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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Orbit functor is not co-representable

Let ${1}\neq H\le G$ be groups. Denote by $G\textit{-}\mathsf{Set}$ the category of sets with a $G$ action, with $G$-equivariant maps as morphisms. Let $(-)/H: G\textit{-}\mathsf{Set}\to \mathsf{Set}$ ...
Robert's user avatar
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colimit with two index category

I want to prove that colimit is commutative with colimit, i.e. $colim_{j}colim_{i}M_{i, j} = colim_{i, j}M_{i, j}$. But I'm a bit confused about how to define $colim_{i}M_{i, j}$? For a single $i$, ...
jhzg's user avatar
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2 votes
1 answer
80 views

limits and colimits under forgetful functor

I'm studying limits and colimits and more precisely I'm looking at forgetful functors and I'm trying to see if they preserve limits and colimits. In order to do that I first look at terminal and ...
bml64's user avatar
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Show that the pushout of this specific simplicial map induces an isomorphism on the $n$th simplicial homotopy group.

Let $n \geq 0$. Consider a pushout square in the category of simplicial sets as follows: $$\begin{array} ~ \Lambda^{n+2}_{i} & \stackrel{}{\longrightarrow} & X \\ \downarrow{} & & \...
udit.m's user avatar
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6 votes
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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
1 vote
2 answers
54 views

Internal hom takes coends to ends

I know that this is a very general fact about limits and colimits, but I would like to prove it directly for ends and coends. If $\mathcal V$ is a closed braided monoidal category, $V$ an object in $\...
Nikio's user avatar
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1 vote
1 answer
35 views

Coequalizer in the category of modules

I am trying to prove that the category of modules is cocomplete. It suffices to show that it has all coequalizers and coproducts. It's relatively easy to show that all coproducts exist, and I am left ...
Squirrel-Power's user avatar
2 votes
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118 views

Given an inverse sequence of functors determined on a subcategory, when is the limit determined on that subcategory?

I will first state the general version of my question, but I do have a specific context in mind in which second I'll dance around. (1.) Let $\mathsf{C}$ be a full subcategory of a category $\mathsf{D}$...
Eric's user avatar
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2 votes
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92 views

Spec of an infinite intersection of ideals, Spec of a colimit

This comes from the study of Krull's Intersection Theorem, and deriving a geometric meaning. Let $I \subset R$ be an ideal of a commutative ring (we shall see the case when $R$ is Noetherian). ...
metalder9's user avatar
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Proof verification: dimension of inverse limit of modules

I'm looking for clarification as to whether or not the following proof is valid. I am unsure in justifying the last paragraph. Thank in advance as always, M Claim: Let $R$ be a ring and $(M_{k} , q_{...
mathieu_matheux's user avatar
1 vote
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29 views

Rank preservation of inverse limit of modules

I am asking about the properties of rank over an inverse limit. Suppose we have a noetherian ring $R$ with uniformizer $\pi$ and noetherian ring of formal power series $R[[X]]$. Let $\mathscr{R}$ be ...
mathieu_matheux's user avatar
11 votes
2 answers
438 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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Prove that an (additive) functor $F$ between abelian categories (categories of modules) that admits an exact left adjoint must preserve injectives

Prove that an (additive) functor $F$ between abelian categories (categories of modules) that admits an exact left adjoint must preserve injectives. State and prove the dual result. I have no idea on ...
Squirrel-Power's user avatar
2 votes
1 answer
69 views

Interpretation of closure in inverse limit

Can one interpret the closure of a set inside an inverse limit as the closure of its individual components? I have not been able to find a source confirming or denying this claim. I have only been ...
mathieu_matheux's user avatar
1 vote
0 answers
117 views

Does profinite completion preserve injectivity?

Let $G$ be an abelian group. Let $\widehat{G}$ be a profinite completion of $G$. Profinite completion means a inverse limit of $G$ by a system given by homomorphisms $G/N\to G/M$ where $N$ and $M$ are ...
Pont's user avatar
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1 vote
1 answer
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Confusion about colimits in the category $\mathbf{Set}$

It is well known that $\mathbf{Set}$ is an $\aleph_0$-accessible category, but I'm very inexperienced and I'm not sure how to prove it in detail. In particular, I need to find a set $\Omega$ of ...
Petersu's user avatar
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3 votes
2 answers
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Difference between different definitions of diagram in a category

I'm currently reading the book "Topoi: The Categorial Analysis of Logic" by Robert Goldblatt, and in chapter 3.11, in order to define limits and co-limits he defines a diagram in a category ...
Eduardo Magalhães's user avatar
1 vote
0 answers
54 views

Non trivial colimit for rings in a finite diagram

I’m trying to understand the concept of colimits for commutative rings, but unable to find a colimit(or at least a compliment) for a finite diagram of rings, is there a(non trivial) example for a ...
Roye sharifie's user avatar
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0 answers
32 views

Sequence of direct summands of modules over a ring

Let $R$ be a ring and suppose that for every $n\in\Bbb Z$ we have a split exact sequence of $R$-modules: $$\{0\}\to E_{n+1}\xrightarrow{\varepsilon_n}E_n\xrightarrow{\pi_n}Q_n\to\{0\}$$ I claim that ...
Fabio Lucchini's user avatar
2 votes
2 answers
82 views

(Co)Products are bifunctors, but are general (co)limits also functors?

