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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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In (relatively) simple words: What is an inverse limit?

I am a set theorist in my orientation, and while I did take a few courses that brushed upon categorical and algebraic constructions, one has always eluded me. The inverse limit. I tried to ask one of ...
Asaf Karagila's user avatar
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38 votes
4 answers
7k views

The "magic diagram" is cartesian

I am trying to solve an exercise from Vakil's lecture notes on algebraic geometry, namely, I want to show that $\require{AMScd}$ \begin{CD} X_1\times_Y X_2 @>>> X_1\times_Z X_2\\ @V V ...
Paul's user avatar
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14 votes
1 answer
1k views

Compact subset in colimit of spaces

I found at the beginning of tom Dieck's Book the following (non proved) result Suppose $X$ is the colimit of the sequence $$ X_1 \subset X_2 \subset X_3 \subset \cdots $$ Suppose points in $X_i$ ...
Luigi M's user avatar
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62 votes
2 answers
7k views

Category-theoretic limit related to topological limit?

Is there any connection between category-theoretic term 'limit' (=universal cone) over diagram, and topological term 'limit point' of a sequence, function, net...? To be more precise, is there a ...
Rafael Mrden's user avatar
  • 2,209
30 votes
6 answers
8k views

Right adjoints preserve limits

In Awodey's book I read a slick proof that right adjoints preserve limits. If $F:\mathcal{C}\to \mathcal{D}$ and $G:\mathcal{D}\to \mathcal{C}$ is a pair of functors such that $(F,G)$ is an adjunction,...
Bruno Stonek's user avatar
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33 votes
1 answer
5k views

On limits, schemes and Spec functor

I have several related questions: Do there exist colimits in the category of schemes? If not, do there exist just direct limits? Do there exist limits? If not, do there exist just inverse limits? With ...
iago's user avatar
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24 votes
3 answers
5k views

Examples of a categories without products

A question was raised in our class about the non-existence of product in a category. The two examples that came up in the discussion was the category of smooth manifolds with boundary and the category ...
ChesterX's user avatar
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3 votes
1 answer
1k views

Do pullbacks commute with filtered limits in this sense?

Let $A_n, B_n, C_n$ be directed systems in some abelian category. Denote by $A \times_C B$ the fibre product of $A$ and $B$ over $C$. Is it true that $(\varprojlim A_n) \times_{\varprojlim C_n} (\...
user110071's user avatar
11 votes
1 answer
2k views

Filtered colimits commute with forgetful functors

In many cases, filtered colimits commute with forgetful functors to $\mathbf{Set}$, for example with $\mathbf{CRing} \to \mathbf{Set}$ or $R-\mathbf{Mod} \to \mathbf{Set}$. Is there a slick way of ...
Andre Knispel's user avatar
7 votes
1 answer
888 views

Infinite tensor product as infinite coproduct in the category of R-algebras

We know that the tensor product is the coproduct in the category of R-algebras for any ring R. What about the infinite tensor product on an index set I, defined as colimit of the finite tensor ...
W. Rether's user avatar
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6 votes
1 answer
1k views

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 ...
Matt's user avatar
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5 votes
1 answer
317 views

Concrete description of (co)limits in elementary toposes via internal language?

In the category of sets, limits and colimits can be concrete described respectively as subobjects of products and quotients of coproducts. It seems like these descriptions make sense in any ...
Arrow's user avatar
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4 votes
2 answers
317 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 $X_i$ and for all $i \leq j$ $$\pi^j_i[U_j]\subseteq U_i$$ where $\pi^j_i: X_j \to X_i$ . This means that ...
ABIM's user avatar
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11 votes
4 answers
7k views

What really is a colimit of sets?

This is probably a question I should have asked myself a bit earlier. For some reason I always thought I knew the answer so I did not bother, but now that I actually need to use it (I am studying the $...
user121314's user avatar
10 votes
1 answer
397 views

Simple example of product not preserving coequaliser in $\mathbf{Top}$

In the category of topological spaces ($\mathbf{Top}$), products do not always preserve colimits. If they did then $\mathrm{Hom}_\mathbf{Top}(-\times X,S)$ would be representable and hence $\mathbf{...
Oscar Cunningham's user avatar
9 votes
2 answers
528 views

Do products preserve colimits in the category of locales?

