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.
925
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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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2
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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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1
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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 ...
- 432k
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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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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 ...
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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 ...
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The universal bundle $\gamma_k\rightarrow BO_k$ is a real vector bundle
For all $n$, let $\gamma_k^n$ bet the tautological bundle over $Gr_k(\mathbb R^n)$, i.e.
$$\gamma_k^n=\{(V,v):V\in Gr_k(\mathbb R^n), v\in V\}$$
This is also naturally identified with the associated ...
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1
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On the topology of $BO_k$
Let $BO_k$ be the classifying space given by:
$$BO_k=\varinjlim_{\mathbb N\ni n}Gr_k(\mathbb R^n)$$
I am trying to determine aspects about the topology of this space, but cannot find any sources that ...
- 3,431
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1
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65
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If a module is a limit of two inverse systems, then the two systems are isomorphic.
The original problem comes from corollary (10.10.6), chapter 10, Volumn I, EGA.
I state it in the language of modules here for convenience.
Claim. If an $R$-module $F$ is a limit of two inverse(or ...
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1
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Reference request for realizing a simplicial set as the homotopy colimit of its simplices
I know that
$$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
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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}$ ...
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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
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1
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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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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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2
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61
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Internal hom takes coends to ends
I know that this is a very general fact about limits and colimits, but I would like to prove it directly for ends and coends. If $\mathcal V$ is a closed braided monoidal category, $V$ an object in $\...
- 1,050
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1
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51
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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
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0
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119
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Given an inverse sequence of functors determined on a subcategory, when is the limit determined on that subcategory?
I will first state the general version of my question, but I do have a specific context in mind in which second I'll dance around.
(1.) Let $\mathsf{C}$ be a full subcategory of a category $\mathsf{D}$...
- 1,632
2
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0
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93
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Spec of an infinite intersection of ideals, Spec of a colimit
This comes from the study of Krull's Intersection Theorem, and deriving a geometric meaning.
Let $I \subset R$ be an ideal of a commutative ring (we shall see the case when $R$ is Noetherian). ...
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473
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Do Wikipedia, nLab and several books give a wrong definition of categorical limits?
It seems unlikely that all these sources are wrong about the same thing, but I can’t find a flaw in my reasoning – I hope that either someone will point out my error or I can go fix Wikipedia and ...
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80
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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 ...
- 3,800
2
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1
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79
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Interpretation of closure in inverse limit
Can one interpret the closure of a set inside an inverse limit as the closure of its individual components? I have not been able to find a source confirming or denying this claim. I have only been ...
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0
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143
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Does profinite completion preserve injectivity?
Let $G$ be an abelian group.
Let $\widehat{G}$ be a profinite completion of $G$.
Profinite completion means a inverse limit of $G$ by a system given by homomorphisms $G/N\to G/M$ where $N$ and $M$ are ...
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1
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Confusion about colimits in the category $\mathbf{Set}$
It is well known that $\mathbf{Set}$ is an $\aleph_0$-accessible category, but I'm very inexperienced and I'm not sure how to prove it in detail. In particular, I need to find a set $\Omega$ of ...
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4
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2
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Difference between different definitions of diagram in a category
I'm currently reading the book "Topoi: The Categorial Analysis of Logic" by Robert Goldblatt, and in chapter 3.11, in order to define limits and co-limits he defines a diagram in a category ...
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57
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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 ...
0
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0
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33
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Sequence of direct summands of modules over a ring
Let $R$ be a ring and suppose that for every $n\in\Bbb Z$ we have a split exact sequence of $R$-modules:
$$\{0\}\to E_{n+1}\xrightarrow{\varepsilon_n}E_n\xrightarrow{\pi_n}Q_n\to\{0\}$$
I claim that ...
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2
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2
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83
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(Co)Products are bifunctors, but are general (co)limits also functors?
In a category with all products or coproducts, the (co)product operation can be understood as a bifunctor. More generally let $\mathcal{C}$ be a category with all limits of shape $D$, where for ...
2
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0
answers
22
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Subtleties in commuting colimits
For context, I am reading Weibel's k-book and I am trying to express the homology of $BS^{-1}S$, the group completion of the classifying space of a symmetric monoidal category, as a colimit. In ...
- 1,495
2
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1
answer
32
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Is a canonical morphism from the wedge sum to the product monic? A section?
Let $C$ be a category with a terminal object $\ast$. For two pointed objects, define their wedge sum $c\vee d$ as the pushout
$$
\require{AMScd}
\begin{CD}
\ast @>>c_0> c \\
@VVd_0V@VV\...
