All Questions
Tagged with limits-colimits equivalence-relations
13
questions
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41
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Equivalence Relations in the colimit of Sets
The Stacks project mentions colimit in the Sets and introduces the following equivalence relationship: $m_{i} \sim m_{i^{'}}$ if $m_{i} \in M_{i}$, $m_{i^{'}} \in M_{i^{'}}$ and $M(\varphi)(m_{i}) = ...
- 301
1
vote
1
answer
51
views
Coequalizer in the category of modules
I am trying to prove that the category of modules is cocomplete. It suffices to show that it has all coequalizers and coproducts. It's relatively easy to show that all coproducts exist, and I am left ...
- 3,800
2
votes
0
answers
109
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Coequalizers in the category of partially ordered sets
Let $X$ and $Y$ be posets and $f, g : X \to Y$ be monotonic functions. I wish to construct the coequalizer $(Z, \; h : Y \to Z)$ of $f$ and $g$.
My attempt
I first define the relation $R \subseteq Y*Y$...
- 1,548
3
votes
1
answer
185
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Difficulty using the (co)limit formulae to construct the $n$-(co)skeleton left and right Kan extensions for truncated simplicial objects
Tl;Dr - I’m struggling to show that the $n$-skeleton is a Kan extension, from the basic limit formula (this should be possible, as it was “left to the reader” in my book). I’m also struggling to even ...
- 42.7k
0
votes
1
answer
47
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Simpler description of pushout in $\mathsf{Set}$ and $\mathsf{Top}$ when one of the two maps is injective
It is known that the pushout in $\mathsf{Top}$ has same underlying set as the pushout in $\mathsf{Set}$ (this is because the forgetful functor $\mathsf{Top}\to\mathsf{Set}$ is a left adjoint). ...
1
vote
1
answer
84
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Mappings out of quotient set by a generated equivalence relation (Colimits in Set)
Let $X$ be a set and $R\subset X\times X$. Let
$$\mathcal{G}:=\{G\subset X\times X\ |\ R\subset G\ \land\ G=G^{-1}\ \land\ G=G\circ G\ \land\ pr_1(G)=X\}.$$ Then $\bigcap_{G\in\mathcal{G}}G\in\...
- 733
0
votes
1
answer
117
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General description of colimits in $\mathbf{Set}$
I'm not sure I can match the statement given here (from https://arxiv.org/abs/1612.09375) with the real results:
Example 5.2.16.
The colimit of a diagram $D \colon \mathbf{I} \to \mathbf{Set}$ is ...
- 12k
4
votes
1
answer
194
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Regular Projective Objects in the Exact Completion of a Finitely Complete Category
Everything I'm talking about is contained in the following:
Carboni & Celia Magno, The free exact category on a left exact one;
Carboni, Some free constructions in realizability and proof theory.
...
- 1,521
3
votes
0
answers
225
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General colimits and filtered colimits in the category of sets
A category $\mathsf{I}$ is filtered if
$\mathsf{Ob(I)} \neq \varnothing$,
for any $i,j \in \mathsf{Ob(I)}$ there is $k \in \mathsf{Ob(I)}$ and morphisms $f\colon i\to k$ and $g\colon j\to k$...
- 4,528
0
votes
1
answer
276
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Pushouts in Top
Consider the following pushout diagram in the category $\text{Top}$:
$$
\require{AMScd}
\begin{CD}
A @>{f}>> C\\
@V{g}VV @VVV \\
B @>{}>> P = B \coprod_A C
\end{CD}
$$
where $f$ is ...
user417289
2
votes
1
answer
260
views
definition of the directed colimit of a functor
Let $(\Lambda,\le )$ be a directed set, which we can understand as a small category: The set of all objects is $\Lambda$ and for $\lambda,\lambda '\in \Lambda$ there exists an unique morphism $i_{\...
- 381
0
votes
1
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94
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These maps from the components into a directed system are injective when the directed system maps are.
Let $I, p_{ij} : A_i \to A_j$ be a directed system of maps such that $p_{jk}\circ p_{ij} = p_{ik}$ whenever $i \leq j\leq k$ and $p_{ii} = \text{id}$. Directed means that for any $i,j \in I$ there ...
2
votes
1
answer
465
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Proving that the direct limit of a directed system is an equivalence relation.
From Dummit & Foote, pg. 268:
Let $I$ be an index set with a partial order. Suppose for every pair of indices $i, j \in I$ with $i \leq j$ there is a map $\rho_{ij} : A_i \to A_j$ such that ...