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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Limits in functor categories

Let $C,C’,D$ be categories and $u:C\to C’$ be a functor. The functor $u^*:\mathbf{Hom}(C’^\circ,D)\to\mathbf{Hom}(C^\circ,D)$ that sends a functor $G$ to $G\circ u$ commutes with limits and colimits, ...
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Example of non-filtered colimit not commuting with a finite limit

I know that the canonical morphism $\mathrm{Colim}_i\mathrm{Lim}_jD(i,j)\to\mathrm{Lim}_j\mathrm{Colim}_iD(i,j)$ for a diagram $D:\textbf{I}\times\textbf{J}\to \textbf{Set}$ is not in general an ...
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Sums factor as products in internal hom functor

In case this is helpful, I am working in the category of condensed abelian groups, but I think this can be phrased in a more general categorical way. Suppose we have a closed symmetric monoidal ...
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Does pushout of schemes along formal neighborhoods exist in the category of schemes?

I have a question about gluing specific types of schemes which doesn't fit into any well-known gluing situation. Assume $C$ is an algebraic curve and $p$ a point on it. The formal completion of $C$ at ...
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Why are there so many different notations for limits and colimits?

My question is relatively simple: Why are there so many different notations for limits and colimits? Is there a reason people don't just use a common convention? Is there any benefit to using one ...
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1answer
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On a definition of Spivak's fuzzy set

In the paper "Metric Realization of Fuzzy Simplicial Sets" of David Spivak it takes $I=(0,1]$ as poset and consider it as a category. He gives it a Grothendieck topology induce it from ...
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Homotopy push-out squares and exact triangles are colimits

I read somewhere that homotopy push-out squares and exact triangles in a triangulated category can both be interpreted as special cases of higher categorical colimits. Why is this true? Please note ...
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Inverse limit without category theory

In Hirsch's book Differential Topology, he by and large does not use any category theory, with the exception of one passage on pg. 52 which I am trying to understand. It is as follows (paraphrased): ...
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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 ...
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Right exactness of projective systems

Suppose that we have a systems of exact sequences $(A_{n}\rightarrow B_{n}\rightarrow C_{n}\rightarrow 0)_{n\in \mathbb{N}}$ together with transitions maps $(A_{n+1}\rightarrow A_{n})_{n\in \mathbb{N}}...
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54 views

A directed inverse limit of finite connected spaces is connected

Apparently, a directed inverse limit of finite connected spaces is connected. Does anyone have a reference?
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General Properties of Direct Limits of Groups

Anyone know a decent reference on the basic theory of direct limits of groups, from an elementary (meaning group theoretic, as opposed to categorical) perspective? Finitely generated abelian groups ...
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Calculating finite inverse limits of Abelian groups

If we are given an infinite system of abelian groups $(\dots \xrightarrow{\varphi_2} A_2 \xrightarrow{\varphi_1} A_1 \xrightarrow{\varphi_0} A_0)$ then I know its inverse limit can be found by $$\lim_{...
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Presheaves are the Free Cocompletion - Proving that the functor preserves colimits

I am trying to understand a proof that, for any small category $\mathcal{C}$, the category $\widehat{C} = [\mathcal{C}^\mathrm{op}, \textbf{Set}]$ is the free cocompletion of $\mathcal{C}$. In ...
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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 ...
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1answer
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Define a sketch $s_{\mathbf{Grp}}$ such that $\mathbf{Grp}\backsimeq \mathbf{Mod}(s_{\mathbf{Grp}},\mathbf{Set})$

I have the following (a) Define a sketch $s_{\mathbf{Grp}}$ and a equivalence functor $$E: \mathbf{Grp}\to \mathbf{Mod}(s_{\mathbf{Grp}},\mathbf{Set})$$ (b) Knowing that finite limits commute with ...
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finite limits and filtered colimits in the category of groups

I have the following problem Describe the finite limits and the filtered colimits in the category $Grp\ $ and show that they commute. This is my first "more practical" exercise on ...
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Directed system by inclusion.

