# Questions tagged [limits-and-colimits]

For questions on limits and colimts in the sense of category theory, and related notions.

171
questions

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### The Stone-Čech compactification of a inverse system

The Stone-Čech compactification of the inverse limit of an inverse system $%
\left\{ X_{i},f_{ij},I\right\} $ of Tychonoff spaces is the limit of the
inverse system $\left\{ \beta X_{i},\beta f_{ij},I\...

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75 views

### Is the largest colimit in a category a special object? [closed]

In a finitely presented category, you could rank the colimits according to the number of objects in their diagram. This means that there is a notion of largest or maximal colimits. Are these largest/...

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161 views

### Colimits in the category of (not necessarily locally convex) topological vector spaces

Are colimits in the category of (not necessarily locally convex) topological vector spaces (over R, C, respectively) exist in general?
If no, is there a well-known condition of when they exist?
If ...

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178 views

### Zero in colimit of sheaves category

This question is motivated by showing that the category $\mathbf{Sheaves} (X)$ from the open subset excluding the empty set category to the category of abelian group $\mathbf{Ab}$ has enough injective ...

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89 views

### Need of filtered indexed categories

Similar questions have already been asked here and here but not exactly in the direction I need.
I have a (small) index category $\mathcal{I}$ which is not cofiltered, and I need to consider ...

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105 views

### Most general context where a “disjoint sum” definition of a direct limit is applicable and always exist

I am a bit out of my element here so I'm hopefully not saying something stupid.
Anyways, wikipedia gives two ways to define direct limits, one for "algebraic structures" and one for general ...

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**1**answer

201 views

### Higher-dimensional version of the “Magic Cube Lemma” for homotopy pushouts/pullbacks

The "Magic Cube Lemma" is a surprising (to me) relationship between (homotopy) pushouts and (homotopy) pullbacks of spaces:
Consider a cubical diagram $I^3\to \mathcal{S}$ in the $\infty$-category of ...

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**1**answer

273 views

### What is the categorical analogue of openness?

Let us say that a category $\mathcal C$ has enough of some class $\mathcal U$ of object if every object in $\mathcal C$ is a colimit of objects of the class $\mathcal U$. The pointset topology ...

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229 views

### Faithfully flat descent for modules from the simplicial point of view

Let $R \rightarrow R'$ be a faithfully flat ring map, let $M$ be an $R$-module, and let $M_n$ be the base change of $M$ to the tensor product of $n + 1$ copies of $R'$ over $R$. One way to formulate ...

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115 views

### Equivalence relations in arbitrary categories

Let $C$ be a category and $A\in\mathrm{ob}(C)$. A relation is a subobject $q:Q\hookrightarrow A^{\times 2}$ and the quotient is defined as the coequalizer
$$A/Q:=\mathrm{coeq}\left(Q\stackrel{q}{\...

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323 views

### Basic example of a formal affine scheme, functorial point of view

$\let\opn=\operatorname$For my BA thesis I have to describe formal groups from the functorial point of view. I am hence reading Strickland - Formal Schemes and Formal Groups, which is apparently the ...

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173 views

### Algorithmically deciding existence of finite limits in a category

Given $\Sigma$ a consistent finite first order theory in vocabulary $L$, one can consider the category of its models $\mathcal{M}(\Sigma)$, its objects are the models of $\Sigma$ and arrows are ...

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111 views

### Two directed colimits of same spaces with different inclusions

For any natural number $n$, let $i_{n},j_{n}:X_{n}\rightarrow X_{n+1}$ be a pair of monomorphisms of simplcial sets.
Define $$X=\operatorname*{colim}_n \{\cdots X_n \rightarrow_{i_n} X_{n+1}\cdots \}...

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233 views

### Sufficient sets of colimits in small categories

Let $C$ be a small category, and consider the class of diagrams $G:D\to C$, with $D$ a small category, that have colimits in $C$. This is a proper class even when $C$ is very small, e.g. whenever $D$ ...

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68 views

### Filtered colimit of a topological space

Suppose that $X$ is a space filtered by closed subspaces $X_{1}\subset X_{2}\subset \dots$.
As topological space $X=\operatorname{colim}_{n}X_{n}$.
We define $Y_{n}=X_{n+1}/X_{n}$, and consider the ...

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113 views

### Inductive limit commutes with topological tensor product

Consider $H \left(U \right); U \subset \mathbb{C}$ - space of holomorphic functions with compact-open topology. In this topology, this space is Montel, nuclear and Frechet. I want to take the ...

