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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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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
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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
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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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34 views

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
43 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
24 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
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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
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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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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
39 views

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
63 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
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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
106 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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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
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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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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
77 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 $\lambda$ ...
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Limits in a Grothendieck fibration

$\newcommand{\E}{\mathcal{E}} \newcommand{\B}{\mathcal{B}}$ I'm currently studying a paper that talks a lot about Grothendieck fibrations and so I'm trying to work with them a bit to get used to them. ...
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1answer
60 views

Filtered colimits commute with finite limits

I am trying to prove the following fact that given $I$ filtered index, $J$ finite index and diagram $F:I\times J \rightarrow \it{Sets}$, $colim_{i\in I} lim_{j\in J}\;F_{ij}=lim_{j\in J}colim_{i\in I}\...
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31 views

Open set in a product of colimits

Suppose that $X=\operatorname{colim} (A_1 \to A_2 \to \dots)$ and $Y=\operatorname{colim} (B_1 \to B_2 \to \dots)$, where $A_i$ and $B_j$ topological spaces. I want to prove that some set $U$ is open ...
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28 views

How to arrive at unique factorization through the limit given naturality compatibility conditions?

If $\alpha : I \to C$ from a small category to any category $C$. Define a functor $\lim\limits_{\rightarrow} \alpha : X \mapsto \lim\limits_{\leftarrow} \text{Hom}_C(\alpha, X)$ from $C^{op}$ to $\...
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1answer
56 views

Why do we need isomorphism between a diagram and a cone L for it to be a limit?

I read from these slides: a limit of a diagram containing just one object A and no morphism is any object L that is isomorphic to A (the isomorphism is part of the limit); and was unsure why ...
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1answer
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VAR, Algebra and local presentability

Here1, on the page 282, I would like to understand why precisely Examples of $k$-ary operations are all $k-\mathrm{lim}$, BUT $k-\textrm{colim}$ must be filtered. Where have we used that $k$ is ...
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Not-quite-preservation of not-quite-filtered colimits

It's well known that if $D\colon J \to \mathbf{Set}$ is a diagram where $J$ is a filtered category, and if $A$ is a finite set, then the natural map $$ \text{colim}_{j}[A,D(j)] \to [A,\text{colim}_{j}...
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Colimit of discrete diagram (coproduct) is the tensor product

Let $C$ be the category of unital, commutative, $k$ algebras, where $k$ is a field. Given a discrete category $I$, with two elements, let $F: I \to C$ be a functor. Denote $I(i_1) = A_i$ for $i = 1,...
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1answer
37 views

Sheaf on basic open sets.

Starting from a $B$-sheaf $\hat{\mathcal{F}}$, a sheaf defined on the basic open sets of a topological space $X$, I am asked to prove that the construction $$\mathcal{F}(U) = \varprojlim_{V \subset U,...
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Coends and adjunctions

I was reading Fosco Loregian's paper This is the co/end, my only co/friend, and here's something that I don't understand in an exercise. The exercise is to prove that given $F: C\to D, U: D\to C$ ...
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1answer
51 views

Direct limit of functions

Let $(f_i)_{i \in I}$ be a directed system of functions $f_j: \mathbb{R} \mapsto \mathbb{R}$ over the directed set $I$. Is there a good way to give a meaning to "direct limit" of this system ? What ...
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1answer
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Colimits where maps in are determined by maps into a component?

I am curious if there is a class of colimits $\mathsf{colim} D$ where maps $A \to \mathsf{colim} D$ must factor through the colimit cocone $D(X) \to \mathsf{colim}D$ for some $X$? For example, would ...
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Simple example of product not distributing over coequaliser in $\mathbf{Top}$

In the category of topological spaces ($\mathbf{Top}$), products do not always commute with colimits. If they did then $\mathrm{Hom}_\mathbf{Top}(-\times X,S)$ would be representable and hence $\...
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1answer
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Limit of $h_{\mathcal C} \circ F$ for $F: I \to \mathcal C$ exists in $\text{Func}(\mathcal C^\text{op}, \text{Set)}$ and is $\text{Hom}(\_,\lim F)$.

