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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When representables are adjoints

Let $\mathbf C$ be a complete category and let $U:\mathbf C\to\mathbf{Set}$ be a representable functor. Show that $U$ preserves limits. In general representables preserve limits, but the hypothesis ...
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Are the cylinders in the sketch for the tensor product of two $\mathcal F$-theories colimiting cylinders?

In section 6.5 of Basic Concepts of Enriched Category Theory Kelly writes If $\lambda : F \to \mathcal A(T-,M)$ is a cylinder in $\mathcal A$, where $F : \mathcal L^{op} \to \mathcal V$ and $T : \...
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When $\mathrm{Hom}$ functor commutes with colimits in a category of modules? [duplicate]

I was looking through this question, and there's a thing I don't understand why it holds. I mean the next statement in the answer by @Peter McNamara: Since $R$ is Noetherian, $I$ is finitely ...
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3 votes
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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},\...
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2 votes
1 answer
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Understanding a product of objects as a limit of a discrete diagram

I have read that a product of objects $\{A_i\}_{i\in I}$ in a category $\mathcal{A}$ can be defined as a limit of a discrete diagram i.e. a diagram $D:\mathcal{I}\to\mathcal{A}$ where the only ...
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1 answer
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Canonical morphism from coproduct to product in a pointed category

Suppose we have a pointed category $\mathcal{A}$ and a collection $\{A_i\}_{i\in I}$ of objects which have both a product $\prod_iA_i$ and a coproduct $\coprod_iA_i$. The product has projections $\...
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3 votes
1 answer
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Why should the target category of a sheaf be complete?

The categorical definition of sheaf given on Wikipedia (https://en.wikipedia.org/wiki/Sheaf_(mathematics)#Complements) requires that the target category $\mathcal{C}$ of the functor $\mathcal{F}:\...
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Compute $\operatorname{Ext}^{1}_{\mathbb{Z}}(\mathbb{Q},\mathbb{Z})$

I know that $\operatorname{Ext}^{1}_{\mathbb{Z}}(\mathbb{Q},\mathbb{Z})=\prod_p({\widehat{\mathbb{Z}_p}/\mathbb{Z}})$. But the following steps took me a long time to figure out what was wrong with ...
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Definition of limit by nLab

Let $d:\mathbf I^{\text{op}}\to \mathbf{Set}$ be a functor, with $\bf I$ a small category. Set $t:\mathbf I^{\text{op}}\to \mathbf{Set}$ the (constant) functor sending every object to the singleton. I ...
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Applications of bounded cocomplete semilattices

Call an ordered set a poset when the ordering is transitive and antisymmetric. Call a poset bounded when it has a top and a bottom element (i.e. a greatest and least element). Call a poset cocomplete ...
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The open $n$-cells of a CW-complex are open in the CW-complex

I am trying to prove that if $X$ is a CW-complex and $e_\alpha^n$ is an open $n$-cell, then $e_\alpha^n$ must be open in $X$. For do so, lets recall the following definitions: Definition 1. Let $\{...
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Definition of finitely generated module versus finite type in category R-Mod

We say an object $X$ in category $C=R$-Mod is of finite type if for any functor $F: I \rightarrow C$ with $I$ a directed poset, the natural map $$\underrightarrow{\lim} Hom_{\mathcal{C}}(X,F(i))\to ...
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Spec commutes with cofiltered limits in concrete example

Currently I am learning some alg. geometry and I would like to show the following claim: Let $\mathfrak{p}$ be some prime ideal of $A$. Then $$ \varprojlim_{f\notin \mathfrak{p}} \operatorname{Spec}...
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2 answers
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Is the limit of a family of sheaves a sheaf?

So, I can prove that the kernel of a morphism of sheaves or a product of sheaves is a sheaf, but I do not know how to prove in general that $lim F_{i}$ is a sheaf for $F_{i}$ sheaves. I know that if ...
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1 answer
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Showing the limit of the discrete category defines the usual product.

How can we see that the categorical limit of the diagram $F$ from the discrete 2 object category to sets is isomorphic to the usual product $\{(x,y)\mid x\in X, y\in Y\}$ defined elementwise on sets. ...
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2 answers
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Limits in the category of topological spaces must come equipped with the initial topology

$ \newcommand{\cat}[1]{\mathsf{#1}} $The title says it all. Let $ \cat J $ be a small category and let $ D $ be a $ \cat J $-indexed diagram of topological spaces. Suppose a limit $ (L,\tau_L) $ of ...
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3 votes
3 answers
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Constructing the topological product space from categorical universal property

Let $X,Y$ be topological spaces. I want to construct the categorical product, i.e. the product space, $X\times Y$ using the universal property of products. To be clear, I don't want to construct the ...
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absolute coequalizers: what do they commute with

Split co-equalizers are preserved by any functor. What can I say on this basis about which limits it commutes with ?
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2 votes
1 answer
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Is the structure presheaf of $\text{Spec}(A)$ a limit of presheaves?

