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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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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
7 votes
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168 views

Ind-objects with "full support"

Now cross-posted to MO. Let $C$ be a small category. Let's say a presheaf $P\colon C^{\mathrm{op}}\to \mathsf{Set}$ has "full support" if $P(X)\neq \varnothing$ for all objects $X$. We ...
Alex Kruckman's user avatar
6 votes
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78 views

Proof of Theorem 3.4.12 in Emily Riehl's "Category Theory in Context"

I have questions about the proof of Theorem 3.4.12 in Emily Riehl's Category Theory in Context. The theorem states that the colimit of a small diagram $F\colon \mathsf J \to\mathsf C$ can be expressed ...
displayname's user avatar
6 votes
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How to understand the effect of adjoint functors?

I have a good grasp of all different definitions/interpretations of adjoint functors, but still do not know have to interpret the left or right adjoint of a give functor, when it exist. It would be a ...
Bumblebee's user avatar
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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 $F: C\to D, U: D\to C$ ...
Maxime Ramzi's user avatar
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6 votes
1 answer
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Colimits for gluing schemes and the functor of points 1

Closely related questions been asked several times in different forms on here but I feel like none really spell out what's going on. I have been looking more at glueing schemes, and particularly ...
Luke's user avatar
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Proving étale maps have discrete fibers by abstract nonsense?

I'm trying to prove that an étale map of spaces has discrete fibers. The first diagram I drew is: $$\require{AMScd} \begin{CD} f^\ast\coprod_i \left\{ x \right\} @>>> f^\ast \coprod_i U_i @&...
Arrow's user avatar
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6 votes
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Why are split coequalizers "contractible"?

In the book Toposes, Triples and Theories by Barr and Wells, the authors define a contractible coequalizer (elsewhere known as a split coequalizer) to be a commutative diagram: $A \rightrightarrows_{d^...
Phil Tosteson's user avatar
5 votes
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1k views

Surjectivity of a map on inverse limits

I found the following statement in P. Gabriel's thesis: Lemma. Let $(I, \leq)$ be a directed poset, and $(M_i, \mu_{ji}:M_j\rightarrow M_i)_{j\geq i},$ $(N_i, \nu_{ji}:N_j\rightarrow N_i)_{j\geq i}$...
Pavel Čoupek's user avatar
4 votes
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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, ...
linkja's user avatar
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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\...
geometricK's user avatar
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What's going on in this notation for the projective limit in Serre?

$\newcommand{\Z}{\mathbf Z}\newcommand{\Q}{\mathbf Q}$I am currently reading Serre's A Course in Arithmetic, and in Chapter 2 where he introduces the $p$-adics, he mentions the projective limit. My ...
YiFan Tey's user avatar
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Sheaf Axioms and Limits - Intuition

The question is based on the following problem from Vakil's notes in Algebraic Geometry: 2.2.C. The identity and gluability axioms (of sheaves) may be interpreted as saying that $\mathcal{F}(\cup_i ...
User20354's user avatar
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311 views

Limits as initial objects

I am relatively new to category theory and was wondering about the following problem: Can I consider a limit as an initial object in some categories? Let $\mathscr{C}$ be a category and $\mathbf{J}$ a ...
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Surjection on limits

Suppose we have a map of diagrams $X \to Y$ of shape $D$ in the category of sets. Suppose further this is an objectwise surjection. That is, $X_d \to Y_d$ is a surjection for all $d \in D$. Are there ...
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4 votes
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180 views

Filtered colimit and functor $\text{Hom}(X,-)$

I would like to prove the following fact: let $X$ a finitely generated $A-$module, $I$ is a filtered category, $F:I\to A-\text{mod}$ a functor. For each $i\in I,$ put $Fi=N_i,$ and $$\lim_{\to}{N_i}=(...
user avatar
4 votes
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Products In the Categary of Skew-Cocommutative Coalgebras Are Skew Tensor Products

Let $A = \bigoplus^n_{i=0} A_i,B = \bigoplus^b_{i=0} B_i $ be a graded modules over the same commutative ring $R$ . The twisting isomotphism $\tau_{A,B} : A \otimes B \to B \otimes A$ is defined on ...
Nik Bren's user avatar
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Stalk of tensor product sheaf is tensor product of stalks via adjunctions and abstract nonsense

Let $(X, \mathcal{O}_{X})$ be a ringed-space with sheaves of modules $\mathcal{F}$ and $\mathcal{G}$. I would like to show that for any point $p \in X$, $$ \mathcal{F}_{p} \otimes_{\mathcal{O}_{X, p}} ...
Luke's user avatar
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4 votes
1 answer
205 views

Will a homotopical right-adjoint preserve homotopy limits?

