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.
2
votes
0answers
25 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 ...
1
vote
1answer
34 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: \...
4
votes
1answer
95 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 ...
2
votes
1answer
28 views
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
$$\mathbb{Z} \xrightarrow{\times 2} \mathbb{Z} \xrightarrow{...
2
votes
1answer
13 views
Can we deduce that $M_0$ is a submodule of the limit of the following diagram?
Let $M_0$ be an R-module, and suppose $M_{n+1}$ is the pushout of the diagram below as shown, for all $n \in \mathbb{N}$:
$$\begin{array}{ccc}M_n&\to& M_{n+1}\\\uparrow &&\uparrow\\A&...
0
votes
0answers
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 ...
1
vote
2answers
68 views
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.
1
vote
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}/\...
0
votes
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 ...
4
votes
2answers
88 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 ...
1
vote
1answer
61 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 ...
3
votes
0answers
43 views
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 ...
2
votes
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 ...
2
votes
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{...
6
votes
1answer
73 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\...
1
vote
2answers
67 views
$\ell^2$ 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 $\ell^2 \cong \varinjlim_{n}\oplus_{i=1}^n\mathbb{R}$?
2
votes
2answers
33 views
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). ...
5
votes
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 ...
2
votes
0answers
42 views
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 ...
2
votes
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 ...
2
votes
0answers
21 views
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 ...
2
votes
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 $\lambda$ ...
3
votes
0answers
43 views
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. ...
1
vote
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}\...
1
vote
0answers
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 ...
1
vote
0answers
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 $\...
0
votes
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 ...
2
votes
1answer
50 views
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 ...
3
votes
0answers
59 views
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
$$
\text{colim}_{j}[A,D(j)] \to [A,\text{colim}_{j}...
1
vote
0answers
47 views
Colimit of discrete diagram (coproduct) is the tensor product
Let $C$ be the category of unital, commutative, $k$ algebras, where $k$ is a field.
Given a discrete category $I$, with two elements, let $F: I \to C$ be a functor.
Denote $I(i_1) = A_i$ for $i = 1,...
1
vote
1answer
37 views
Sheaf on basic open sets.
Starting from a $B$-sheaf $\hat{\mathcal{F}}$, a sheaf defined on the basic open sets of a topological space $X$, I am asked to prove that the construction
$$\mathcal{F}(U) = \varprojlim_{V \subset U,...
6
votes
0answers
82 views
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 $F: C\to D, U: D\to C$ ...
0
votes
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 ...
3
votes
1answer
37 views
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 ...
5
votes
0answers
82 views
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 $\...
0
votes
1answer
32 views
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^{\...
1
vote
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 ...
0
votes
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_{\...
2
votes
1answer
51 views
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 ...
1
vote
0answers
77 views
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 ...
6
votes
1answer
89 views
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 ...
2
votes
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 ...
0
votes
0answers
36 views
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?
1
vote
1answer
27 views
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 ...
2
votes
0answers
88 views
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) $\mathbb{Z} \mapsto e$,
...
2
votes
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 ...
0
votes
0answers
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
...
3
votes
1answer
54 views
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} \...
3
votes
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 $\mathcal D$ ...
1
vote
0answers
41 views
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 $(S_i, f_{ij})$ ...
