Questions tagged [limits-colimits]
For questions about categorical limits and colimits, including questions about (co)limits of general diagrams, questions about specific special kinds of (co)limits such as (co)products or (co)equalizers, and questions about generalizations such as weighted (co)limits and (co)ends.
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When colimit of subobjects is still a subobject?
What are the conditions on a category (or on a certain object) that will guarantee that the colimit of a family of subobjects of a given object is a subobject of the same object?
Update: To clarify ...
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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{...
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subgroups of $(\mathbf Q, +)$ as direct limits
This is a follow-up to this question.
A finitely generated subgroup of $(\mathbf Q, +)$ is isomorphic to the direct limit of the system
$$\mathbf Z\xrightarrow{1}\mathbf Z\xrightarrow{1}\mathbf Z\...
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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, $[...
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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 ...
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Infinite tensor product of Hilbert spaces.
I was reading Chapter 6.2 of Martingales in Banach Spaces by Gilles Pisier. The result is used in the context: $L_2(G) = \bigotimes\limits_{k\geq0}L_2(\mathbb{T})$, where $G=\prod_{k\geq0}\mathbb{T}$ ...
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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 ...
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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}$...
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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 ...
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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 ...
- 334k
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Inverse image sheaf and espace étalé
Let $f\colon X \rightarrow Y$ be a continuous map of topological spaces.
Let $\mathcal{F}$ be a sheaf of abelian groups on $Y$.
The inverse image sheaf $f^{-1}(\mathcal{F})$ is the sheaf associated to ...
1
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1
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Inverse image functor and restriction of colimits
Let $f:X\longrightarrow Y$ be a morphism of topological spaces. I want to prove, that the inverse-image functor of (set-valued) presheaves
\begin{equation}
f^{-1} : \mathrm{PSh}(Y)\longrightarrow\...
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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}) = ...
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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 ...
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On the topology of $BO_k$
Let $BO_k$ be the classifying space given by:
$$BO_k=\varinjlim_{\mathbb N\ni n}Gr_k(\mathbb R^n)$$
I am trying to determine aspects about the topology of this space, but cannot find any sources that ...
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