In a category with all products or coproducts, the (co)product operation can be understood as a bifunctor. More generally let $\mathcal{C}$ be a category with all limits of shape $D$, where for ...
Zoltan Fleishman's user avatar
1 vote
0 answers
20 views

Subtleties in commuting colimits

For context, I am reading Weibel's k-book and I am trying to express the homology of $BS^{-1}S$, the group completion of the classifying space of a symmetric monoidal category, as a colimit. In ...
DevVorb's user avatar
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2 votes
1 answer
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Is a canonical morphism from the wedge sum to the product monic? A section?

Let $C$ be a category with a terminal object $\ast$. For two pointed objects, define their wedge sum $c\vee d$ as the pushout $$ \require{AMScd} \begin{CD} \ast @>>c_0> c \\ @VVd_0V@VV\...
Milten's user avatar
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0 votes
1 answer
42 views

Colimits of full subdiagrams vs topological subspaces

Let $F:J \rightarrow \mathrm{Top}$ be a diagram in the category of topological spaces and $X:=\mathrm{colim} F$ be its colimit. For each $a \in \mathrm{ob}(J)$, denote by $\imath_a:F(a)\rightarrow X$ ...
Grabovsky's user avatar
  • 151
2 votes
1 answer
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$\mathcal{O}(U)$ as a projective limit of Hilbert spaces

It is well-known that the space of holomorphic functions $\mathcal{O}(U)$ (with the standard topology of compact-uniform convergence) on an open set $U \subset \mathbb{C}$ is a projective limit of ...
Invincible's user avatar
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-1 votes
1 answer
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Is there a direct limit in the category of rings for hypercomplex numbers [closed]

I recently learned about the concept limits in categories. From R we can construct C the H etc... by iterating the Cayley-Dickson construction. My question is: Can we construct a (non-associative)ring ...
Pielcq's user avatar
  • 27
2 votes
2 answers
84 views

Showing that the diagonal functor $\Delta:\mathbb{C} \to \mathbb{C} \times \mathbb{C}$ having a right adjoint implies $\mathbb{C}$ having products.

I started brushing up on my understanding of adjunctions and came across this well-known fact (rephrased in my own words): Let $\mathbb{C}$ be a category, and let $\Delta:\mathbb{C} \to \mathbb{C} \...
user11718766's user avatar
1 vote
0 answers
61 views

Inverse limit of a quotient space (simple question)

Setup: I have a tower of abelian groups $\hspace{1em} \cdots \to A_3 \stackrel{f_2}{\to} A_2 \stackrel{f_1}{\to} A_1 \stackrel{f_0}{\to} A_0$. There are similar towers for $B_i, C_i$, and $D_i$. There ...
June in Juneau's user avatar
1 vote
2 answers
86 views

About definition of direct limit

In the definition of direct limit of abelian groups (or modules or ...), one takes abelian groups $G_i$ ($i\in I)$ with a morphism $f_{ij}:G_i\rightarrow G_j$ with following conditions: for every $i\...
Maths Rahul's user avatar
  • 2,887
0 votes
1 answer
32 views

Direct and inverse limit after deleting some groups

Suppose we have a preordered set $(I,\le)$ and a sequence of abelian groups $\{G_i\}$ with a homomorphism $\alpha_{i,j}:G_i\rightarrow G_j$ if $i\le j$ in $I$. Let $G$ be the direct limit of this ...
Maths Rahul's user avatar
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2 votes
0 answers
132 views

Cech cohomology on infinite open cover commutes with colimit on Noetherian space? (Exercise 5.2.6 in Qing Liu's book)

This is Exercise 5.2.6 in Qing Liu's book Algebraic Geometry and Arithmetic Curve. In part (b), I can show (b) if the open covering has only finitely many open subsets, since the the colimit of the ...
Z Wu's user avatar
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0 votes
2 answers
60 views

Direct limit of a system: computation

I am considering a homomorphism from $\mathbb{Z}_4\rightarrow \mathbb{Z}_6$ given by $\bar{1}\mapsto \bar{3}$. My question is: What is the direct limit of this system in the category of abelian ...
Maths Rahul's user avatar
  • 2,887
4 votes
1 answer
63 views

A locally $\kappa$-presentable category is also locally $\lambda$-presentable for $\lambda>\kappa$? (Typo?)

In Riehl's Category Theory in Context, Sect. 4.6, we find the following: Definition 4.6.16. Let $\kappa$ be a regular cardinal.¹ A locally small category $\mathsf{C}$ is locally $\kappa$-presentable ...
Elías Guisado Villalgordo's user avatar
3 votes
0 answers
48 views

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
6 votes
3 answers
156 views

Are finite colimits closed under finite colimits?

