All Questions
Tagged with abelian-categories limits-colimits
24 questions
0
votes
1
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119
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Prove that an (additive) functor $F$ between abelian categories (categories of modules) that admits an exact left adjoint must preserve injectives
Prove that an (additive) functor $F$ between abelian categories (categories of modules) that admits an exact left adjoint must preserve injectives. State and prove the dual result.
I have no idea on ...
- 3,830
1
vote
0
answers
81
views
Non trivial colimit for rings in a finite diagram
I’m trying to understand the concept of colimits for commutative rings, but unable to find a colimit(or at least a compliment) for a finite diagram of rings, is there a(non trivial) example for a ...
6
votes
1
answer
670
views
Is taking (co)limits exact in an Abelian category?
Let $\mathcal{A}$ be a complete and cocomplete Abelian category.
Let $J$ a small category, and let $F_1 \xrightarrow{r} F_2 \xrightarrow{s} F_3$ be a sequence of diagrams/functors $F_i : J\rightarrow \...
- 2,753
0
votes
0
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157
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Meaning and examples of Grothendieck condition AB4*
What are some examples of AB4* categories? In particular, for a thesis I am writing I need to know if cochain-complexes form an AB4* category
- 578
2
votes
0
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107
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Objects in Ind-category are filtered colimits of compact subobjects
In his paper 'Categories Tensorielles' in section 2.2 Deligne states that if in a tensor category $\mathcal{A}$ all objects are of finite length, then every object of the Ind-category $\text{Ind}\...
- 1,190
0
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0
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275
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Calculating finite inverse limits of Abelian groups
If we are given an infinite system of abelian groups $(\dots \xrightarrow{\varphi_2} A_2 \xrightarrow{\varphi_1} A_1 \xrightarrow{\varphi_0} A_0)$ then I know its inverse limit can be found by
$$\lim_{...
- 265
2
votes
1
answer
146
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Define a sketch $s_{\mathbf{Grp}}$ such that $\mathbf{Grp}\backsimeq \mathbf{Mod}(s_{\mathbf{Grp}},\mathbf{Set})$
I have the following
(a) Define a sketch $s_{\mathbf{Grp}}$ and a equivalence functor $$E: \mathbf{Grp}\to \mathbf{Mod}(s_{\mathbf{Grp}},\mathbf{Set})$$ (b) Knowing that finite limits commute with ...
- 1,422
1
vote
1
answer
106
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Cocomplete $R$-linear categories are tensored : adjoint functor theorem?
Let $B$ be an abelian category which is actually $Mod_R$-enriched for some ring $R$ (say unital commutative ring).
For $b\in B$, we have a functor $\hom(b,-) : B\to Mod_R$ which preserves limits, so ...
- 44.4k
1
vote
1
answer
290
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On covariant linear functors $T: R$-Mod $\to R$-Mod which preserves direct-limits or inverse-limits
Let $R$ be a commutative ring with unity. Let $R$-Mod denote the category of $R$-modules, and $Ab$ denote the category of Abelian groups. Now, it is known that a covariant additive functor
$T: R$-...
- 1,453
1
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1
answer
153
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Does every covariant, additive, faithfully-exact functor $T:R$-Mod $\to Ab$ preserve either direct sum or direct product?
Let $R$ be a commutative Noetherian ring. Let $Ab$ denote the category of abelian groups.
Let $T:R$-Mod $\to Ab$ be a covariant, additive functor such that for any sequence of $R$-modules, $A \...
- 1,453
3
votes
1
answer
237
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Constructing limits in an additive category given the existence of products and kernels
The title says it all really. Given an additive category $\mathcal{A}$, is having all kernels and arbitrary products sufficient to conclude that it has all limits? Dually, is having all cokernels and ...
- 3,693
4
votes
1
answer
696
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Vanishing of $\varprojlim^1$ on Mittag-Leffler sequences story.
I'm trying to clarify myself on some points about the story of the first derived functor $\varprojlim^1$ of the projective limit functor vanishing on some kind of filtered inverse systems in arbitrary ...
- 342
5
votes
1
answer
2k
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Limits and $Hom(-,Y)$-functor in abelian categories
Let $\cal C$ be an abelian category and let $(\{x_i\} , \{\phi_{ij}:x_i\rightarrow y_j\}_{i\leq j})$ be a direct system with direct limit $(\varinjlim x_i, \phi_i: x_i \rightarrow \varinjlim x_i)$. ...
- 155
1
vote
1
answer
285
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Colimit of submodules
I was going through a proof in the paper "Local unit versus local projectivity", where I came across the fact that for an $R$ module $P$ if $P = \operatorname{colim}\limits_{i\in I} P_i$ where $I$ is ...
- 23
0
votes
1
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169
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Does the kernel of a morphism equal to its source object in an abelian category imply it is a zero map?
Does the kernel of a morphism equals to its source object in abelian category implies this is a zero map?
Namely let $C$ be an abelian category, $X,Y$ are two objects, $f\in Hom_{C}(X,Y)$ is a ...
3
votes
2
answers
285
views
Union in an abelian category
I'm working on the following exercise. I have most of it, but there's one detail at the end that I can't work out.
Let $\{A_i\}$ be a family of subobjects of an object $A$. Show that if $\mathcal{A}...
- 307
7
votes
1
answer
218
views
On relationship of two categorical characterization of finitely generated objects.
I've encountered The following categorical characterization of finitely generated modules:
A $R$-module $M$ is finitely generated iff it satifies one of the following properties:
a): for any family ...
- 6,015
9
votes
1
answer
972
views
Colimit of a direct system of monomorphisms
Let $\mathcal A$ be an abelian category, $\{X_i,f_{ij}\}_{i\leqslant j\in I}$ a direct system of $\mathcal A$ such that for any $i\leqslant j\in I$, $f_{ij}:X_i\to X_j$ is an monomorphism.
Suppose ...
- 6,015
2
votes
1
answer
132
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Are binary unions of regular monomorphisms in regular categories effective?
In the presence of pullbacks, the intersection of subobjects is given by their pullback. Whenever image factorization exist, the union of subobjects is given by the image of their coproduct. Sometimes,...
- 14.2k
5
votes
1
answer
141
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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?
- 685
8
votes
1
answer
2k
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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 ...
0
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1
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158
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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.
1
vote
1
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131
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Wiki on exact sequences in regular categories
In a regular category, an exact sequence is a diagram which is both a coequalizer and a kernel pair: $$R\overset r{\underset s\rightrightarrows} X\to Y$$
Wiki says that in the abelian case, the above ...
user153312
2
votes
1
answer
602
views
Pullback preserves cokernel
Is that true that in an abelian category $\mathcal{C}$, if I have the pullback diagram:
$$
\require{AMScd}
\begin{CD}
P @>{p_1}>> C\\
@V{p_2}VV @V{g}VV \\
A @>{f}>> B
\end{CD}
$$
...
- 305