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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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271 votes
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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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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
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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
  • 1,252
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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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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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
  • 2,216
14 votes
1 answer
996 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
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
  • 3,899
13 votes
3 answers
2k views

Atiyah Macdonald – Exercise 2.15 (direct limit)

Atiyah-Macdonald book constructs the direct limit of a directed system $(M_i,\mu_{ij})$, (where $i\in I$, a directed set, and $i\leq j$) of $A$-modules as the quotient $C/D$, where $C=\bigoplus_{i\in ...
gradstudent's user avatar
  • 3,372
11 votes
2 answers
769 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
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11 votes
2 answers
459 views

Do Wikipedia, nLab and several books give a wrong definition of categorical limits?

It seems unlikely that all these sources are wrong about the same thing, but I can’t find a flaw in my reasoning – I hope that either someone will point out my error or I can go fix Wikipedia and ...
joriki's user avatar
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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
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
11 votes
1 answer
265 views

Can the fundamental group and homology of the line with two origins be computed as a direct limit?

Let $X$ be the line with two origins, the result of identifying two lines except their origins. Let $X_n$ be the result of identifying two lines except their intervals $(-\frac{1}{n},\frac{1}{n})$. $...
183orbco3's user avatar
  • 1,451
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
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11 votes
1 answer
308 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
10 votes
1 answer
1k views

When do coproducts map canonically to products?

I have noticed that in certain categories (e.g. $k$-vector spaces, pointed sets, ...), given an indexed family $\{x_i\}_i$ of objects, we always have a canonical map $$\bigsqcup_ix_i\longrightarrow\...
Daniel Robert-Nicoud's user avatar
10 votes
1 answer
703 views

Any criteria for a category to have all connected limits?

There are many theorems in category theory that give criteria for the existence of a large class of limits in terms of the existence of a few special, more manageable types of limits. For instance: ...
Tomo's user avatar
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10 votes
1 answer
399 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
10 votes
1 answer
1k 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 ...
Maxime Ramzi's user avatar
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9 votes
2 answers
2k views

What is the intuition behind pushouts and pullbacks in category theory?

What is the intuition behind pullbacks and pushouts? For example I know that for terminal objects kind of end a category, they are kind of last is some sense, and that a product is a kind of pair, but ...
geckos's user avatar
  • 225
9 votes
2 answers
1k views

Is direct limit of local rings a local ring? [closed]

Let $\{R_i\}_{i\in A}$ be a directed set of commutative local rings with directed index set $A$, and let $R$ be the direct limit of this set. I want to know if $R$ is a local ring (we know that $R$ is ...
saha stanly'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
9 votes
1 answer
915 views

Colimit of a direct system of monomorphisms

Let $\mathcal A$ be an abelian category, $\{X_i,f_{ij}\}_{i\leqslant j\in I}$ a direct system of $\mathcal A$ such that for any $i\leqslant j\in I$, $f_{ij}:X_i\to X_j$ is an monomorphism. Suppose ...
Censi LI's user avatar
  • 5,945
9 votes
1 answer
178 views

Why are (co-)ends called "(co-)ends"?

Briefly put, the (Co-)end is the universal wedge of a diagram. Why is it called (co-)end? What is it an "end" of? By the universal property, it is in some sense the "universal end/...
Qi Zhu's user avatar
  • 8,314
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
9 votes
0 answers
189 views

Fundamental groupoid of a filtered union

Let $X$ be a topological space and let $(X_i)_{i\in I}$ be a filtered family of subspaces. Let $X =\bigcup_{i \in I} X^°_i$ be the union of the interiors of the $X_i$. I want to prove the following ...
Alice in Wonderland's user avatar
8 votes
5 answers
687 views

What are the end and coend of Hom in Set?

A functor $F$ of the form $C^{op} \times C \to D$ may have an end $\int_c F(c, c)$ or a coend $\int^c F(c, c)$, as described for example in nLab or Categories for Programmers. I'm trying to get an ...
Hew Wolff's user avatar
  • 4,114
8 votes
3 answers
1k views

Why does a union have to be disjoint to constitute a coproduct?

