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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References for Learning Inverse Limits (for Profinite Groups)

I'm doing some independent study on Profinite Groups this summer and, as I understand it, it is important to be familiar with the notion of an inverse limit before doing so. The trouble for me is that ...
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Question on a cone of an inverse system.

Let $(X_i,f_{ij})$ be an inverse system of objects and morphisms (say over $\mathbb{N}$) in a category $C$ and let $X$ in $C$ together with morphisms $\pi_i:X\rightarrow X_i$ be the inverse limit of $...
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The initial object in the index category corresponds to the limit [duplicate]

In page 33 of Pierre Schapira's Algebra and Topology, it is briefly mentioned that given functor $\alpha:I\rightarrow C$, if $i_0$ is initial in $I$, then there is an isomorphism $\lim_\...
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If one vertical arrow in a pullback is an iso, then so is the other

Consider the pullback square: $\require{AMScd}$ \begin{CD} A @>f>> B\\ @Vg VV @VV h V\\ C @>>j > D \end{CD} Suppose $h$ is an isomorphism. I'm trying to show that $g$ ...
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Do people use other limits than products, equalizers and pullbacks?

This is a somewhat vague question. I've seen several introductions to category theory, and when someone presents (co)limits, the typical examples are always (co)products, (co)equalizers and pullbacks/...
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Fundamental Group functor has no left Adjoint

I have a question about a remark done by Martin Brandenburg on Tyler Lawson's answer in this MO discussion: https://mathoverflow.net/questions/10364/categorical-homotopy-colimits/10399#10399 The ...
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Dualizing the definition of the kernel relation in $\mathbf{Set}$ as a pullback

Given a set function $f : A \rightarrow B$, my book notes that the kernel relation $R_f = \{ (a_1, a_2) | f(a_1) = f(a_2) \}$ along with the corresponding projections is a pullback: $$\require{AMScd} \...
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Hom functor preserves limits [duplicate]

I was looking to prove that $Hom_C(Y,limX) \simeq lim Hom_C(Y,X)$. I saw something in https://ncatlab.org/nlab/show/hom-functor+preserves+limits, but I didn't really understood the proof. Can someone ...
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Laurent series ring as ${\lim}^1$

Let $R_*$ be a graded ring concentrated in even degrees. I was presented a construction of $R_*((x))$, the ring of Laurent series in the variable $x$ with degree $-d$, as follows. For every $i \in \...
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Colimits in triangulated categories

Let $C$ a triangulated category and $(Y_a)$ a direct system of objects of $C$. Is it true that $colimY_a\cong \oplus Y_a$? My supervisor told me that he thinks this is not true, but didn't tell me ...
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Cofibrations in diagram category

Let $\mathcal{C}$ be a model category and $\mathcal{I}$ ba a small category. Then we have the projective model category structure on the diagram category $\mathcal{C}^\mathcal{I}$ where fibrations and ...
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Colimit of Disjoint Metric Spaces vs Topological Coproduct

Let $\{X_i\}_{i=1}^{\infty}$ be a sequence of pairwise disjoint metric spaces. Here we use the convention that a metric space can assume infinite distance. Let Met be the category with metric spaces ...
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Cosheaf on a base

There is the well-known construction of sheaves on a base, i.e. rather than specifying a sheaf $S$ on all open sets of a topological space $M$, specify its data only on a topological base $\mathcal{B}$...
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51 views

Equivalence of categories preserves (co)products

Let $A$ and $B$ be categories with (arbitrary) products and coproducts and $F : A \rightarrow B$ is an equivalence of categories, then $F$ preserves limits and colimits, hence $F$ preserves arbitrary ...
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Non-mechanical proof in direct limit (or colimit)

Let $I$ be a directed set and for the sake of simplicity, let us work with a directed system of modules over some commutative ring $A$ (although I am looking for an answer which can be extended to a ...
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33 views

Is an equalizer really a special case of a limit?