In Shafarevich's Basic Algebraic Geometry, he defines the structure presheaf on $\text{Spec}(A)$ first by $\mathcal{O}(D(f))\cong A_f$ and then $\mathcal{O}(U)=\lim_{D(f)\subset U}\mathcal{O}(D(f))$, ...
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Does the (pseudo)functor that assigns a commutative monoid $M$ to the topos of $M\text{-Sets}$ preserve limits? [closed]

Let $\mathrm{CMon}$ be the category of commutative monoids, and $\mathrm{Topos}$ be the bicategory of (Grothendieck) toposes with geometric morphisms. Consider the (pseudo)functor $\mathrm{CMon}\to\...
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Existence of Limits in Category theory in Roman's book

I am self-reading An Introduction To The Language Of Category by Roman and having some questions about the set-up for proving the existence of limits. In the book, the hom-classes are assumed to be ...
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2 answers
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Counterexample: existence of categorical product of family does not imply existence of categorical product of subfamily [duplicate]

Assumptions/context: Let's say we have some objects $X_i$ in some category, with $i$ belonging to some index set $\mathbf{I}$, such that there exists some object, call it $X_{\mathbf{I}}$, in the ...
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1 answer
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Are singleton sets nullary products or unary products in the category of sets? (Or are they both nullary and unary simultaneously?)

I understand why terminal objects are "nullary products" in any category, and I understand why singleton sets are terminal objects in the category of sets. So I understand why singleton sets ...
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1 answer
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Simpler description of pushout in $\mathsf{Set}$ and $\mathsf{Top}$ when one of the two maps is injective

It is known that the pushout in $\mathsf{Top}$ has same underlying set as the pushout in $\mathsf{Set}$ (this is because the forgetful functor $\mathsf{Top}\to\mathsf{Set}$ is a left adjoint). ...
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1 vote
0 answers
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Colimits of a diagram over an ordinal number

The context for my question is the following: The class $l(F)$ of morphisms which have the left lifting property with respect to $F$ is stable under transfinite compositions. I was able to prove that $...
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2 votes
1 answer
86 views

Where does the map $X_1\times_ZX_2\rightarrow Y\times_ZY$ come from in the Magic Diagram? Why is it commutative?

Given a morphism $Y\rightarrow Z$ and morphisms $X_1\rightarrow Y$, $X_2\rightarrow Y$, the magic diagram is given by the diagram The question I have is about the map on the right. There are three ...
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1 vote
1 answer
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Adjunctions b/w constant diagram functor and limit/colimit functors for fixed index category

Let $\mathcal{C}$ be a locally small category and let $\mathcal{J}$ be a small category. Assume that $\mathcal{C}$ has all $\mathcal{J}$-shaped limits and colimits. Describe the unit and counit for ...
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2 votes
3 answers
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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 ...
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Inverse limit of projections of a submodule

Suppose we have an inverse system $(M_i,\phi_{ij})$ and suppose $M=\lim_{i}M_{i}$ and $N$ is a submodule of $M$. Let $\pi_i\colon M\to M_i$ be the projections and let $N_i=\pi_i(N)$. What are ...
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1 vote
1 answer
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Is the unit compact in a closed monoidal (co)complete category?

Let $\mathcal{M}$ be a complete and cocomplete closed symmetric monoidal category, with unit $I.$ Does the functor $\text{Hom}_{\mathcal{M}}(I,-)$ always preserve filtered colimits? A reference or a ...
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4 votes
1 answer
166 views

Constructing counit in adjoint functor theorem for total categories

The theorem I am referring to is, Let $C,$ $D$ be locally small categories. Assume $C$ is a total category (i.e. the Yoneda functor $Y : C \to \operatorname{PreSh}(C)$ has a left adjoint $Y^L$). Let $...
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2 votes
0 answers
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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}}\...
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7 votes
2 answers
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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 ...
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1 vote
1 answer
73 views

Left Kan Extension along identity?

