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Questions tagged [limits-colimits]

For questions about categorical limits and colimits, including both questions about limits and colimits of general diagrams and questions about specific special kinds of (co)limits such as (co)products or (co)equalizers.

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How to Find limits and co-limits of diagrams over Vector?

I am having trouble understanding how to find limits and colimits of specific diagrams over the category of finite dimension vector spaces. I understand the definitions of cones, terminal objects, ...
3
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0answers
55 views

I am stuck with an example of direct limits and need help to prove $\mathbb{Q/Z} = \varinjlim \mathbb{Z}/(i)$.

Let $N$ denote the set of positive integers that are ordered as $m \le n \iff m | n$. Let $X_m = \mathbb{Z}/(m)$ denote the set of integers $x \mod m$. I want to show that the direct limit of $X_i$, ...
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1answer
46 views

How does a map between inverse systems induce the inverse limit of its components?

I encountered the following definition in the book Profinite Groups by Ribes and Zallesski: Let $\{X_i, \varphi_{ij} \}$ and $\{ X_i', \varphi_{ij}' \}$ be inverse systems of topological spaces over ...
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1answer
48 views

Inverse limit of compact Hausdorff spaces is nonempty and compact

Suppose that $\left\{ X_n \right\}_{n=1}^{\infty}$ is a sequence of compact Hausdorff spaces and for each $n, f_n : X_{n+1} \rightarrow X_n$ is a continuous function (not necessarily onto). Show ...
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1answer
98 views

Question about $Q = \varinjlim \frac{1}{n} \mathbb{Z}$

I am a little confused by one of Vakil's exercises, which reads Interpret the statement $Q = \varinjlim \frac{1}{n} \mathbb{Z}$ Does it mean $Q = \bigcup \frac{1}{n} \mathbb{Z}$? But what is the ...
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1answer
63 views

Example of computing a direct limit

Let $(\mathbb{N},\leq$) be a directed set where $m\leq n$ if an only if $m$ divides $n$. We define a directed system of groups where $G_{n}=\mathbb{Z}$ for all $n\in \mathbb{N}$ and $f_{mn}\colon \...
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1answer
45 views

Is the transpose of the projection under the exponentiation adjunction a constant morphism?

Consider a cartesian closed category $\mathbf{C}$ and fix an object $B \in \mathbf{C}$. For any $X$, we have the product $X \times B$ and a projection $\pi_B : X \times B \rightarrow B$. Under the ...
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1answer
69 views

Is a limit of profinite spaces profinite?

Here is the statement of Lemma 5.22.3 in the Stacks Project: Lemma. A cofiltered limit of profinite spaces is profinite. And here is the proof: Proof. Let us use the characterization of profinite ...
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35 views

Cohomology of colimit is limit of cohomology ? (group cohomology)

In Homotopy theoretic methods in group cohomology, Henn's part, section 1.2, the example following definition 1 has the following sentence "the cohomology $H^*(G,\mathbb{F}_p)$ of a group $G$, which ...
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34 views

Preservation of small limits implies preservation of all limits

Let $\Phi:\mathscr B\to\mathscr A $ a functor between categories which preserves small limits (colimits). There are non-trivial conditions on $\Phi $, $\mathscr A $ or $\mathscr B $ which makes true ...
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1answer
25 views

On why categories with coseparating sets and intersections have initial objects [Proof Explanation]

I'm trying to understand the proof of the following result from Category Theory in Context, p. 148: Lemma 4.6.11. Suppose $C$ is locally small, complete, has a small coseparating set $\Phi$, and ...
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2answers
159 views

Category with colimits but no limits

(I suspect this is a very easy question: I haven't spent much time thinking about category theory.) $\DeclareMathOperator{\colim}{colim}\DeclareMathOperator{\Dom}{Dom}\DeclareMathOperator{\im}{im}$ ...
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1answer
25 views

A question about limit cones and isomorphism

A consider a limit cone on a diagram $D: \mathbf{I} \rightarrow \mathbf{C}$: $$ \left( L \xrightarrow{p_i} D(I) \right)_{I \in \mathbf{I}} $$ Now suppose that $L' \in \mathbf{C}$ is some object ...
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2answers
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Limits and colimits preservation under adjoint equivalence of categories

If $F\colon\mathcal{A}\to\mathcal{B}$ is left adjoint to $U\colon\mathcal{B}\to\mathcal{A}$, then $U$ preserves limits and $F$ preserves colimits. Can we say something more if $F$ is left adjoint to $...
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1answer
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Universal property of the homotopy limit/colimit.

I have been trying to find a reference for what I have heard is a universal property which defines homotopy limits and colimits. In the category Top, colimits can be defined using the following ...
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45 views

Showing that the projections of a product are jointly monic

Suppose that we have a category $\mathcal{C}$ which "has binary products" (full definition provided here: https://en.wikipedia.org/wiki/Product_(category_theory)). We want to show that given $A, B, C \...
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1answer
42 views

Projection maps in the product of an object with itself.