Let $C$ be a cocomplete category and $S$ a set of objects of $C$. We may assume, if need be, that the objects of $S$ are compact. Consider $S'$ the class of objects spanned by finite colimits of ...
qualcuno's user avatar
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-1 votes
2 answers
87 views

Coequalizer that is not absolute

A coequalizer is called absolute when it is preserved by each functor. Could somebody give me an example of a coequalizer that is not absolute (with proof) ? If possible it would be great if the ...
Richard Southwell's user avatar
0 votes
0 answers
67 views

Is the category of graded modules over a graded-commutative ring an AB5 category?

Is the category of $\mathbb{Z}$-graded modules over a graded-commutative ring an AB5 category? It is abelian, the subobjects of each object form a set, and it admits arbitrary coproducts. But I don't ...
user829347's user avatar
  • 3,422
0 votes
0 answers
30 views

Proof verification computing pushout (i.e. colimit) of the diagram $\langle x \rangle \hookrightarrow D_{2\cdot n}, \langle x \rangle \to 1, |x| = n$

If you could verify my solution, that would be great. Compute the colimit of the following diagram. \begin{array}{c c c} \langle x \rangle & \xrightarrow{} & D_{n} \\ \downarrow & & \\ ...
love and light's user avatar
0 votes
1 answer
57 views

Prove that the forgetful functor $U: \textbf{Ab} \to \textbf{Set}$ preserves all filtered colimits

I realize that my question is exactly the same as this post here. However, I tried finding the book that was mentioned, Borceux's Handbook of Category Theory I, but my efforts to find the book here ...
love and light's user avatar
3 votes
1 answer
63 views

Reference for the alternative construction of the free cocompletion

Given any small category $\mathcal{C}$, the Yoneda embedding $y:\mathcal{C} \to \mathbf{Set}^{\mathcal{C}^{\operatorname{op}}}$ is well-known to represent the free cocompletion of $\mathcal{C}$. That ...
Geoffrey Trang's user avatar
2 votes
1 answer
95 views

Generalizing pullbacks

A cospan is two morphisms having the same codomain. A pullback is a limit of a cospan. If instead of having two morphisms, we have a set of morphisms having the same codomain, is there a name ...
Bruno's user avatar
  • 278
3 votes
0 answers
177 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
  • 289
4 votes
1 answer
114 views

Cartesian products in categories of subobjects

Let $\mathcal{C}$ be a category and $A$ be an object of $\mathcal{C}$. If the inclusion functor from the category $Sub_{\mathcal{C}}(A)$ of subobjects of $A$ (objects are monomorphisms of $\mathcal{C}$...
Bruno's user avatar
  • 278
1 vote
1 answer
69 views

Properties of the Projective Limit (in $\textbf{Set}$)

Let $I$ be an index set with a preorder $\leq$ and let $(G_i)_{i \in I}$ be a family of sets. Furthermore, for all $i,j \in I$ with $i \leq j$ let $f_{ij} \colon G_i \longrightarrow G_j $ be maps such ...
puck29's user avatar
  • 438
3 votes
1 answer
152 views

On kernels and stalks of sheaves.

Suppose we're given sheaves $F,G$ on a space $X$ and a morphism of sheaves (of abelian groups) $\phi:F\to G$. I want to prove two things : the presheaf $\ker \phi$, defined by $(\ker\phi)(U):=\ker(\...
t_kln's user avatar
  • 1,048
2 votes
0 answers
45 views

Representable ind-objects

Let $C$ and $I$ be categories and $I$ is filtered. Let $F$ be an inductive system indexed by $I$ in $C$. Then we have the ind-object $$X\to \varinjlim\limits_{i\in I} \mathrm{Hom}_C(X, F(i)),$$ which ...
Sergey Guminov's user avatar
2 votes
0 answers
26 views

2-pullbacks in 2-functor 2-categories

Let $\mathcal{C}$ and $\mathcal{D}$ be 2-categories, and let $[\mathcal{C}, \mathcal{D}]$ be the 2-category of 2-functors from $\mathcal{C}$ to $\mathcal{D}$. If $\mathcal{D}$ has 2-pullbacks, does it ...
User7819's user avatar
  • 1,621
3 votes
0 answers
34 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
  • 169
1 vote
1 answer
52 views

$F : I → C$ is a functor from a filtered category with any colimit. $ι : J → I$ is the embedding with $J$ cofinal. Then colim $F \cong$ colim($F ◦ ι$)

Let $I$ be a filtered category. We say a full subcategory $J$ is cofinal if for every object $A \in I$, there is an object $B \in J$ so that $\text{Hom}(A, B) \neq \emptyset$. Let $F : I \to \mathcal ...
love and light's user avatar
9 votes
0 answers
183 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
3 votes
1 answer
131 views

Writing $\operatorname{Spec}\mathbb{Z}$ as an inverse limit of finite $T_0$-spaces

I want to show that $\operatorname{Spec}\mathbb{Z}$ can be written as an inverse limit of finite $T_0$-spaces. First off, $\operatorname{Spec}\mathbb{Z} = \{(0), (2),(3),(5),...\}$, so the closed sets ...
IstEsOverFurMich's user avatar

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