prove that $\operatorname{Ext}^{n}(A,B)$, is ...
6
votes
1answer
82 views

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, ...
1
vote
1answer
35 views

A linear category is a Vect-module

I would like to know how to show that any linear category is a $\mathrm{Vec}$-module. Here $\mathrm{Vect}$ denotes a category of finite dimensional vector spaces. More general statement can be found ...
0
votes
2answers
43 views

Subcategory of category of Module satisfies SSA?

Let $\mathcal{D}=Mod-A$ be a category of module over an Algebra $A$. and $\mathcal{C}$ be a subcategory of $\mathcal{D}$, I have the following questions:- Is $C$ complete category? Is $\mathcal{C}$ ...
2
votes
1answer
46 views

adjoint functor and subcategories

First at all i have to say I am very new to categories (just basic definition). My Question:- We have two categories ( $\mathcal{C},\mathcal{R}$) object in both of these two categories are module ( ...
1
vote
0answers
35 views

When are sheafification and the embedding of sheaves into presheaves exact functors?

Let $\text{PreSh}(X, \mathsf{A})$ and $\text{Sh}(X, \mathsf{A})$ be the categories of $\mathsf{A}$-valued presheaves and sheaves on $X$, respectively. Here, $\mathsf{A}$ is a category such that we ...
2
votes
0answers
48 views

Are product / coproduct projections / inclusions 'semistrict'?

Let $\mathbf{C}$ be a category with zero object, kernels, and cokernels. Then, a morphism $f\colon A\rightarrow B$ in $\mathbf{C}$ is semistrict iff the canonical map $\operatorname{Coker}(\ker (f))\...
0
votes
0answers
41 views

Establish canonical isomorphism $Z^{Y \times X} \cong (Z^Y)^X$ for objects $X, Y$ and $Z$ from $\mathcal{AB}$ category of Abelian groups

Let $X^Y := \mathsf{Hom}_{\mathcal{AB}}(Y, X)$ be a set of all morphisms from objects $Y$ to $X$ from Abelian groups category $\mathcal{AB}$. Let $X \times Y$ be a product and $X + Y$ be a coproduct ...
1
vote
1answer
46 views

Homotopy of chain complexes (category theoretic proof)

I know the usual proof of the fact that if a morphism between chain complexes $f$ is homotopic to zero then it induces the $0$ map on cohomology. I was wondering if there is an easy proof of this ...
4
votes
1answer
124 views

Sheaf cohomology via resolutions vs. derived categories

So I know that when introducing sheaf cohomology, there are two main approaches via derived categories, and a perhaps more "down to earth" method of resolving by acyclic, fine, soft, sheaves. I'm ...
1
vote
1answer
28 views

Composition of stable-pseudomonomorphisms

Terminology Let $\mathbf{C}$ be a finitely-complete finitely-cocomplete category with zero object (not necessarily additive!). A morphism $f\colon A\rightarrow B$ is a pseudomonomorphism iff $(0\...
0
votes
0answers
38 views

Equalizers in abelian categories

I'm trying to prove that hom-sets in an abelian category have a canonical abelian group structure, working with this definition of an abelian category: A category is abelian if It has a ...
0
votes
1answer
39 views

A question about filtered colimits in a category of representations

For $k$ a field, are filtered colimits exact in the category $\mathbf{Rep}_k(G)$ of (finite-dimensional) $k$-representations of a group $G$? I can neither prove it nor find a counterexample.
4
votes
2answers
185 views

Flat Modules are Filtered Colimits of Free Modules

A result by Wraith and Blass states that every flat module is a filtered colimit of free modules (see nLab, Thm 1). I am wondering if this is simply a corollary of Yoneda's density theorem which ...
0
votes
0answers
58 views

An exact sequence of inverse systems of $R$-modules

Let $$0\longrightarrow \big\{A_n,f_{mn}\big\}_{m \leq n} \overset{\Phi}\longrightarrow \big\{B_n,g_{mn}\big\}_{m \leq n} \overset{\Psi}\longrightarrow \big\{C_n,h_{mn}\big\}_{m \leq n} \...
3
votes
1answer
44 views

An identity map which is not null-homotopic

I have some problems in understanding how the definition of a null-homotopic cochain map actually works. Maybe I lack concrete examples. Let $f^{.}:X^{.}\longrightarrow Y^{.}$ a cochain map of ...
4
votes
0answers
47 views

When modular tensor categories are equivalent?