- 7,021
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votes
1
answer
49
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Colimits of full subdiagrams vs topological subspaces
Let $F:J \rightarrow \mathrm{Top}$ be a diagram in the category of topological spaces and $X:=\mathrm{colim} F$ be its colimit. For each $a \in \mathrm{ob}(J)$, denote by $\imath_a:F(a)\rightarrow X$ ...
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1
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91
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$\mathcal{O}(U)$ as a projective limit of Hilbert spaces
It is well-known that the space of holomorphic functions $\mathcal{O}(U)$ (with the standard topology of compact-uniform convergence) on an open set $U \subset \mathbb{C}$ is a projective limit of ...
- 2,636
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1
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69
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Is there a direct limit in the category of rings for hypercomplex numbers [closed]
I recently learned about the concept limits in categories. From R we can construct C the H etc... by iterating the Cayley-Dickson construction.
My question is: Can we construct a (non-associative)ring ...
- 27
2
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2
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105
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Showing that the diagonal functor $\Delta:\mathbb{C} \to \mathbb{C} \times \mathbb{C}$ having a right adjoint implies $\mathbb{C}$ having products.
I started brushing up on my understanding of adjunctions and came across this well-known fact (rephrased in my own words):
Let $\mathbb{C}$ be a category, and let $\Delta:\mathbb{C} \to \mathbb{C} \...
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0
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67
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Inverse limit of a quotient space (simple question)
Setup:
I have a tower of abelian groups $\hspace{1em} \cdots \to A_3 \stackrel{f_2}{\to} A_2 \stackrel{f_1}{\to} A_1 \stackrel{f_0}{\to} A_0$.
There are similar towers for $B_i, C_i$, and $D_i$.
There ...
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1
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2
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108
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About definition of direct limit
In the definition of direct limit of abelian groups (or modules or ...), one takes abelian groups $G_i$ ($i\in I)$ with a morphism $f_{ij}:G_i\rightarrow G_j$ with following conditions:
for every $i\...
- 3,047
0
votes
1
answer
35
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Direct and inverse limit after deleting some groups
Suppose we have a preordered set $(I,\le)$ and a sequence of abelian groups $\{G_i\}$ with a homomorphism $\alpha_{i,j}:G_i\rightarrow G_j$ if $i\le j$ in $I$.
Let $G$ be the direct limit of this ...
- 3,047
2
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0
answers
138
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Cech cohomology on infinite open cover commutes with colimit on Noetherian space? (Exercise 5.2.6 in Qing Liu's book)
This is Exercise 5.2.6 in Qing Liu's book Algebraic Geometry and Arithmetic Curve. In part (b), I can show (b) if the open covering has only finitely many open subsets, since the the colimit of the ...
- 1,785
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2
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68
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Direct limit of a system: computation
I am considering a homomorphism from $\mathbb{Z}_4\rightarrow \mathbb{Z}_6$ given by $\bar{1}\mapsto \bar{3}$. My question is:
What is the direct limit of this system in the category of abelian ...
- 3,047
4
votes
1
answer
78
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A locally $\kappa$-presentable category is also locally $\lambda$-presentable for $\lambda>\kappa$? (Typo?)
In Riehl's Category Theory in Context, Sect. 4.6, we find the following:
Definition 4.6.16. Let $\kappa$ be a regular cardinal.¹ A locally small category $\mathsf{C}$ is locally $\kappa$-presentable ...
3
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0
answers
50
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Writing an enriched graph as a colimit
I am working with enriched directed graphs (aka, directed graphs/quivers such that the edges are objects in a category V). I can write every graph as a filtered colimit of finite graphs, and I can ...
- 31
6
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3
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169
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Are finite colimits closed under finite colimits?
Let $C$ be a cocomplete category and $S$ a set of objects of $C$. We may assume, if need be, that the objects of $S$ are compact. Consider $S'$ the class of objects spanned by finite colimits of ...
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-1
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2
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89
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Coequalizer that is not absolute
A coequalizer is called absolute when it is preserved by each functor. Could somebody give me an example of a coequalizer that is not absolute (with proof) ? If possible it would be great if the ...
- 1,659
0
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0
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73
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Is the category of graded modules over a graded-commutative ring an AB5 category?
Is the category of $\mathbb{Z}$-graded modules over a graded-commutative ring an AB5 category? It is abelian, the subobjects of each object form a set, and it admits arbitrary coproducts. But I don't ...
- 3,440
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1
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78
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Prove that the forgetful functor $U: \textbf{Ab} \to \textbf{Set}$ preserves all filtered colimits
I realize that my question is exactly the same as this post here. However, I tried finding the book that was mentioned, Borceux's Handbook of Category Theory I, but my efforts to find the book here ...
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