Suppose $\mathscr{C}$ is a category and $(A \times A,p,q)$ be the product of $A$ with itself, where $p$ and $q$ are the projection morphisms. My question is: can we always take $p=q$? If so, is this ...
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35 views

Existence of terminal object in cocomplete category

Let $C$ be a cocomplete category and $S$ be a set such that $S \subset \operatorname{Ob} C$ and for any object $a$ in $C$ there is at least one arow from $a$ to some object in $S$. Is there terminal ...
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1answer
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Commutative rings as co-limit of Noetherian rings?

Question 1: Does there exist a small category $\mathcal J$ such that for every commutative ring $A$, there is a functor $F :\mathcal J \to \mathcal CRing$ such that $ F$ takes every object to a ...
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3answers
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$\Lambda = \varprojlim\Lambda_n$ (ring of symmetric functions)

This question is related to this other question. When understanding how it is defined the ring of symmetric functions, I can not see why is so much important to take the inverse limit in the category ...
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1answer
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Understanding the inverse limit and universal property of topological spaces

Let $\{X_i, \varphi_{ij},I\}$ be an inverse system of topological space index by a directed poset $I$. Now I would like to understand the proof for the existence of an inverse limit $(X,\varphi_i)$. ...
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Is every commutative ring a limit of noetherian rings?

Let $\mathsf{Noeth}$ be the category of noetherian rings, viewed as a full subcategory of the category $\mathsf{CRing}$ of commutative rings with one. Let $A$ be in $\mathsf{CRing}$. Question 1. ...
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1answer
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Notation: Definition of little $d$-disk operad $D_d$ for $d=\infty$

Let $D_d$ be the little $d$-disk operad as outlined in Fresse's book Homotopy of Operads and Grothendieck-Teichmuller Groups. We have the sequence of inclusions of operads $$D_1 \to D_2 \to \cdots \...
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1answer
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Does the category of artinian rings admit finite limits?

Let $\mathsf{Artin}$ be the category of artinian rings, viewed as a full subcategory of the category $\mathsf{CRing}$ of rings. (Here "ring" means "commutative ring with one".) Question 1. Does $\...
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1answer
70 views

Category of Abelian Groups: Limits

Let the category of Abelian Groups. I know that product and coproduct of a finite number of objects are the same in this category. Then, it follows that the projective and injective limits of finite ...
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0answers
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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 $g\colon j\to k$...
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1answer
42 views

Finding a colimit in the category of presheaves

I have the following problem as part of my exam preporation and I need an idea how to approach it at least: Let $\mathfrak C$ be a small category and $F: \mathfrak C^{opp} \rightarrow \mathfrak S ...
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1answer
98 views

Example of a finite category without inverse or injective limit

I recently started studying categories theory and i need help to understand the concept of limit. Please, tell me an example of a finite category without inverse or injective limit and why. Thanks.
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1answer
47 views

About a specific step in a proof of the fact that filtered colimits and finite limits commute in $\mathbf{Set}$

I'm currently working on the following theorem from Emily Riehl's Category Theory in Context: Theorem 3.8.9. Filtered colimits commute with finite limits in $\mathbf{Set}$. I understand most of ...
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1answer
44 views

On the canonical map $\text{colim}_I \text{lim}_JH(i,j) \longrightarrow \text{lim}_J\text{colim}_IH(i,j)$

I'm working through a proof of the existence of a canonical mapping $$ \mu: \text{colim}_I \text{lim}_JH(i,j) \longrightarrow \text{lim}_J\text{colim}_IH(i,j) \tag{1} $$ induced by a cone $(\mu_i: \...
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1answer
108 views

The forgetful functor from a category of elements strictly creates limits and connected colimits

This will be an exercise (3.4.iii) from the book "Category Theory in Context" by Emily Riehl. First, let me fix notation. Let $F\colon\mathsf{C}\to\mathsf{Set}$ be a set-valued functor. Its ...
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1answer
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What is the direct limit of $\mathbb Z$ with $f_{ij}(x)=x \cdot j$?

I'm just learning the concept of direct limit. I did problem $8$a fromt Dummit Foote section $7.6$. As an example, someone suggested I look $$\mathbb{Z} \xrightarrow{\times 2} \mathbb{Z} \xrightarrow{...
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1answer
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Can we deduce that $M_0$ is a submodule of the limit of the following diagram?

Let $M_0$ be an R-module, and suppose $M_{n+1}$ is the pushout of the diagram below as shown, for all $n \in \mathbb{N}$: $$\begin{array}{ccc}M_n&\to& M_{n+1}\\\uparrow &&\uparrow\\A&...
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0answers
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What does it mean for an object to be a colimit of another object in a category?