Let $M$ and $N$ be homotopical categories (both of them complete) and $I$ a small category. Assume that we have right deformations for the limit functors $M^I \to M$ and $N^I \to N$, so we have the (...
Lukas Woike's user avatar
4 votes
0 answers
126 views

What does the "Eternal Round" topological space look like?

I shall identify points on a circle with their angle in this question. For example, $0$ and $2\pi$ both correspond to the top point of the circle. Take the function $d(\theta)=2\theta$, which is a ...
Christopher King's user avatar
4 votes
0 answers
373 views

Visualizing co-Yoneda lemma

I am studying the Yoneda and co-Yoneda lemmas, and in order to understand them well I am trying to develop particular cases. This one is getting me in trouble (it must be easy, but I cannot get it): ...
Luoisv's user avatar
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148 views

Looking for an example of a concrete calculation of a certain direct limit to get my hands dirty with generalized fractions

I am going to give a talk on the Koszul complex and its connection with local cohomology. We are using the book Residues and Duality for Projective Algebraic Varieties by Kunz and in the chapter that ...
Lennart's user avatar
  • 537
3 votes
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Writing an enriched graph as a colimit

I am working with enriched directed graphs (aka, directed graphs/quivers such that the edges are objects in a category V). I can write every graph as a filtered colimit of finite graphs, and I can ...
Dimitriadis's user avatar
3 votes
0 answers
190 views

$\mathrm{Ext}$ and direct limit

Let $R$ be a commutative Noetherian ring. Then, for $R$-modules $\{X_i\}_i$ and $Y$, do we always have $$\mathrm{Ext}^n_R(\varinjlim X_i,Y) \cong \varprojlim \mathrm{Ext}^n_R( X_i,Y)$$ for all $n\geq ...
Alex's user avatar
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0 answers
36 views

Dual Concept of a Well-Powered Category

I was studying the SAFT theorem (Special Adjoint Functor Theorem), using Leinster's Basic Category Theory and I had the homework to dualize it. So far I have the following, Consider a category $\...
babu's user avatar
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Category is equivalent to the colimit of its finite subcategories and colimit in Cat

$\DeclareMathOperator{\colim}{colim}$Let $I$ be a (small) category, and take $S(I)$ to be the poset of finite subcategories of $I$, ordered by inclusion, treated as a category. One can show $S(I)$ ...
t_kln's user avatar
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Are the limits of Euclidean spaces Hilbert spaces?

Consider each Euclidean space $\mathbb{R}^n$ as a topological vector space. I wondered what I get by taking $n \to \infty$ in the category of topological vector spaces over $\mathbb{R}$. There are ...
Dannyu NDos's user avatar
  • 2,059
3 votes
0 answers
88 views

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 ...
Andrea Marino's user avatar
3 votes
0 answers
176 views

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\...
MrPajeet's user avatar
  • 325
3 votes
0 answers
421 views

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{...
yaden's user avatar
  • 97
3 votes
0 answers
77 views

The difference between totally (large) cocontinuous functors and small cocontinuous functors

$\newcommand{\cat}{\mathbf}\newcommand{\op}{\mathrm{op}}\newcommand{\Hom}{\operatorname{Hom}}\newcommand{\cSet}{\cat{Set}}$A category $\cat C$ is total if the Yoneda embedding $\cat C→[\cat C^{\op},\...
Dabouliplop's user avatar
  • 2,061
3 votes
0 answers
111 views

Localization of cocomplete categories and right orthogonality: does the equivalence always hold?

In Handbook of Categorical Algebra, Volume 1: Basic category theory, Borceux proves the following (around theorem 5.4.7 page 198). Definition. An object $x$ of a category $\newcommand{\cC}{\mathsf{C}}\...
Dabouliplop's user avatar
  • 2,061
3 votes
0 answers
62 views

Does the forgetful functor from the category of models of a cartesian theory preserve and create limits?