Let $I$ be a small category and $\mathcal C$ a complete category. For $F: I \to \mathcal C$, show that the limit of the functor $h_{\mathcal C} \circ F$ exists in $\operatorname{Func}(\mathcal C^{\...
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1answer
54 views

Show that $\lim$ is a right adjoint to the constant functor

A problen in category theory: Let $I$ be a small category and $\mathcal C$ be a comolete category, and consider the functor $\lim : Func(I, \mathcal C) \to \mathcal C$. Show that $\lim$ is right ...
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1answer
26 views

Let $t: I \to \mathcal C$ be a funtor with $I$ small, and suppose that $t$ has a colimit. Prove that $\lim_{\to} (F \circ t)$ exists

A problem in category theory: Suppose $F : \mathcal C \to \mathcal D$ has a right adjoint. Fix a functor $t: I \to \mathcal C$ with $I$ small, and suppose that $t$ has a colimit. Prove that $\lim_{\...
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1answer
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Catgorical sequential limit of $… \to \mathbb Q \to \mathbb Q \to \mathbb Q$

Find the categorical sequential limit of $... \to \mathbb Q \to \mathbb Q \to \mathbb Q$ in the category of abelian groups, where all arrows are multiplication by a natural number $n$, with $n$ fixed ...
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Show that $F$ preserves direct limits (resp. inverse limits)

I am trying to prove the next results, Let $F: {}_{R}\mathfrak{M}\longrightarrow {}_{S}\mathfrak{M}$ a covariant funtor (between $R$-modules and $S$-modules categories). Suposse that such functor is ...
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1answer
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Categories for which every contiuous sheaf is representable

I'm interested in locally small, cocomplete categories $\mathbf{C}$ such that every limit preserving functor $$\mathbf{C}^\mathrm{op}\to\mathbf{Set}$$ is representable. Is there a name for such ...
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1answer
52 views

What is the difference between a directed set and a filtered category?

This may seem like a stupid question, but these two concepts seem to be identified so often that it's just a detail I've overlooked. Apparently a filtered category is a generalisation of a directed ...
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Regarding direct limit of C* algebra

I am trying to understand direct limit in category of $C^*$ algebras. Is it well known that direct limit behaves well with double dual and quotient of $C^*$ algebras? Any references or ideas?
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1answer
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Constructing limits in an additive category given the existence of products and kernels

The title says it all really. Given an additive category $\mathcal{A}$, is having all kernels and arbitrary products sufficient to conclude that it has all limits? Dually, is having all cokernels and ...
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Why $\{e\} \ast_{\mathbb{Z}} \mathbb{Z} \cong \mathbb{Z}/2\mathbb{Z}$? [closed]

Am I correct? Following Serre's "Trees", I can conclude that $\{e\}\ast_{\mathbb{Z}} \mathbb{Z} = \varinjlim (\{e\},\mathbb{Z},\mathbb{Z})$ with respect to homomorphisms: a) $\mathbb{Z} \mapsto e$, ...
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1answer
60 views

What is the (co)limit of a Quiver?

Given that a Quiver (also known as directed multigraphs) is so important in category theory, being that every Quiver produces a free Category, and every Category has an underlying Quiver, I was ...
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23 views

On the limit of a directed system of spectral sequences

Suppose that we have a directed system $(E_N^{pq}, f_{N,N'})_{N,N' \in \mathbb{N}}$ of spectral sequences, and that, moreover, for any $N$, the spectral sequence $E_N^{**}$ collapses at its $E_2$-page ...
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1answer
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Recovering a terminal object from categorical definition of limit

Question: I want to define terminal objects from the categorical definition of limit, but I cannot. What is my mistake in the following argument? Limit: Given a functor $F: \mathcal{D} \...
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1answer
104 views

Filtered vs Directed colimits

I am having trouble with Adamek and Rosicky "Locally presentable and Accessible categories", specifically with the proof of theorem 1.5, namely For every small filtered category $\mathcal D$ ...
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Inverse limits over coinitial sets.

Given a directed set $A$, a subset $B \subset A$ is said to be coinitial if for every $a \in A$ there is some $b \in B$ such that $b \leq a$. Now consider an inverse system of rings $(S_i, f_{ij})$ ...