I understand what colimits are, and what it means for an object to be a colimit, however I have come across the expression "$x$ is a colimit of $y$", where $x$ and $y$ are both particular objects in a ...
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2answers
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Example of category in which direct limit does not exist

I am trying to understand concept of direct limit.I am looking for An example of a category and directed sequence whose direct limit does not exist. I am unable to find one please help.
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1answer
45 views

equality of inverse limits in $R=k[x_1,x_2,…]$

Let $k$ be a field, $R=k[x_1,x_2,x_3...]=k[x_i]_{\mathbb{N}_0}$ and $\mathfrak{m}=(x_1,x_2,x_3,...)$. I want to check whether the following equality is true or not: $\varprojlim{\mathfrak{m}/\...
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1answer
27 views

Conservative functor in a category of co-limit preserving functors between $R$ modules and vector spaces

Let $R$ be a commutative algebra over some field $k$, and let $RM$ denote the category of $R$ modules. Let $C =$ Fun$_{co}(RM,$ Vect$)$ be the category of colimit preserving functors, where Vect is ...
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2answers
108 views

Left adjoint to the tensor functor

Let $V$ be a vector space over a field $k$. Define $V \otimes - :$ Vect $\to$ Vect the tensor endofunctor on the category of $k$ vector spaces. Assume that $V \otimes - $ preserves limits. It can be ...
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1answer
80 views

Riehl's “Category Theory in Context” - Exercise 3.4.i

Let $\mathsf{I}$ be a small category, let $\mathsf{C}$ be a locally small category and let $F\colon\mathsf{I\to C}$ be a functor. Emily Riehl in her book "Category Theory in Context" defines a limit ...
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0answers
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Limits in a product category: proof verification

First, I will fix a relevant definition. Given a small category $\mathsf{I}$, a diagram $\mathcal{D}\colon\mathsf{I}\to\mathsf{C}$, a functor $F\colon\mathsf{C}\to\mathsf{D}$ and a cone $\lambda\colon ...
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1answer
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On why equivalent categories $E : I \simeq J$ have equivalent categories of cones $\int \mathbf{Cone}(\_,F) \simeq \int\mathbf{Cone}(\_,FE)$

I'm currently working on the following exercise from Emily Riehl's Category Theory in Context, Exercise 3.1.xii. Suppose $E:I \stackrel{\simeq}{\to} J$ defines an equivalence between small ...
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1answer
65 views

Category theory: ex. 3.3.vi from Riehl's “Category Theory in Context”

Prove that for any small category $\mathsf{A}$, the functor category $\mathsf{C^A}$ again has any limits or colimits that $\mathsf{C}$ does, constructed objectwise. That is, given a diagram $\mathcal{...
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1answer
78 views

Is there a concept of soft limit?

Let $\mathcal{C}$ and $\mathcal{D}$ be categories, $\mathcal{F}\colon\mathcal{C}\to\mathcal{D}$ be a functor. There is a notion of limit of $\mathcal{F}$, namely a pair $(\ell,\varphi)$, where $\ell\...
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2answers
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$\ell^2$ as colimit in $\mathbf{TopVect}_{\mathbb{R}}$

Let $\mathbf{TopVect}_{\mathbb{R}}$ be the category of topological vector spaces with continuous linear maps as morphisms. Is it ineed true that $\ell^2 \cong \varinjlim_{n}\oplus_{i=1}^n\mathbb{R}$?
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2answers
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Inverse Limit of Dense Subsets is Dense

Suppose that $(X_i,\leq)$ is an inverse system in Top, and $U_i$ is a dense subset of each $X_i$. This means that $(U_i,\leq)$ is an inverse system in Top also (by considering the relative topology). ...
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1answer
111 views

Hawaiian earring as inverse and direct limit

Let be $C_n$ a circle of radius $1/n$ and center $(1/n,0)$. Let be $X_n$ the union of the first n $C_k$ and H the Hawaiian earring, i.e. the union of all $C_k$. Is true that H is both the direct limit ...
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0answers
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Trying to understand a proof from math overflow regarding direct limit

Direct limit of $\mathbb {C^*}$ behaves well with quotients. The following solution is from mathoverflow: Direct limits do behave well with respect to quotients. Suppose $A$ is the direct limit of a ...
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0answers
43 views

Which simplicial sets are filtered colimits of standard simplices?

The question is all in the title : every simplicial set is a colimit of the standard simplices $\Delta^n$, but I'm wondering which ones are filtered or directed colimits of these, if there's a nice ...
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0answers
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Analytic functionals carried by $K$

Let $K$ be a compact subset of $\mathbb{C}$. By definition, one has $$\mathcal{O}'(K) = \left( \varinjlim_{U\supset K} \mathcal{O}(U) \right)',$$ where $U$ are open neighborhoods of $K$. My question ...
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1answer
78 views

$\lambda$-accessible categories, unclear proof

In the context of $\lambda$-accessible categories consider the proof of the proposition $1.22$ here. How/where did we use the fact that $\cal D$ in the last but one line has less than $\lambda$ ...