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43 views

### direct limit in locally convex modules and continuous map

Let we have short exact sequences of LCM over LC algebra $A$ with continuous linear maps
$$
0\to B_j\;{\xrightarrow {\ f_j\ }}\;C_j\;{\xrightarrow {\ g_j\ }}\;D_j\to 0.
$$
We can take inductive limit (...

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118 views

### When does the canonical $t$-structure restrict to perfect complexes?

I am interested in non-Noetherian(!) rings such that the canonical $t$-structure on $D(R)$ (the derived category of left $R$-modules) restricts to perfect complexes i.e. to the subcategory of ...

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177 views

### When do Kan extensions preserve colimits?

Assume that we have a pair of functors $Y:A \to B$ and $F:A \to C$ where $A$ is an essentially small category, $B,C$ are cocomplete categories and $Y,F$ preserve colimits. Assume also that for some ...

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226 views

### Weighted (co)limits as adjunctions

It's well known that a category $\mathcal{C}$ having (conical) limits/colimits of shape $\mathcal{D}$ is equivalent to the diagonal functor $\Delta^\mathcal{D}_\mathcal{C}:\mathcal{C}\to\mathcal{C}^\...

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1k views

### Is every commutative ring a limit of noetherian rings?

Edit of Feb. 14, 2019. After Laurent Moret-Bailly's accepted answer, only Questions 4 and 5 remain open. I don't care that much about Question 4, but I'm very curious about Question 5, which is
Do ...

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107 views

### Coend of full subcategory

$\require{AMScd}$Let $F:\mathcal{C}^{op}\times \mathcal{C} \to \mathcal{D}$ be a functor and $\mathcal{C}' \subseteq \mathcal{C}$ a full subcategory. Assume that the coends $C$ over $F$ and $C'$ over $...

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53 views

### Does a cartesian transformation induce a cartesian transformation on absolute limit cones?

Fix a category or $\infty$-category $C$ with all small limits.
We call a natural transformation $\alpha\colon f\to g$ between two functors $f,g\colon K\to C$ Cartesian, if for every arrow $k\colon a\...

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108 views

### Is the projection onto the regular image an epimorphism?

Let $f:X\to Y$ be a morphism in a category $\mathcal{C}$.
Let $m:I\hookrightarrow Y$ be the regular image of $f$. This means that $f$ can be written as $f=m\circ e$, with $m$ regular mono (i.e. being ...

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39 views

### Inductive limits of unitary groups and quantum mechanics

I'm curious if someone has seen concrete applications of $U(\infty)$ in quantum mechanics. Is it possible, for example, in some particular cases to write down the propagator as a limit of a sequence ...

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362 views

### Yoneda Embedding and pull back

Given a manifold $M$ we have a geometric stack associated to it namely $\underline{M}$ whose objects are smooth maps to $M$. For the sake of consistency I am writing $BM$ for $\underline{M}$.
Given a ...

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391 views

### Monoidal functors $\mathcal C \to [\mathcal D,\mathcal V]$ are monoidal functors $\mathcal C \otimes \mathcal D \to \mathcal V$?

It is well known (e.g., Reference for "lax monoidal functors" = "monoids under Day convolution" ) that if $\mathcal C$ is a monoidal $\mathcal V$-enriched category, then a monoid ...

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68 views

### Reflexive coequalizer and singular functor.

Does the singular functor $sing: Top\rightarrow SimpSet$ form the category of topological spaces to simplicial sets commutes with reflexive coequalizers?
Recall that the singular functor $sing$ is ...

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315 views

### Functoriality of (co)limits in $\infty$-categories

I have some questions about the functoriality of (co)limits in $\infty$-categories, say in the framework of Lurie's Higher Topos Theory.
From the general stuff about Kan-extensions (HTT 4.3.2.6) ...

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61 views

### Is there an analogue of final functors for genuine 2-categorical limits

A functor $I\to J$ of $1$-categories is called final, if each undercategory $(j, I)$ is connected.
More generally, for $(\infty,1)$-categories there is an analogous notion where one requires the ...

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316 views

### Limits, colimits and universes

For many purposes in category theory, we consider limit and colimits of diagrams $F\colon\mathsf{J\to C}$ where $\mathsf{J}$ is small category, that is, a category where the classes of objects and ...

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159 views

### Behaviour of direct limit with matrices

I am trying to understand direct limits in the category of $C^*$-algebras by self reading. My last question was also related to direct limits. Here is another of my doubts:
Let $(A_n,f_n)$ be a ...

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548 views

### Defining abstract varieties and their morphisms over a finitely generated subfield of the base field

Let $k$ be an algebraically closed field.
By a finitely generated subfield of $k$ I mean a subfield $k_0\subset k$ that is finitely generated over the prime subfield of $k$ (that is, over $\mathbb Q$ ...