Let $I$ be a category with $2$ objects $A,B$ and four morphisms $1_A, 1_B, u,v$ where $u,v \in Hom_I(u,v)$. Define a functor $F: I \to \mathcal{C}$ by $$F(A) = X, F(B) = Y, F(u) = f, F(v) = g$$ My ...
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Creation of limits and diagram chasing

Let T be a monad in X. I want to prove that the forgetful functor G from category of T-algebras on X to X creates limits. I have read that it can be done via "diagram chasing". At this point, I am ...
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Uniqueness and existence of an infinite probability distribution extending finite probability distributions

Let $I$ be a (possibly infinite) set. To each finite subset $J$ of $I$, we associate a joint probability distribution $X_J$ whose sample space is $\mathbb R^J$ (i.e. the outcomes are $|J|$-tuples). ...
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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}^{\...
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eqaulizer is a monomorphism [duplicate]

How do I prove that an equalizer $e:E\to A \rightrightarrows B$ of a pair $f,g$ is a monomorphism ?
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a small technical detail

I would like to understand what it technically means that regular monomorphisms are stable under pushouts. And even yet more trivially what does it mean that $h:...
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Construction of a corner of a diagram out of the homotopy pushout

Let \begin{array}{ccc} X & \xrightarrow{} & Y \\ \downarrow & & \downarrow\\ Z & \xrightarrow{} & W\\ \end{array} be a homotopy commutative diagram in a proper model catgeory $\...
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38 views

A question about homotopy pushout

Let \begin{array}{ccc} X & \xrightarrow{} &Y \\ \downarrow & & \downarrow \\ Z & \xrightarrow{} & W\end{array} be a commutative diagram in a proper model catgeory and $P$ be ...
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100 views

Questions about pushout in a suitable model category

Let us consider a diagram in any suitable( proper) model category as follows: Where $X' \to X,$ $Y' \to Y$ and $Z' \to Z$ are weak equivalences and P and P' are the pushout of the diagram $Z' \...
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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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Final Topology Generated by Inclusion Map

Let $Y\subset X$ (strict inclusion) and let $\tau^Y$ be a topology on $Y$. Then, we may endow $X$ with the final topology induced by the inclusion (map) $i:Y\rightarrow X$. Question: Is there an ...
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What is the colimit of this diagram of binary sequences?

I'm working in the category $C$ of sets of (possibly but not necessarily infinite) binary sequences. Let $E_n$ be the object of $C$ consisting of sequences of all length $n$. I consider the diagram in ...
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66 views

Pullback of 2 functors one of which creates limits and the other preserves

I have been solving this problem and do not know what to do next and if I am thinking in the right direction. Suppose we have this pullback square and know that $H$ creates and $G$ preserves limits. ...
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Problem with HTT's definition of homotopy limits

In the appendix to HTT, Lurie defines homotopy limits in a combinatorial model category $\mathbf A$ as follows : he takes $f: C\to C'$ a functor between small categories, and defines $f^* : Fun(C',\...
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Empty projective limit of finite sets

Let $I$ be a preordered set and $(A_i)_{i\in I}$ be a collection of non-empty finite sets. I want to find a pair $(I,(A_i)_{i\in I})$ such that the projective limit $\varprojlim A_i$ is empty. I know ...
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Searching for inverse limit candidates-initial verification.

Let $(I,\leq)$ is some directed poset and let us assume that we are in the category of rings. We know that, in the category of rings, we can always construct the inverse limit of a given inverse ...
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Inverse limit of rings of polynomials in many variables [duplicate]

I am trying to understand inverse limits and so I decided to try and compute one of such limits. We are going to work in the category of rings. Let $R$ be a ring. First, I set up an inverse system. ...
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Comma category: completeness

Let $C$ be a category and $c\in C$ an object. What are the necessary and sufficient conditions for the comma category $$(c\downarrow C)$$ to be (co)complete ?
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Limits via universal arrows and functor categories

I would like to understand is some detail the connection between the 2 snippets taken from McLane's book CWM. Namely, I do not follow the connection between the functor $S$ and categories and arrows $...
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1answer
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Weighted limits in ordinary category theory are ordinary limits from the category of elements

In this question, the OP says that In $\mathsf{Set}$-enriched category theory, one can say that the limit of $\mathbf{J} \xrightarrow{D} \mathscr{A}$ weighted by $\mathbf{J} \xrightarrow{W} \...
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1answer
84 views

Dieudonné module associated to the dual of a $p$-divisible group

Let $k$ be a perfect field of characteristic $p>0$, and consider $X=(X_m,i_m)$ a $p$-divisible group of height $h$ over $\operatorname{Spec}(k)$: it is an inductive system where $X_m$ is a finite ...
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Construction of diagram for given pushout

Let $\mathcal{C}$ be a proper model category and $P$ be the pushout of the diagram $Z \leftarrow X \rightarrow Y$ in $\mathcal{C}.$ Now consider $P' \in \mathcal{C}$ such that there is an weak ...
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1answer
54 views

Does Direct limit/union of subspaces commute with sheaf cohomology

Let $X$ be a topological space and $\mathcal{F}$ an abelian sheaf on $X$. Furthermore let $0=X_0 \subset X_1 \subset X_2 \subset \ldots$ be an increasing sequence of subspaces of $X$ such that $X=\...
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226 views

Show that $\Gamma$, $\Lambda$, and the associated sheaf functor are all left exact.

