Tagged Questions
2
votes
0answers
38 views
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 ...
5
votes
2answers
126 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 ...
1
vote
1answer
53 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 ...
5
votes
1answer
50 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.
...
16
votes
0answers
309 views
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 ...
0
votes
0answers
88 views
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 ...
3
votes
1answer
52 views
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 ...
1
vote
1answer
39 views
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, ...
2
votes
2answers
80 views
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 ...
2
votes
1answer
58 views
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) ...
4
votes
1answer
37 views
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 ...
1
vote
0answers
70 views
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, ...
0
votes
1answer
108 views
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.
...
2
votes
1answer
40 views
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 ...
2
votes
1answer
60 views
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 ...
1
vote
0answers
33 views
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 $F$ ...
5
votes
2answers
170 views
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 : ...
1
vote
1answer
56 views
Exactness of Colimits
Let $\mathcal A$ be a cocomplete abelian category, let $X$ be an object of $\mathcal A$ and let $I$ be a set. Let $\{ X_i \xrightarrow{f_i} X\}_{i \in I}$ be a set of subobjects. This means we get an ...
1
vote
1answer
41 views
Smallest subobject in an abelian category containing a set of objects
Let $\mathcal A$ be an abelian category. Let $A$ be an object in $\mathcal A$ and let $(A_i)_{i\in I}$ be a set of subobjects of $A$. Then there is a subobject $\sum_{i \in I} A_i$ of $A$ which has ...
2
votes
1answer
67 views
Set of generators in an abelian category - two definitions
Let $\mathcal C$ be a category. We say that $\mathcal C$ has a set of generators $\{ G_i\}_{i \in I}$ if whenever we take two distinct morphisms $f, g \colon A \to B$ in $\mathcal C$ there exists some ...
3
votes
1answer
68 views
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? ...
3
votes
2answers
156 views
An exact sequence of homology in abelian categories
$\renewcommand{\im}{\mathop{\rm im}}\DeclareMathOperator{\coker}{coker}$Let $A\xrightarrow{f}B\xrightarrow{g}C$ be a complex in an abelian category, I.e. $gf=0$.
Let $H:=\coker(\im(f)\to \ker(g)).$
...
0
votes
0answers
59 views
Image in abelian categories
$\def\im{\operatorname{im}}\def\coker{\operatorname{coker}}$For a morphism $ f: A\to B$ in an abelian category, we let $\im f:=\ker(\coker f)$.
Then the morphism $A\to \im f$ is an epimorphism and ...
7
votes
1answer
260 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 ...
5
votes
1answer
202 views
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:
...
12
votes
1answer
297 views
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 ...
0
votes
0answers
47 views
What is the cohomological dimension of a functor?
Let $F:\mathcal{C}\rightarrow \mathcal{D}$ be a functor between abelian categories. Could anyone explain what the cohomological dimension the functor $F$ is?
We may need some additional condition to ...
3
votes
1answer
138 views
Definition of a Functor of Abelian Categories
What is the precise definition of a functor of abelian categeries. I've looked on the internet but can't find one. From the Wikipedia definition of an abelian category, I'm guessing that, for two ...
3
votes
0answers
72 views
A finite diagram in an abelian category which may not be locally small
This question is motivated by this.
I will use the notations of my answer to this.
We say a category $\mathcal C$ is locally small if Hom($X, Y$) is small for any $X, Y \in$ Ob($C$).
Let $\mathcal ...
0
votes
0answers
59 views
invert Grothendieck spectral sequence
I have 4 topoi $A,B,C,D$ (these are associated to abelian sheaves on some sites) and functors (which corresponds to some push-forward of sheaves)
$F: A\rightarrow B$
$G: B \rightarrow C $
$H: A ...
1
vote
1answer
92 views
Rows have the same kernel in a pullback square.
Suppose for simplicity that we are in an abelian category. Suppose the following square is a pullback:
...
2
votes
2answers
123 views
$\operatorname{Func}(J,Ab)$ has enough injectives.
I am trying to show that the functor category $\operatorname{Func}(J,Ab)$ has enough injectives (meaning that for each $F\in \operatorname{Func}(J,Ab)$ there is an injective object $I\in ...
3
votes
2answers
157 views
Morita equivalence of acyclic categories
(Crossposted to MathOverflow.)
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 ...
1
vote
1answer
77 views
Equivalent module categories
Let $A$ and $B$ be rings and let $A\text{-mod}$ and $B\text{-mod}$ be their abelian module categories. Let $F:A\text{-mod}\to M\text{-mod}$ and $F':B\text{-mod}\to A\text{-mod}$ be functors which ...
1
vote
0answers
79 views
Image of projective objects.
Let $A$ and $B$ be two abelian categories. Assume that there exist a functor $F$ between them which is exact, full and essencially surjective.
If $x$ is a projective object in $A$, then $F(x)$ is a ...
4
votes
1answer
180 views
Is quasi-isomorphism an equivalence relation?
Let $E^\bullet$ and $F^\bullet$ be complexes on an abelian category; what does it mean to say that $E^\bullet$ and $F^\bullet$ are quasi-isomorphic?
Does it only mean that there is a map of complexes ...
17
votes
4answers
758 views
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' ...
2
votes
1answer
78 views
The smallest subobject $\sum{A_i}$ containing a family of subobjects {$A_i$}
In an Abelian category $\mathcal{A}$, let {$A_i$} be a family of subobjects of an object $A$. How to show that if $\mathcal{A}$ is cocomplete(i.e. the coproduct always exists in $\mathcal{A}$), then ...
14
votes
1answer
1k views
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, ...
5
votes
2answers
346 views
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 ...
2
votes
0answers
169 views
kernel of cokernel is cokernel of kernel [duplicate]
Possible Duplicate:
Equivalent conditions for a preabelian category to be abelian
Let $\mathcal{C}$ be an abelian category, and consider an arrow $f:A\rightarrow B$. In a number of sources ...
8
votes
1answer
403 views
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 ...
2
votes
2answers
135 views
Uniqueness of Kernels in Abelian Categories
Refering to Serge Lang's "Algebra" pp. 133-134 on Abelian Categories, the following is unclear to me. Let $Q$ be an additive category and $F\stackrel{f}\rightarrow E$ a morphism. Let $A ...
0
votes
1answer
144 views
Additive functors and zero homomorphisms
The question:
Let $F:\mbox{Ab} \to \mbox{Ab}$ be an additive functor; if $f$ is a zero homomorphism, then so is $F(f)$; if $A$ is the zero group, then so is $F(A)$.
This boils down to showing that, ...
3
votes
0answers
147 views
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 ...
6
votes
1answer
213 views
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 ...
8
votes
3answers
614 views
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 ...
10
votes
1answer
273 views
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 ...
1
vote
1answer
157 views
Complement of a submodule
Let $N \subseteq M$ be a subobject in an abelian category (say, modules). A complement of $N$ in $M$ is then defined to be a subobject $Q \subseteq M$ which is maximal with respect to the condition $Q ...