I would like to know when we say that two modular tensor categories are equivalent. Is it true that two modular tensor categories are equivalent if they are equivalent as monoidal categories? Or do ...
4
votes
2answers
144 views

What is the most general category in which exist short exact sequences?

Let $A,B,C$ be objects, $0$ the final object, and $f:A\to B$ and $g:B\to C$ morphisms in some category. Consider the sequence: $$ 0 \to A \to B \to C \to 0\;. $$ I would like to say something ...
1
vote
1answer
24 views

Converse to: Equivalent conditions for a preabelian category to be abelian

In the following question: Equivalent conditions for a preabelian category to be abelian. How is the converse easily shown? I see why every monomorphism, f, is the kernel of the cokernel of f, but why ...
1
vote
2answers
41 views

Adjoint pair of functors and cogenerator elements

Let $F:\mathcal{A} \rightarrow \mathcal{B}$ and $G:\mathcal{B} \rightarrow \mathcal{A}$ be additive functors between abelian categories, such that $(F,G)$ is an adjoint pair. If $B \in \mathcal{B}$ is ...
1
vote
1answer
54 views

Exactness of a right adjoint functor

Let $F: \mathcal{A} \longrightarrow \mathcal{B}$, $G: \mathcal{B} \longrightarrow \mathcal{A}$ be two additive functors between abelian categories, such that $(F, G)$ is an adjoint pair. I want to ...
3
votes
1answer
72 views

Direct sums of semisimple objects

Let $\mathcal{A}$ be an abelian category. Call an object $M\in\mathcal{A}$ semisimple if every exact sequence $0\longrightarrow M'\longrightarrow M\longrightarrow M''\longrightarrow 0$ splits. Is it ...
2
votes
2answers
54 views

Subobject of a direct sum in terms of components?

Let $\mathbf C$ be an abelian category containing arbitrary direct sums and let $\{X_i\}_{i\in I}$ be a collection of objects of $\mathbf C$. Consider a subobject $Y\subseteq \bigoplus_{i\in I}X_i$ ...
0
votes
1answer
43 views

Intuition behind $Ext^1(A,\,C)$

So I recently asked a question concering $Ext^1(A,\,C)$ regarding the connection between isomorphism and the congruence '$\equiv$' (Where am I making a mistake with $Ext^1(A,C)$?). Suppose, for ...
2
votes
0answers
32 views

Is $\Lambda$ essentially the unique solution to $F(M)\cong\frac{F(M\oplus R)}{F(M)}$?

Let $R$ be a commutative ring and let $F$ be a functor $\mathbf{Mod}_R\rightarrow \mathbf{Mod}_R$. Then for a module $M$ the split mono $M\rightarrow M\oplus R$ gives a split mono $F(M)\rightarrow F(M\...
0
votes
1answer
37 views

Where am I making a mistake with $Ext^1(A,C)$?

I am learning about $Ext^1(A,C)$ and how it forms a group under '$+$', the Baer sum and I am clearly missing the point somewhere. So, let us suppose for simplicity that $Ext^1(A,C)\cong\mathbb{Z}/3$. ...
2
votes
1answer
70 views

Five Lemma with category theory

I know how to prove the 5 Lemma by diagram chasing. I would be interested in seeing a proof which only uses category theory. Does anybody know some reference where this is done?
0
votes
0answers
10 views

Proving that $(Tor_n(\_\ ,N))_n$ is a universal homological $\delta$ functor

Problem: Let $N$ be a left $R$-module, for some ring $R$. Let $T_n$ denote $Tor^R_n(\_\ , N)$. Let $(S_n)$ be another homological delta-functor from $mod$-$R$ to $Ab$, with a natural transformation $\...
0
votes
0answers
27 views

Derived functors on 2 term exact sequence

Suppose we have a not-short exact sequence 0 -> A -> B -> 0 in some abelian category. Now let us apply right exact functor F: F(A) -> F(B) -> 0. So could I consider a left derived functor H to obtain ...
1
vote
1answer
61 views

Isomorphic kernels imply pullback?

In Hilton/Stammbach's A Course in Homological Algebra, they are treating the Ext functor, and they give the following lemma: [][2 He implies (but doesn't say) that the same is not true if we ...
1
vote
1answer
55 views

Coproducts and products are same in any preadditive category

Here is the proof that coproducts and products are same in any preadditive category from the Stack project. I have few questions regarding the above proof. I don't understand what do they mean by ...