In the wikipedia article of the coproduct it says that in the category of sets the a coproduct is a disjoint union of sets. It is not obvious to me that the union must be disjoint.
fakedrake's user avatar
  • 183
8 votes
2 answers
386 views

Continuous function space on a profinite group as a direct limit

I would greatly appreciate any help with the following problem. If there are existing references related to this, kindly provide them. If not, any help in this matter would be highly valued. Problem: ...
John's user avatar
  • 137
8 votes
2 answers
343 views

Proving adjoint functors preserve limits by lifting the adjunction to cone categories

Let $C$ and $D$ be categories and let $I$ be any (maybe small?) category. $\newcommand{\Hom}{\operatorname{Hom}}$ $\newcommand{\Ob}{\operatorname{Ob}}$ $\newcommand{\Ar}{\operatorname{Ar}}$ $\...
k.stm's user avatar
  • 18.7k
8 votes
1 answer
264 views

What are the categories whose sheaves are representable?

Let $\mathcal{A}$ be a category. The Yoneda embedding $Y : \mathcal{A} \hookrightarrow \mathrm{Hom}(\mathcal{A}^{\mathrm{op}},\mathbf{Set})$ corestricts to an embedding $$y : \mathcal{A} \...
Martin Brandenburg's user avatar
8 votes
1 answer
135 views

"Perfecting" an endomorphism in a category

Let $\mathcal{C}$ be a complete category and suppose $f: X \to X$ is an endomorphism in $\mathcal{C}$. Associated to $f$ is an inverse system, $$X_\bullet: \dots \to X \to X \to X \to X,$$ where every ...
Mr. Chip's user avatar
  • 5,009
7 votes
4 answers
503 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
2 answers
2k views

Quotient group as colimit

I have been wondering for a while about the following question without getting anywhere: Let $G$ be a group, $N$ a normal subgroup. Can the quotient group $G/N$ be seen as the (category theoretical) ...
Daniel Robert-Nicoud's user avatar
7 votes
1 answer
450 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
7 votes
1 answer
2k views

Why is $\mathbb{Z}[1/p]$ the direct limit of $\mathbb{Z}\xrightarrow{p}\mathbb{Z}\xrightarrow{p}\mathbb{Z}\to...$?

This is an example from Algebraic Topology, by Hatcher. As far as I understand, I have to take the direct sum of all the $G_i$s (in this case, $\mathbb{Z}\oplus\mathbb{Z}\oplus...$) and quotient out ...
man_in_green_shirt's user avatar
7 votes
1 answer
889 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
  • 3,120
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
469 views

Why is $\operatorname{colim} F \cong \pi_0\left (\int F\right )$?

Given a small functor $F:\mathsf{C \to Set}$, I need to prove that $\operatorname{colim} F$ is isomorphic/in bijection with the connected components of the category of elements $\int F$. It's not the ...
Mnifldz's user avatar
  • 12.9k
7 votes
2 answers
341 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
1 answer
166 views

Is $\underset{i}\varprojlim R/(I+J)^i \cong \underset{i}\varprojlim \left( \underset{j}\varprojlim R/(I^i + J^j)\right)$?

I am studying $I$-adic completions of a ring and I was wondering if the following holds for any commutative ring $R$ and ideals $I, J \subset R$ $\underset{i}\varprojlim R/(I+J)^i \cong \underset{i}...
user313212's user avatar
  • 2,226
7 votes
1 answer
208 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
  • 5,945
7 votes
1 answer
1k views

What is a direct limit of exact sequences?

(Hatcher Section 3.3, page 243) First, recalling the definition of a directed system of groups: Suppose one has abelian groups $G_\alpha$ indexed by some partial ordered index set $I$ having the ...
user169845's user avatar
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
  • 1,128
7 votes
1 answer
283 views

Show that $\widehat{\mathbb Z}\cong \prod_{p\;\text{prime number}}\mathbb Z_p$

Let $\widehat{\mathbb Z}=\varprojlim_n \; \mathbb Z/n\mathbb Z$ be the inverse limit of the inverse system $(\mathbb Z/n\mathbb Z)_{n\in \mathbb N}$ and let $\mathbb Z_p=\;\varprojlim_n\; \mathbb Z/p^...
eddie's user avatar
  • 1,063
7 votes
2 answers
154 views

A more succinct group object diagram (all axioms in one connected diagram), questions about its properties...

Here is the definition of group object from nLab. They give 3 associated maps $* \xrightarrow{1} G$, $m: G^2 \to G$, and $-^{-1}: G \to G$ and require 3 commutative diagrams to complete the axioms ...
HighAsAKiteOnMath's user avatar
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
1 answer
431 views

Relationship between two definitions of pro-representable functors

Edit: I'm pretty sure that my conjecture $$ \operatorname{Hom}(\varprojlim_i R/\mathfrak{m}^i, A) = \operatorname{colim}_i \operatorname{Hom}(R/\mathfrak{m}^i, A), $$ is true. To prove it, just use ...
Sebastian Monnet's user avatar
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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