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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Colimit of categories don't preserve equivalences [duplicate]
I have a question about the following problem: Find two diagrams of shape J in Cat $F,G:J\rightarrow Cat$ and a natural transformation $\eta:F\rightarrow G$ such that $\eta_i:F(i)\rightarrow G(i)$ is ...
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Canonical morphism of inverse limits of inverse systems of modules
Let $A$ be a commutative ring and $I$ an index set. Let $(M_i, f_i)_{I\in I}$ be an inverse system of $A$-modules and let $N$ be an $A$-module. There is a canonical map
$$\phi: (\mathrm{lim}_{\...
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1
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Equivalence of categories and colimits
Consider functors $F,G:J\to Cat$ which admit colimits and $\eta:F\to G$ a natural transformation such that $\forall i\in J$ $\eta_i:F(i)\to G(i)$ is an equivalence of categories. Is it true that then ...
- 240
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Limit of sequential diagram
Consider $X_0\leftarrow X_1\leftarrow \dots$ be a sequential diagram in a category $C$ and suppose it has a limit. I want to show that if $(n_k)$ is a sequence of increasing natural integers, then the ...
- 240
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1
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A name for 'lax' colimit for poset-valued functors?
Let $F,G\colon C\to \mathsf{POS}$ be functors, where $C$ is a category and $\mathsf{POS}$ denotes the category of posets and monotone maps.
By a lax natural transformation $\alpha\colon F\Rightarrow G\...
- 23
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1
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31
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Categorical products as inverse limits of finite subproducts
Let $\mathfrak C$ be a category with products. Given a family $(X_i)_{i \in I}$ of objects of $\mathfrak C$, we get their product $\prod_{i \in I} X_i$ with projections $\pi_i : \prod_{i \in I} X_i \...
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Dense, codense, finite subcategory
I found in some lecture notes a statement, that there exists an inclusion of full subcategory $\mathcal C \hookrightarrow \mathcal D$ which is both dense and codense, and moreover $\mathcal C$ is ...
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Question about an exercise on showing why the colimit for Poset as a category can not be constructed in Arbib and Manes text
The following question is taken from $\textit{Arrows, Structures and Functors the categorical imperative}$ by Arbib and Manes
$\text{(1):}$ Given a relation ${\stackrel{\small{R}\stackrel{\small\small ...
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Colimit in the category of Sets.
Below $\mathcal{I}\to \textbf{Set}$ is a functor(diagram) between two categories while $M_i$ are all sets. There are no restrictions on $\mathcal{I}$. I cannot see why the equivalence of elements $m_i\...
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Question about the concept of limits/colimits
Recently, I asked in a post, questions concerning the concept concerning concerning "limit for the diagram". For reasons probably due to my questions not clear or there were too many in ...
- 2,119
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1
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Is inverse limit topology closed in product topology?
Suppose $\{X_i\}_{i\in I}$ is an inverse system of topological spaces. Denote $X=\varprojlim X_i$. It's naturally a subspace of the product topology $\prod_{i} X_i$. I want to ask: is $X$ closed in ...
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Questions about the specifics in the definitions for cone, and limits in Category theory
The following are taken from $\textit{Arrows, Structures and Functors the categorical imperative}$ by Arbib and Manes
$\quad$$\textbf{Definition 1}$ A $\textbf{directed graph}$ is an arbitrary class ...
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is there a standard projective limit system that covers Sharkovskii's ordering?
Let $Y\subsetneq X$
You might assume both are continuous functions on the real line, with $Y$ being a segment and $X=\Bbb R$ but not necessarily.
Let $Y$ have fixed points and periodic points of ...
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Filtered colimits and the free frame construction
Let $F : C \to D$ be a functor between categories with filtered colimits, and let Cat be the category of categories. Let $(M, μ, η)$ be a monad on Cat. What properties of $M$ are necessary to ensure ...
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Example of surjective inverse system where projections from limit are not surjective
I am reading "Profinite Groups" by Ribes and Zalesskii and on page 9 it says that the projections of the nonempty inverse limit of a surjective inverse system are not necessarily surjective, ...
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1
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Colimit of symmetric groups
I don't yet know much about categorical limits and colimits, I have just started learning about them, and so I wanted to experiment a bit with this concept. And to that end, my first natural attempt ...
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2
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Finding an example of a functor $\alpha:\mathbb{N}\to Mod(\mathbb{Z})$ such that the colimit of $\alpha$ is $0$.
$\DeclareMathOperator{\colim}{colim}$ $\DeclareMathOperator{\Hom}{Hom}$ $\DeclareMathOperator{\Mod}{Mod}$ As described in the title, I'm trying to construct an example of a functor $\alpha: \mathbb{N}...