Does the functor $X\times-:\mathbf{Loc}\to\mathbf{Loc}$ preserve small colimits for all locales $X$? The reason that I'm interested in this question is that the same property fails in the category of ...
Oscar Cunningham's user avatar
7 votes
1 answer
247 views

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 ...
DevVorb's user avatar
  • 1,443
7 votes
2 answers
232 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 ...
qualcuno's user avatar
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6 votes
1 answer
211 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\...
Oskar's user avatar
  • 4,373
6 votes
2 answers
4k views

Category Theory: homset preserves limits

edit I updated my question at the end, I think the claim may be false? Let $(L,\lambda)$ be a limit cone of a diagram $D$ in a category. For any object $X$ it is said that the hom functor $Hom(...
user avatar
5 votes
1 answer
594 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 ...
Giorgio Mossa's user avatar
5 votes
1 answer
570 views

Modules finitely generated and of finite type (categorical meaning)

An object $C$ in an additive category admitting all filtered direct limits $\mathcal{C}$ is called "of finite type" if the canonical map $$\underrightarrow{\lim} Hom_{\mathcal{C}}(C,F(i))\to Hom_{\...
DDT's user avatar
  • 747
4 votes
1 answer
577 views

Homotopy colimit

Some time ago I asked this question. I am trying again to get an understanding of the definition of the homotopy colimit of a diagram of topological spaces. One of the answers at the above says ...
Matt's user avatar
  • 3,316
4 votes
3 answers
185 views

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 ...
FShrike's user avatar
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4 votes
1 answer
730 views

Direct limit isomorphism

Suppose I have a directed system $(V_i, \phi_i: V_i \rightarrow V_{i+1})$, say of vector spaces $V_i$. Let $\psi_i: V_i \rightarrow V_i$ be isomorphisms. I can construct the related directed system $(...
user39598's user avatar
  • 1,554
2 votes
1 answer
165 views

Are poset-shaped limits of finite groups profinite?

My basic question is Question 0: Let $\mathcal{C}$ be a complete category and $\mathcal{D}$ its full subcategory such that $\mathcal{D}$ is closed under finite limits (in $\mathcal{C}$). Let $I$ be ...
Pavel Čoupek's user avatar
1 vote
1 answer
139 views

Equivalent definitions of preserving limits

On page 137, Leinster gives two equivalent characterizations of limit preservation: Is it supposed to be obvious that they are the equivalent? If so, how to see that? (When I tried to prove that, I ...
user557's user avatar
  • 11.9k
1 vote
1 answer
83 views

Topology on $R((t))$, why is it always the same?

Let $R$ be a topological ring (commutative with $1$) and let $R((t))$ the ring of Laurent power series. So, it is the ring containing the formal power series: $$\sum_{i\ge m} a_i t^i,\quad m\in\...
manifold's user avatar
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1 vote
0 answers
66 views

Equivalent definitions of preserving limits - 2

I've already asked here about the following confusing restatement of the definition of limit preservation from Leinster's book (p. 137): And now a new question has arisen. As far as I understand from ...
user557's user avatar
  • 11.9k
1 vote
1 answer
400 views

Direct Limit of Both Rings and Their Modules

Suppose we have a directed set $\langle I,\leq\rangle$, with a direct system $\langle A_i,f_{ij}\rangle$ of rings and a direct system $\langle M_i,g_{ij}\rangle$ of abelian groups, such that each $M_i$...
Dave's user avatar
  • 1,373
0 votes
1 answer
252 views

Rephrasing a particular lemma on limits for general fibrations

In the book Galois Theories by Borceux and Janelidze appears the following lemma. Lemma 6.2.3. Let $I:\mathsf{Fam}(\mathsf A)\longrightarrow \mathsf{Set}$ be a family fibration and let $D:\mathsf K\...
Arrow's user avatar
  • 13.9k
0 votes
1 answer
468 views

On the proof of the density theorem

I'm trying to understand Leinster's proof of the density theorem. Here's the terminology and the statement. Below is his proof. Here are some things that I don't understand: This must be silly, but ...
user557's user avatar
  • 11.9k
14 votes
1 answer
994 views

Calculating (co)limits of ringed spaces in $\mathbf{Top}$

Let $\mathbf{Top}$ be the category of topological spaces, $\mathbf{RS}$ the category of ringed spaces and $\mathbf{LRS}$ the category of locally ringed spaces. There are forgetful functors $$ U_{\...
user8463524's user avatar
  • 2,177
11 votes
2 answers
768 views

Example of a functor which preserves all small limits but has no left adjoint

The General Adjoint Functor Theorem (Category Theory) states that for a locally small and complete category $D$, a functor $G\colon D \to C$ has a left adjoint if and only if $G$ preserves all small ...
Conan Wong's user avatar
  • 2,233
11 votes
2 answers
3k views

Meaning of pullback

I was wondering if the following two meanings of pullback are related and how: In terms of Precomposition with a function: a function $f$ of a variable $y$, where $y$ itself is a function of ...
Tim's user avatar
  • 47.6k
11 votes
1 answer
307 views

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. ...
Pierre-Yves Gaillard's user avatar
9 votes
2 answers
364 views

In the absence of limits from the codomain, do evaluation functors still preserve limits?