$\newcommand{\cSet}{\mathsf{Set}}$ I'd like a verification of my computation of the Left Kan Extension of a functor along the identity functor. Consider the group $G \equiv \mathbb Z/2\mathbb Z \equiv ...
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3 votes
1 answer
80 views

Forgetful functor from the slice category creates limits

I am reading "Category theory in context" and I'm having difficulties with proposition 3.3.8, top of page 92, pdf. Theorem: The forgetful functor $F : c/C \to C$ strictly creates all limits ...
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4 votes
0 answers
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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 ...
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Notation for limit in category theory

Really quite a silly question, but I'm confused about the notation around limits in category theory. Given a diagram of shape $I$, say $F$, I would write the limit of $F$ in some category as $\...
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2 votes
1 answer
39 views

Products in quotient category

Suppose $\mathcal{C}$ is a category, $\sim$ is a congruence relation on $\mathcal{C}$, and $[-]: \mathcal{C} \to \mathcal{C}/{\sim}$ is the quotient map. I'm able to prove that if $\mathbf{0}$ is ...
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3 votes
0 answers
38 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, ...
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2 votes
0 answers
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Notation: Interpolation between functors in extension/lifting problems with simple categories

In chapter 6 on homotopy (co-)limits of Jeffrey Strom's Modern Classical Homotopy Theory he gives the following definition on p. 156 (I'm explaining the terminology at the bottom): [Let $\mathscr{C}$ ...
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0 votes
2 answers
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Prove the forgetful functor $U: Grp \to Set$ preserves limits

Let $F: A \to B$ be a functor. F preserves limits iff any for any diagram $ D: I \to A$ s.t. $\operatorname{lim}_I D $ exists, $\operatorname{lim}_I FD$ exists and $\operatorname{lim}_I FD \cong F( \...
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1 vote
1 answer
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Why is the limit the product, but not the coproduct?

On page 489 of Algebra, Paolo gives an example of a limit: Let $I$ be the discrete category with 2 objects with only identity morphisms, let $\alpha$ be a functor from $I$ to any category $C$, and let ...
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2 votes
1 answer
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Limits/Colimits as representing objects and size of functor category

Let $J,C$ be categories and $T\in C$, we define the functor $\Delta_J(T):J \to C, j \to T $ to be the constant $T$-valued $J$-diagram. Let $F:J \to C$ be a functor, my professor defined the limit of $...
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1 vote
1 answer
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Equivalent definitions of weighted limits

Consider a symmetric monoidal closed, complete and cocomplete category $\mathcal{V}$. Let $\mathcal{A,C}$ be $\mathcal{V}$-categories and $\mathcal{D}:\mathcal{A} \rightarrow \mathcal{C}$, $\mathcal{W}...
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0 answers
10 views

Inverse limit and crossed product

Let $p$ be a prime and $H$ be a uniform, pro-$p$ group. Then the Iwasawa algebra $\mathbb{F}_p[[H]]$ can be seen as the $I$-adic completion of the group algebra $\mathbb{F}_p[H]$ for $I$ the ...
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1 vote
1 answer
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Do ample sheaves descend along limits / noetherian approximation?

Suppose $X \to S$ is a proper (projective) morphism of schemes, $S$ is quasi-compact and quasi-separated, and $\mathcal L$ is a relatively ample sheaf on $X$. By stacks, tag 01ZA, the scheme $S$ is a ...
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0 votes
0 answers
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Proving $\operatorname{dir.lim}(M_i\otimes_{A_i}N_i)\cong \operatorname{dir.lim} M_i\otimes_{\operatorname{dir.lim} A_i}\operatorname{dir.lim} N_i$

If $\left ((G_i)_{i\in I},(\varphi_{ij})_{i\le j}\right )$ is an inductive system of abelian groups then its limit $G=\varinjlim G_i$ is the abelian group whose underlying set is the quotient of the ...
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1 vote
1 answer
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Doubt about the colimit of a sequence of topological spaces

Let $X_1\stackrel{j_1^2}{\hookrightarrow }X_2 \stackrel{j_2^3}{\hookrightarrow}\dots$ be a sequence of topological embeddings (i.e., $j_n^{n+1}$ is s sequence of injective continuous map such that $...
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  • 493
1 vote
1 answer
67 views

Direct limit of $\mathrm{Hom}$

Let $E$ be a module over a ring. Let $\{ M_i \}$ be a directed family of modules. If $E$ is finitely generated, then the natural homomorphism $$\varinjlim \mathrm{Hom}(E, M_i) \to \mathrm{Hom}(E, \...
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3 votes
2 answers
50 views

Inverse limit of a bi-inverse system and limit of the "diagonal" inverse system.

I've a question concerning inverse limits, since I don't usually work with them this extensively. I'm considering the inverse limit of the following "bi-inverse system" of $R$-modules and ...
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