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Union of an ascending chain of submodules
Let $M$ be a module and $N_1\subset N_2\subset\cdots$ be an ascending
chain of submodules of $M$. Denote $\bigcup_{i=1}^{\infty}N_i$ by $N$. Prove
that:
$N$ is a submodule of $M$, and
$N = \...
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1
answer
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Proving a formula involving Hom-set go colimit and constant functor
Let $I$ and $C$ be categories, assume $I$ is small and denote $\Delta$ the functor $C \to C^I$ that sends $Y\in C$ to the constant functor $I\to C$, i.e. $\Delta(Y)(i) = Y$ and $(i \to j) \mapsto \...
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1
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Intuition behind cokernel pair
I'm trying to find some insights about cokernel pairs. I understand the definition and I know how to calculate them in concrete categories, but while the kernel pairs seem to have a very sensible ...
- 264
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2
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Inverse limit and direct limit of an infinite sequence of sets
Let $S_1 \subset \cdots \subset S_n \subset \cdots$ be an infinite sequence of finite sets. We can define both inverse limit and direct limit of this sequence. What is the difference between them? ...
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Inductive limit in category of (small) categories
Recently I attended a talk where the speaker mentioned about inductive limit in the category of (small) categories. I have never seen anything like this so I was wondering how does one constructs ...
- 3,330
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2
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Is the inclusion into a coproduct an equalizer?
Let $\mathcal{C}$ be a category with finite limits and binary coproducts. Let $1$ denote the terminal object, and let $\langle\rangle_A : A \rightarrow 1$ denote the unique map from $A$ to $1$. If we ...
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1
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Direct limit of nested fundamental groups
Let $M$ be a compact submanifold (with boundary) of $S^n$ realised as the intersection of some other compact manifolds $M_i\subset S^n$ so $M=\cap M_i$ and $M_i\subset Int(M_{i-1})$. Then we have:
$\...
- 157
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2
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A functor continuous with respect to a cylinder
In the erratum of their paper: “Categories of continuous functors, I”, (Journal of Pure and Appplied Algebra 2 (1972) 169–191), the authors, P.J. Freyd and G.M. Kelly, define the continuity of a ...
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Topology determined by cubes
Which spaces $X$ have the property that $X \to Y$ is continuous if and only if $I^n \to X \to Y$ is continuous for all $I^n \to X$?
Example: manifolds and CW complexes have this property, since we ...
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Can the topological spaces $\mathbb{R}$ and $\mathbb{Q}$ be expressed as a colimit of a diagram of discrete/finite spaces?
I am interested in knowing if $\mathbb{R}$ and $\mathbb{Q}$ are colimits in the category of topological spaces of a diagram $J$ of discrete or finite spaces. I would like to know also if it is ...
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How to define the natural topology on $A[a^{-1}]$?
Suppose $A$ is a topological commutative ring, which is also a domain, and $a\in A$. Then how to define the topology on $A[a^{-1}]$?
Since $A[a^{-1}]=\varinjlim a^{-n}A$, my idea is to define the ...
- 1,190
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1
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Constructing finite limit from terminal object and pullbacks
I'm trying to understand, why in Proposition 3.1 in Finite Limit in ncatlab (3) implies (1). My intuition tells me to use induction to show that the limit for any diagram having $n$ vertices (objects) ...
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Projective Limits of Compact Groups: Exact or Not?
I am reading the following lemma from Washington's book "Introduction to Cyclotomic Fields":
On the other hand, there is a counterexample, given by this answer. The comments below this ...
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1
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Difficulty using the (co)limit formulae to construct the $n$-(co)skeleton left and right Kan extensions for truncated simplicial objects
Tl;Dr - I’m struggling to show that the $n$-skeleton is a Kan extension, from the basic limit formula (this should be possible, as it was “left to the reader” in my book). I’m also struggling to even ...
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Confusion about direct limit of vector spaces
Consider the sequence of vector spaces
$i_n : \mathbb R^n\to \mathbb R^{n+1}$
given by the inclusions $x\mapsto (x,0)$.
We can consider the direct limit of this sequence, call it
$\mathbb R^\infty$.
...
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2
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What are the inclusion arrows in the coproducts of the category of algebras for a monad?
$\newcommand{\A}{\mathscr{A}}\newcommand{\C}{\mathsf{C}}\newcommand{\T}{\mathcal{T}}\newcommand{\id}{\operatorname{id}}$Riehl, proposition $5.6.11$, from Category Theory in Context:
Suppose $\C$ is a ...
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Density of a subspace in certain topological spaces.
Let $(X_i)_{i \in \mathbb{N}},(f_{i,j})_{i\leq j \in \mathbb{N}})$ be an inverse system in $Top$ with inclusion $\iota_i:U \hookrightarrow X_i$ for all $i$, such that $f_{i-1,i}\circ\iota_i=\iota_{i-1}...