My book[Algebra - Lang, pg no 161] says the following, " Let $x$ be a point in a Topological space $X$. The Open neighborhood form directed system by Inclusion.", It went on saying that &...
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$\ker \varphi_p \subset (\ker \varphi)_p$ where $(\cdot)_p$ is taking the stalk of sheaves at the point $p$ (Diagram inside!).

I've already proven that $(\ker \varphi)_p \subset \ker \varphi_p$ using a commutative diagram and the definition $F_p = \lim\limits_{\longrightarrow \\ U \ni p} F(U) = \bigsqcup\limits_{U \ni p} F(U)/...
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Strange inverse limit of infinite direct sums $S_n$

Let $p_j$ be the prime numbers in order, indexed by $j$. Define $S_n$ as follows: \begin{equation} \begin{aligned} S_1 &:= \mathbb Z / 2 \mathbb Z &\oplus &\mathbb Z / 2 \mathbb Z &\...
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Limits in the localization of a category of fibrant objects

Suppose we have category $\mathcal{C}$ which has the structure of a category of fibrant objects, and suppose we have a functor $F:I\to \mathcal{C}$ with a limit $\lim F$ in $\mathcal{C}$. If we have ...
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Stable homotopy groups as a generalized (reduced) homology theory

It is known that $\pi_*^{st}$ defines a generalized homology theory, where $\pi_n^{st}(X) = \text{colim}_{k \geq 0} \pi_{n+k}(\Sigma^k X)$ is the $n$th stable homotopy group of the based space $X$. ...
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Segre Embedding and Group Actions

The segre embedding takes two schemes $X, Y$ with $G_m$ and (I think) produces an embedding $X/G_m \times Y/G_m \hookrightarrow (X \times Y)/G_m$, with $G_m$ acting diagonally on $X \times Y$. My ...
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Colimit of a (relatively) complicated diagram

I want to compute the colimit of the following diagram, where $A$ is some set. I was also wondering about how to approach these complicated diagrams in general. Thanks!
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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 ...
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Limit in $\mathbb{Z} / P^{2} \to \mathbb{Z} / P $

Consider the commutative diagram in Ab, where the morphisms are the canonical ones. Let us denote the top horizontal sequence by A and the bottom one by B. Then the vertical maps induce a morphism of ...
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1answer
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Inverse limit of $\left(\mathbb{Z}/p^n\mathbb{Z}\right)_{n \in \mathbb{N}}$

In the wikipedia article about the inverse limit it is stated that for a prime number $p$ $$\varprojlim_{n \in \mathbb{N}} \mathbb{Z}/p^n\mathbb{Z} = \mathbb{R}/\mathbb{Z},$$ where the arrow between ...
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32 views

Bounded lattice that's isomorphic to proper subcategories of itself

Suppose I have a bounded lattice $L$ and a full subcategory $C\subseteq L$ containing exactly the elements of $L$ that are both join prime and meet prime. Now suppose $C$ can be partitioned into $n>...
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1answer
36 views

Find the product and coproduct of the category of Set with a given set

I am learning Category theory and I've found a problem : Let $S$ be a fixed set. Define a category $\textbf{Set}_S$ , where collection of object is a set map $ f: X \rightarrow S$. Let $f':X' \to S$ ...
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34 views

Vanishing direct limit of vector spaces

Let $k$ be a field. Let $\{V_n\}_{n\in \mathbb{N}}$ be a direct system of (possibly infinite-dimensional) $k$-vector spaces. Is it true that if $\varinjlim_{n\in \mathbb{N}}V_n=0$, then $V_m=0$ for ...
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When is a bounded lattice isomorphic to some $(\mathbf{2}^S,\subseteq)$?