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184 views

### Behaviour of Direct limit with quotient and double dual

I am trying to understand direct limit in category of $C^*$ algebras.
Is it well known that direct limit behaves well with double dual and quotient of $C^*$ algebras?
Any references or ideas?
P....

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177 views

### Are sifted (2,1)-colimits of fully faithful functors again fully faithful? (And a de-categorified variant)

1) Suppose that I have a sifted diagram of categories $\mathcal{C}_i$, another of the same shape $\mathcal{D}_i$, and that I have a system $F_i:\mathcal{C}_i\to\mathcal{D}_i$ commuting with the ...

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343 views

### Surjectivity of a map on inverse limits

(The following is crossposted from Math.SE, where the question did not receive any answers.)
I am looking for a proof of the following lemma from P. Gabriel's Des catégories abéliennes (Chap. IV, §3, ...

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82 views

### When is a nested sequence of closed sets a colimit?

Let $X$ denote a topological space and $X_0\subset X_1\subset \ldots\subset X$ a nested sequence of closed subsets of $X$ such that $$ \bigcup_i X_i =X$$
It is easy to see that in the general case $X$...

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219 views

### Example of an $\omega_1$ decreasing chain of dense semicontinua?

In his well-known paper Bellamy constructs an indecomposable continua with exactly two composants. The setup is as follows:
We have an inverse-system $\{X(\alpha); f^\alpha_\beta: \beta,\alpha < \...

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551 views

### Calculating limits progressively

Consider the problem of finding the limit of the following diagram:
$$ \require{AMScd} \begin{CD}
& & & & E
\\ & & & & @VVV
\\ && C @>>> D
\\ & &...

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129 views

### Pushout of Nisnevich sheaves

Let us consider the projective line $\mathbb{P}^1$ over a field $k$ and take the following open embeddings
$$j_{\epsilon}\colon \mathbb{P}^1\setminus\{0,\infty\} \to \mathbb{P}^1\setminus\{\epsilon\}$$...

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357 views

### Grothendieck construction and coends

In category theory, both the Grothendieck construction and coends are represented by a sort of "integral sign", respectively:
$$
\int F
$$
for a functor $F:C\to\mathbf{Cat}$,
and:
$$
\int^x G(x,x)
$$
...

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172 views

### Homology of a limit of semidirect products

Suppose I have two families of groups $A_k$ and $B_k$ indexed by the natural numbers and suppose $B_k$ acts on $A_k$. Suppose there are groups homomorphisms $A_{k+1} \rtimes B_{k+1} \to A_k \rtimes ...

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253 views

### About pointwise Kan extension

Suppose that you want to look at the left Kan extension of a functor $F : \mathcal{C} \to \mathcal{A}$ along a functor $K : \mathcal{C} \to \mathcal{B}$. It is widely known that if the colimit of the ...

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390 views

### Must an inverse limit of simply connected groups be simply connected?

While the fundamental group $\pi_1$ preserves products, it is not true in general that an inverse limit of simply connected topological spaces is simply connected. I would like to know if similar ...

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96 views

### Conditions on a diagram category such that constant limits always exist and is constant

I am looking for a good set of sufficient conditions to put on a category $C$ such that the following becomes true:
Given any $\infty$-category $S$ and a functor $f\colon N(C)\to S$ which is ...

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309 views

### maximal tensor product commutes with inductive limits

Let $(A_n, \phi_n)$ be an inductive system of $C^*$ algebras and let $B$ be an arbitary $C^*$ algebra.
I want to prove $(\varinjlim A_n)\otimes_{max} B \cong \varinjlim (A_n \otimes_{max} B)$. This ...

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147 views

### A strict directed colimit of Hausdorff locally-convex spaces that is not Hausdorff

We work in the category of locally-convex spaces (morphisms are the continuous linear maps). Let $\Lambda$ be a directed set, for every $\lambda \in \Lambda$ let $V_{\lambda}$ be a locally-convex ...

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183 views

### Loop space functor and sequential colimits of inclusions

The question is about a fact that is mentioned as "evident" everywhere in the literature, so my guess is that some small detail is passing over my head. Here it is:
Let $X_0\hookrightarrow X_1 \...

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109 views

### terminology problem related to finitely generated objects

If $x$ is an object of a category $C$, one usually says that $x$ is if finite presentation (or compact) if for any direct filtered system ($y_i$) in $C$, the canonical map
$$f : \text{colim}_i \, \...

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112 views

### Colimits of small categories in the category of big categories

This is a question regarding set-theoretic issues in higher category theory. Let $\text{Cat}_{\infty}^{big}(\omega)$ be the $\infty$-category of (not necessarily small) $\infty$-categories with ...