All Questions
Tagged with limits-colimits category-theory
687
questions
2
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1
answer
55
views
Does the category $\mathbf{Hilb}_m$ contain directed colimits?
I'm reading the paper "Hilbert spaces and $C^*$-Algebras are not finitely concrete" by Lieberman et al. (https://doi.org/10.48550/arXiv.1908.10200). When discussing the category $\mathbf{...
- 41
1
vote
2
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60
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Is it true that $A[2]\cong \varinjlim_i (A_i[2])$ if $A\cong \varinjlim_{i\in I} A_i$?
Let $I$ be a directed set. Let consider the direct limit in the category of abelian groups. Suppose $A\cong \varinjlim_{i\in I} A_i$. Then, is it true that $A[2]\cong \varinjlim_i (A_i[2])$ ? Here, $[...
- 6,351
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1
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47
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When proving that colimits are universal (stable under pullback), why is it sufficient to prove it for coproducts and coequalizers?
I am trying to understand Borceux's proof that colimits are universal in Set. He opens by saying that it is sufficient to prove this for coproducts and coequalizers. I saw this answer, but I am ...
- 121
3
votes
1
answer
57
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Does a functor which reflects limits also reflect cones?
Following Borceux's Categorical Algebra Definition 2.9.6:
Let $F: \mathcal{C}\to\mathcal{B}$ be a functor. $F$ reflects limits when, for every functor $G: \mathcal{D}\to\mathcal{A}$ with $\mathcal{D}$...
- 121
0
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0
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49
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Why restricted product $\prod'$ is $\varinjlim_{S\subset I \text{ runs finite subset of} I} (\prod_{i\in S} X_{i}\times \prod_{v\in I-S}Y_i)$
This is a question related to this page.
https://ncatlab.org/nlab/show/restricted+product .
Let $I$ be a directed set.
Let $X_i(i\in I)$ be a group.
Let $\prod'_{i\in I}(X_i,Y_i)$ be a restricted ...
- 6,351
6
votes
1
answer
132
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What is the subcategory of Top generated by the discrete spaces wrt limits and colimits?
In the category $\text{Top}$ of topological spaces, start with the subcategory $\text{Disc}$ of spaces equipped with the discrete topology (which is equivalent to $\text{Set}$). Then take its closure ...
- 432k
0
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0
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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
6
votes
1
answer
112
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Are all unions in a topos with complete subobject lattices secretly colimits? On a logical analogue of the AB5 axiom
To clarify, here “topos” always means an elementary topos; I do not assume sheaves on a site, where I already knew my question to have a positive answer.
It is known but not so immediate from the ...
- 42.7k
3
votes
1
answer
44
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A sort of Day convolution without enrichment
Some time ago I was trying to define a monoidal structure on a functor category $[\mathcal{C},\mathcal{D}]$ between two monoidal categories $\mathcal{C}$ and $\mathcal{D}$, such that the monoid ...
- 26.3k
2
votes
0
answers
86
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What morphism is sent to a monomorphism by the left Kan extension ${\rm Lan}_{\Delta}\colon{\bf Set^\Delta\to\bf Set^{\hat\Delta}}$ along Yoneda?
For any small category $C$, let us write $\hat{C} = \mathbf{Set}_C$ for the presheaf category $\mathbf{Set}^{C^{\mathrm{op}}}$, and $y=y_C\colon C\to \mathbf{Set}_C$ for the Yoneda embedding. Consider ...
- 152
2
votes
1
answer
33
views
Reference request for realizing a simplicial set as the homotopy colimit of its simplices
I know that
$$X\simeq hocolim_{Simp(X)}\Delta^n,$$
where $Simp(X)$ is the category of simplices of $X$, I know this for example because of proposition 7.5 of the nLab's page for homotopy limits. ...
- 1,495
3
votes
1
answer
69
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Orbit functor is not co-representable
Let ${1}\neq H\le G$ be groups. Denote by $G\textit{-}\mathsf{Set}$ the category of sets with a $G$ action, with $G$-equivariant maps as morphisms. Let $(-)/H: G\textit{-}\mathsf{Set}\to \mathsf{Set}$ ...
- 596
0
votes
0
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27
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colimit with two index category
I want to prove that colimit is commutative with colimit, i.e. $colim_{j}colim_{i}M_{i, j} = colim_{i, j}M_{i, j}$. But I'm a bit confused about how to define $colim_{i}M_{i, j}$? For a single $i$, ...
- 301
2
votes
1
answer
95
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limits and colimits under forgetful functor
I'm studying limits and colimits and more precisely I'm looking at forgetful functors and I'm trying to see if they preserve limits and colimits. In order to do that I first look at terminal and ...
- 682
6
votes
0
answers
81
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Proof of Theorem 3.4.12 in Emily Riehl's "Category Theory in Context"
I have questions about the proof of Theorem 3.4.12 in Emily Riehl's Category Theory in Context.
The theorem states that the colimit of a small diagram $F\colon \mathsf J \to\mathsf C$ can be expressed ...
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