If I have a set $S$, I can turn it into a bounded lattice in any number of ways, the simplest of which is to designate an initial object and a terminal object, adding a single object if necessary. ...
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1answer
55 views

Inverse limit of (sub)sets

Let $(X_i)_{i\in I}$ be a family of subsets $X_i\subset X$ partially ordered by inclusion. If $X_j\subseteq X_i$ let $\iota_{ij}\colon X_j\hookrightarrow X_i$ be the inclusion and write $i\le j$. This ...
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5answers
112 views

A category which direct limits but no general colimits

I am looking for a (at best, real life) category that has direct limits, but no general small colimits, or a category that has inverse limits, but no general small limits. Are there any interesting ...
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Composition of $h_1\cdot h_0$ [closed]

How can I see both directions for the case $\lambda=2$ below ? A picture would help.
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90 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 ...
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Generalization of $\lambda$-directedness

It is well known that objects in $\mathbb{Set}$ whose hom-functors preserve $\lambda$-directed colimits,i.e. colimits whose schemes are $\lambda$-directed posets are those sets whose cardinality is $&...
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Embedding the vertex of a cone inside inverse limit

Let $(R_i,f_{ij})_{i \in I}$ be an inverse system of rings and ring homomorphisms. Let $(L,p_i)_{i \in I}$ be the inverse limit of $(R_i,f_{ij})_{i \in I}$ and let $(K,f_i)_{i \in I}$ be a cone into $...
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Solving a (fun!) coequalizer problem for $\mathrm{SL}_n(\mathbb{R})\rightarrow\mathrm{SL}_n(\mathbb{C})$ in $\mathbf{Grp}$

First off, the problem posed below is mostly arbitrary; it's just for my own education. (And maybe for yours, as well.) It's fairly clear to me what the (co)equalizers of abelian groups in $\mathbf{...
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Colimit of coproduct

Let $g := \underset{c\in ob(\mathcal{C})}{\coprod} \frac{\mathcal{F}(c)}{\sim}$ for a small category $\mathcal{C}$ and $g \in Top$ where $\sim$ is the equivalence relation $x\sim\mathcal{F}(f)x$ for $...
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When must a specific product map exist?

Suppose that it's unknown whether a category $\mathcal{C}$ contains all products, but that it happens to have the products $a\times b$ and $c\times d$. Further, $\mathcal{C}$ has morphisms $f:a\...
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Why $S^n$ is the pushout of the inclusion $S^{n-1} \rightarrow D^n$?

What will be the pushout for the following : where $i:S^{n-1} \rightarrow D^n$ is the inclusion of the boundary $S^{n-1}$ to the n-disk $D^n$. According to Pg 40 in Julia E. Bergner's The Homotopy ...
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Computing $\Bbb F_p[t]^{perf}$

This is the second example of 1. in Ex. 2.0.3 of Bhatt's notes in perfectoid space. We define $R^{perf}:= \varprojlim ( \cdots R \xrightarrow{\phi} R)$ where $\phi$ is the Frobenius map. He claims ...
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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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1answer
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How do I see that the map $X^n \to X^{n+1}$ of CW-building blocks is an embedding?

In his book “A concise course in algebraic topology”, May defines a CW complex inductively as being the union of increasing subspaces $X^n$, where $X^0$ is discrete $X^{n+1}$ is the simultaneous ...
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1answer
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Every $R$-module is an iterated colimit of $R$

Let $R$ be a commutative algebra over a field $k$. My problem set asks me to Show that every $R$-module is an iterated colimit of $R$. My idea Let $A$ be a free $R$-module with basis $M$, and let $\...
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Is $M_\infty \simeq M_\infty \otimes M_\infty \simeq M_n \otimes M_\infty$?

All matrix rings are over the integers, colimits are colimits of $\mathsf{Rng}$s and $\otimes$ means $\otimes_{\mathbb{Z}}$. My guess is that yes, indeed, and my reasoning is as follows: since $M_\...
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2answers
121 views

Whats wrong with this argument that $\operatorname{Spec}(\prod A_i) = \bigsqcup\operatorname{Spec}(A_i)$ infinite product.

We have the spec functor $\text{CRng}^\text{op} \rightarrow \text{Aff}$.$\DeclareMathOperator{\Spec}{Spec}\DeclareMathOperator{\Hom}{Hom}$ Then $$\Hom _{\text{Aff}}(\Spec(\lim A_i), \Spec B) = \Hom_{\...
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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 ...
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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 $(-\frac{1}{n},\frac{1}{n})$. $...
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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/...

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