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Statement in Riehl's book of the result "a diagram of functors has a limit if it has objectwise limit"
$\def\A{\mathsf{A}}
\def\C{\mathsf{C}}
\def\ob{\operatorname{ob}}
\def\ev{\operatorname{ev}}
\def\J{\mathsf{J}}
$
I am having a little trouble understanding the last sentence of this result in p. 93 ...
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1
answer
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the hom-functor is left exact: an equivalent condition
I would like to understand the equivalence of the 2 facts that hom is left exact in the sense that it carries short exact sequences to left exact sequences and that it preserves all small existing ...
- 3,889
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2
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any $\lambda-$pure morphism in a $\lambda$-accessible category is a monomorphism
I'm trying to understand the proof of the proposition 2.29 in the book
Locally presentable and accessible categories which is also given in the snippet below.
I've got stuck in the -2nd paragraph ...
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Intuition for density formula for Pre-sheaves/Co-Yoneda lemma
I was reading Fosco's notes coend calculus, so far I am at Ninja-Yoneda lemma.
The density formula for pre-sheafs says that,
$${K} V\cong \int^{U\in C} K U\times \textstyle\text{Hom}_{C}(V,U)\cong\...
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Is a scalar extension of an $(t)$-adically complete module complete?
Let $M$ be a $(t)$-adically complete $\mathbb C[[t]]$-module, say $M$ is a topologically free $\mathbb C[[t]]$-module. Let $\mathbb C((t))$ be the field of formal Laurent series. Is then the scalar ...
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Sheaves valued in a $k$-category
Let $\mathcal{C}$ be a $k$-category which we regard as an $\infty$-category whose objects all happens to be $k$-truncated.
A sheaf $F$ valued in $\mathcal{C}$, in the $\infty$-categorical sense, is a ...
- 1,475
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1
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Are inverse limits just special cases of limits?
I have a quick question
Are inverse limits as they are defined for inverse systems just a special case of a limit? I am not really proficient in category theory. I guess for a given inverse system we ...
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1
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75
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Is the ring of formal power series in $n$ variables the colimit of the powers of its maximal ideal
Let $\mathbb C[[t_1, ..., t_n]]$ be the noetherian local ring of formal power series in $n$ variables with a unique maximal ideal $\mathfrak{m}=(t_, ..., t_n)$. Then, there is a descending chain of ...
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Functoriality of limit
The object map $\mathrm{lim}:\sf C^J\to C$ is known to extend to a functor; does this fact is already proven by the fact that the limit of a diagram of functors is computed objectwise?
If $F:\sf I\to ...
- 179
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115
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Is the inverse limit of quotient the quotient of inverse limits?
I am recently studying inverse limits and I have the following issue :
Le $V$ be a vector space and $W \subset V$ a subspace of $V$.
Assume that $V$ is equipped with a decreasing filtration $(\mathcal{...
- 87
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0
answers
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Direct limit of sequences induced by fusing together copies of $\mathbb{Z}$
Let $A=l^\infty(\mathbb Z,\mathbb Z)$ be the abelian group of bounded sequences $\mathbb Z\to\mathbb Z$. Define a homomorphism $f\colon A\to A$ by
$$f(a)(n)=a(2n)+a(2n+1),$$
for $a\in A$ and $n\in\...
- 4,520
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0
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Showing the Krull topology is the same as the categorical limit topology of (infinite) Galois extensions - is my solution correct?
$\newcommand{\gal}{\operatorname{Gal}}$This question is about showing the two definitions of Krull topology over an infinite Galois group are the same. I would greatly appreciate any feedback on the ...
- 22k
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votes
2
answers
77
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When is $f$ not the coequalizer of its kernel pair?
Let $f : X \rightarrow Y$ be an arrow, and suppose it has a kernel pair $p_1, p_2 : P \rightarrow X$. I know that in general, $f$ may not be the coequalizer of $p_1, p_2$, but what is a simple example ...
- 5,223
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0
answers
15
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Field of formal Laurent series as a colimit of fractions
Let $\mathbb C[[x]]$ be the ring of formal power series in one indeterminate $x$. Let $\mathbb C((x))$ be the field of formal Laurent power series. We know that that $\mathbb C((x))$ is the fraction ...
- 775
1
vote
1
answer
65
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Inverse limit of compact hausdorff spaces
Suppose that $(X,\varphi_i)$ is an inverse limit of an inverse system of compact hausdorff spaces $\{X_i,\varphi_{ij}\}$. Let $Y$ be a subspace of $X$, I would like to proof that $\overline{Y}$ (...
- 13
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0
answers
40
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Hom set in coslice category as coends
Let $\def\catC{\mathcal{C}}\catC$ be a (locally small) category and let $A\xrightarrow{\;f\;}B\xrightarrow{\;g\;}C$ be two composable morphisms. This question is about morphisms in coslice categories.
...
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