It is well-known that a monadic functor preserves and creates limits, which in particular shows that any algebraic category, or equivalently, any category of models of an algebraic theory, has limits, ...
Lingyuan Ye's user avatar
3 votes
0 answers
611 views

Direct limit, inverse limit and Spec

The set-up $k$ is a field $T_i = \operatorname{Spec} A_i$ is an inverse system of affine $k$-schemes, where $i<j$ if $\operatorname{Spec} A_j \subset \operatorname{Spec} A_i$ (inclusion). $X$ is a ...
rationalbeing's user avatar
3 votes
0 answers
64 views

Argumentation via "Limits are constructed object-wise"

How do the "limits are constructed objectwise thus a property about limits true in $\rm Set$ is also true in $ {\rm{Set}}^I$" argument works? For example, I encountered the following two ...
newbie's user avatar
  • 307
3 votes
0 answers
658 views

Inverse Limit of Complexs and Homology

Let $\mathcal{A}$ be an abelian category and denote by $D(\mathcal{A})$ its derived category. Let $K_n\in D(\mathcal{A})$ be an inverse system of complexes in $\mathcal{A}$ viewed as elements of the ...
curious math guy's user avatar
3 votes
0 answers
61 views

On extension of endofunctors to pro/ind-objects and preservation of (co)limits

Suppose that $\mathsf{C}$ is a cocomplete category and $F : \mathsf{C} \to \mathsf{C}$ is and endofunctor together with a natural transformation $\eta : \mathsf{C} \to 1$. Then, for any $x \in X$ one ...
qualcuno's user avatar
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3 votes
0 answers
194 views

Base point of BU/BO, classifying space of U/O.

There are principal bundles $$U(n) \to V_k(\mathbb{C}^n) \to G_k(\mathbb{C}^n)$$ and $$O(n) \to V_k(\mathbb{R}^n) \to G_k(\mathbb{R}^n),$$ where $V_k(\mathbb{F}^n)$ and $G_k(\mathbb{F}^n)$ are the ...
JohnDonski's user avatar
3 votes
1 answer
395 views

Using Yoneda's Lemma to show that limits are unique

I recall my professor mentioning that Yoneda's Lemma can be used to show that limits of a functor are unique up to isomorphism. Here is my attempt: Let $F:J\to\mathcal{C}$ be a functor and let $X$ ...
Anonymous's user avatar
  • 2,660
3 votes
1 answer
143 views

How can I prove the following isomorphism?

A profesor of mine said the following: Let $\mathbb{A}_{\mathbb{Q}}$ be the adele group of $\mathbb{Q}$. There exist a isomorphism of topological groups $$\frac{\mathbb{A}_{\mathbb{Q}}}{\mathbb{Q}}\...
HeMan's user avatar
  • 3,139
3 votes
0 answers
664 views

Inverse Limit of Finite Direct Product of Groups/TopSpaces

I’m wondering if the inverse limit of a finite direct product of groups $G_{i,j}$ or topological spaces $X_{i,j}$ is isomorphic to the direct product of the inverse limits of $G_{i,j}$ or $X_{i,j}$, ...
categoricalimperative's user avatar
3 votes
0 answers
187 views

Non-existence of pushout in homotopy category

I want to show that $S^1_{(0)}\leftarrow *\to S^1_{(1)}$ has no pushout in the homotopy category without using Eilenberg–MacLane spaces. In a first step, I want to show that if there is such a ...
FKranhold's user avatar
  • 759
3 votes
0 answers
224 views

General colimits and filtered colimits in the category of sets

A category $\mathsf{I}$ is filtered if $\mathsf{Ob(I)} \neq \varnothing$, for any $i,j \in \mathsf{Ob(I)}$ there is $k \in \mathsf{Ob(I)}$ and morphisms $f\colon i\to k$ and $g\colon j\to k$...
Jxt921's user avatar
  • 4,518
3 votes
0 answers
230 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 ...
Jxt921's user avatar
  • 4,518
3 votes
0 answers
32 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 ...
Noether's user avatar
  • 382
3 votes
0 answers
324 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. ...
Maxime Ramzi's user avatar
  • 43.9k
3 votes
0 answers
72 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}...
John Gowers's user avatar
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3 votes
0 answers
1k views

Limits in functor categories can be computed pointwise

Let $[\mathcal{C},\mathcal{D}]$ denote the category of functors $\mathcal{C}\to\mathcal{D}.$ My notes say that if $\mathcal{D}$ has limits of shape $I$, then the composite $$[I, [\mathcal{C},\mathcal{...
quasicoherent_drunk's user avatar
3 votes
0 answers
259 views

Topology on the Schwartz space over local fields and over the adeles

There is a standard way of endowing the Schwartz space $\mathcal{S}(\mathbb{R}^d)$ on $\mathbb{R}^d$ with a topology via semi-norms, turning it into a Fréchet space. Now let $F$ be a non-archimedean ...
m.s's user avatar
  • 2,238
3 votes
0 answers
69 views

$Pro$ objects as presheaves

Let $\mathcal{C}$ be a category, and $Pro(\mathcal{C})$ the category of small cofiltered limits $I \to \mathcal{C}$. Dually, we have the category $Ind(\mathcal{C})$ of filtered colimits in $\mathcal{C}...
Exit path's user avatar
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