This is Exercise II.6 of Mac Lane and Moerdijk's, "Sheaves in Geometry and Logic [. . .]". According to the first few pages of this Approach0 search, it is new to MSE. The Details: The functors $\...
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A presheaf $P$ on $X$ is a sheaf iff for every covering sieve $S$ on an open set $U$ of $X$ one has $PU=\varprojlim_{V\in S}PV.$

This is Exercise II.2 of Mac Lane and Moerdijk's, "Sheaves in Geometry and Logic [. . .]". The Details: Adapted from Adámek et al.'s, "Abstract and Concrete Categories: The Joy of Cats", p. 48 . . ....
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canonical isomorphism $Y\times_{Y\times_Z Y}X\times_Z X\overset{\sim}\longrightarrow X\times_Y X$

How is the canonical isomorphism defined here ? $Y\times_{Y\times_Z Y}X\times_Z X\overset{\sim}\longrightarrow X\times_Y X$ additionally given are the maps $f:X\to Y$ and $g:Y\to Z$ both being ...
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Having binary products, terminal object, and equalizers implies having finite limits

The answers to this questions prove that if a category has all products and equalizers, then it has all limits. How to modify that proof to show that if a category has binary products, a terminal ...
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82 views

If $\mathscr A$ has all products and equalizers, then it has all limits

This question is about part (a) of this proposition: Here is a plan of the proof. Here's what the picture looks like, from what I understand: But I don't understand what the condition $s\circ p=t\...
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The set $5^{-\infty}\mathbb{Z}$ is a colimit

I am trying to understand why the set of rational numbers whose denominators are powers of $5$, $5^{-\infty}\mathbb{Z}$, is a colimit. Specifically, why is $5^{-\infty}\mathbb{Z}$ the colimit of the ...
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1answer
97 views

Preservation of weak pullback

A weak pullback is defined in the same way as a pullback, but the arrow to the vertex of the limit cone is not required to be unique. Here's the problem: Let $\mathscr P:\mathbf {Set}\to\mathbf{...
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55 views

An arrow is monic iff the square is a pullback

Here's a lemma: Is the following proof correct? Suppose the square is a pullback. Then for all objects $Z$ and arrows $\alpha,\beta: Z\to X$ such that $f\alpha=f\beta$, there is a unique $\Gamma:Z\...
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1answer
54 views

Proving that this set is a limit in $\mathbf{Set}$

As a continuation of this question, I'm trying to prove that $$L=\{(x_I)_{I\in \mathbf I} : x_I\in D(I)\text{ for all } I\in\mathbf I \text { and } (Du)(x_I)=x_J \text{ for all } u:I\to J \text{ in } \...
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1answer
54 views

Proving that $p_I\circ h=p_I\circ h'$ implies $h=h'$

An exercise from Leinster: I was wondering if my solution is correct? (a) Assume $h,h':A\to L$ are arrows such that $p_I\circ h=p_I\circ h'$ for all $I$. Define $f_I=p_I\circ h$, as shown on the ...
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1answer
24 views

Homotopy cofinality of $\Delta^{op}$ in $\Delta^{op}\times \Delta^{op}$

There is the usual diagonal inclusion $i:\Delta^{op}\to\Delta^{op}\times \Delta^{op}$ which is easily seen to be cofinal in the $1$-categorical sense, and so one can compute colimits on $\Delta^{op}\...
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43 views

The forgetful functor $U:\mathbf{B}G\to\mathbf{Sets}$ need not preserve infinite limits.

This is Exercise I.7 of Mac Lane and Moerdijk's, "Sheaves in Geometry and Logic [. . .]". Here $\mathbf{B}G$ is the category of all continuous $G$-sets, where $G$ is a topological group. The ...

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