Consider a diagram $K:\mathsf J\longrightarrow [\mathsf C,\mathsf D]$ as well as the evaluation functors $\operatorname{ev}_C:[\mathsf C,\mathsf D]\to \mathsf D$. The fact limits in functor categories ...
Arrow's user avatar
  • 13.9k
7 votes
1 answer
1k views

Limit functor is right adjoint to diagonal functor

Let $C$ and $J$ be categories. Suppose that the limit functor, $\varprojlim$ say, of $C^{J}$ exists, i.e. the limit of any functor $J \to C$ exists. I would like to prove that $\varprojlim$ is right ...
Plankton's user avatar
  • 1,118
7 votes
1 answer
3k views

Representable functors preserve limits. [duplicate]

Theorem : Representable functors preserve limits. I'm struggling to see why this is true. It's not obvious to me where I should be actually using the fact that functor in question is representable. ...
user avatar
7 votes
2 answers
339 views

Does $\operatorname{Spec}$ preserve pushouts?

The spectrum-functor $$ \operatorname{Spec}: \mathbf{cRng}^{op}\to \mathbf{Set} $$ sends a (commutative unital) ring $R$ to the set $\operatorname{Spec}(R)=\{\mathfrak{p}\mid \mathfrak{p} \mbox{ is a ...
user8463524's user avatar
  • 2,177
7 votes
4 answers
501 views

$\Lambda = \varprojlim\Lambda_n$ (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 ...
idriskameni's user avatar
  • 1,428
7 votes
1 answer
207 views

On relationship of two categorical characterization of finitely generated objects.

I've encountered The following categorical characterization of finitely generated modules: A $R$-module $M$ is finitely generated iff it satifies one of the following properties: a): for any family ...
Censi LI's user avatar
  • 5,925
7 votes
1 answer
447 views

Example of complete category with no initial object

My original question is this. I found Zhen Lin's answer very useful, but I couldn't think of a category which is complete but has no initial object. The first category that I thought that has no ...
Marcelo's user avatar
  • 1,372
6 votes
1 answer
2k views

Inverse vs Direct Limits

This is probably a basic question but I haven't found anything satisfying yet. I'm trying to understand the difference between inverse and direct limits other than the formal definition. In my mind, ...
Supersingularity's user avatar
6 votes
1 answer
718 views

Possible adjoint to Yoneda embedding and Repeated Yoneda embedding?

While thinking about Yoneda embedding, I came up with following two questions (I should apologies, if those are too vague): Does the Yoneda embedding $y :\mathcal{C}\to\mathbf{Set}^{\mathcal{C}^{\...
Bumblebee's user avatar
  • 18.3k
6 votes
1 answer
320 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 ...
CNS709's user avatar
  • 1,667
6 votes
1 answer
257 views

Is there a simple abstract reason why a profinite group is an inverse limit of finite groups?

Let $G$ be a profinite group (defined as a Hausdorff, compact, totally disconnected topological group). Suppose you know that as a profinite set, it's an inverse limit of finite sets. Is there an ...
gothicvague's user avatar
6 votes
1 answer
320 views

Is there a coproduct in the category of path connected spaces?

Well, first of all, does the coproduct exist in the category of path-connected spaces, and if not how would you prove it? If it does exist what is it and how do you find it? The usual coproduct of ...
Visitor's user avatar
  • 83
6 votes
2 answers
1k views

Is a direct limit of Noetherian rings necessarily Noetherian?

Is the direct limit of Noetherian rings necessarily Noetherian? And if it is, how to prove this? If it is not, what is a counterexample? (I was thinking this question: if $A_{m}$ are Noetherian for $...
Alex's user avatar
  • 2,613
5 votes
5 answers
314 views

A category which direct limits but no general colimits

I am looking for a (at best, real life) category that has direct limits, but no general small colimits, or a category that has inverse limits, but no general small limits. Are there any interesting ...
Bubaya's user avatar
  • 2,214