All Questions
Tagged with limits-and-colimits reference-request
37 questions
1
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1
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234
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Notion of $\kappa$-sifted categories?
Let $\kappa$ be a regular cardinal. It seems reasonable to introduce the following definition:
Definition. A simplicial set $K$ is $\kappa$-sifted if, for every set $E$ with $\lvert E\rvert<\kappa$...
- 2,806
1
vote
0
answers
54
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contravariant finiteness and limit closure: is there dual to a result of Crawley-Boevey?
Let $\mathcal A$ be a locally finitely presented category. Theorem 4.2 of https://doi.org/10.1080/00927879408824927 says that given a full additive subcategory $\mathcal D$ of finitely presented ...
- 480
8
votes
1
answer
253
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Compact objects in slice categories of finitely presentable categories
Given a locally finitely presentable category $\mathscr C$ and an object $X \in \mathscr C$, it is not so hard to show that a morphism $(X \to Y)$ is compact in $\mathscr C_{X/}$ if it can be obtained ...
- 28.7k
4
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0
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131
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Weakening of terminal object in a category
I’ve come across a category $\mathcal{C}$ recently with an object $T$ such that any other object $X$ has a map $f:X\rightarrow T$, and for any two maps $f,g:X\rightarrow T$, there exists a (not ...
- 1,949
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1
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498
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What does it mean for a category to be generated under (some) colimits?
This is going to be a long post, so let me state my question first and then explain what I am interested in by way of examples.
Question.
Is there any literature studying notions of generation under ...
- 15.9k
8
votes
2
answers
306
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Reference for certain categorical limits
I would like to know if there is a special name for the following concept, papers that feature something similar or a general reference. Let $\mathcal{C}$ be a category and $\mathcal{D}$ a subcategory ...
- 167
2
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0
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108
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Characterization of inverse limits of finite-dimensional convex cones
Consider a countable inverse system $C_1\substack{f_1 \\ \leftarrow} C_2 \substack{f_2 \\ \leftarrow} C_3 \substack{f_3 \\ \leftarrow} \ldots$ where the $C_i$ are finite-dimensional convex cones of ...
- 21
6
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1
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243
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Stability properties of essential geometric morphisms
Notation.
$\mathsf{Topoi}$ is the bicategory of topoi, geometric morphisms and natural transformations between left adjoints.
$\mathsf{Topoi}_{\text{ess}}$ is the bicategory of topoi, essential ...
- 9,111
16
votes
2
answers
736
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Original reference for categories of presheaves as free cocompletions of small categories
It is well known that, for a small category $\mathbf A$, the category $\widehat{\mathbf A} = [\mathbf A^\circ, \mathbf{Set}]$ of presheaves on $\mathbf A$ together with the Yoneda embedding $\mathbf A ...
- 10.6k
1
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0
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90
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Dual of essentially compactly supported functions on a hemi-compact Radon space
Let $X$ be a hemicompact Radon space and fix a $\sigma$-finite Radon measure $\mu$ on $X$. Let $L(X_n)$ denote the subspace of $L_{\mu}^1(X)$ of "functions" which are $\mu$-essentially ...
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2
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339
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Strongly compact categories (reference request)
The notion of a "compact category" was introduced by Isbell$\color{red}{^{1,2}}$. A locally small category $\mathcal{C}$ is called compact when every functor $\mathcal{C} \to \mathcal{D}$ ...
- 63.1k
8
votes
0
answers
165
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Original reference for the Fam construction
For a category $\mathbf C$, the category of families of $\mathbf C$, denoted $\mathrm{Fam}(\mathbf C)$ is the free coproduct completion of $\mathbf C$. Explicitly, the objects of $\mathbf C$ are given ...
- 10.6k
14
votes
2
answers
761
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Is there a large colimit-sketch for topological spaces?
Question. Is there a large colimit-sketch $\mathcal{S}$ such that $\mathrm{Mod}(\mathcal{S}) \simeq \mathbf{Top}$?
In other words, is there a category $\mathcal{E}$ with a class of cocones $\mathcal{S}...
- 63.1k
7
votes
1
answer
236
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Free extension of algebra for an operad
I fix $C$ a symmetric monoidal model category (with a cofibrant unit if it helps). I'm assuming that it is closed, or at least that the tensor product commutes to colimits in each variable.
If $X$ is ...
- 42.4k
9
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0
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393
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When is an increasing union a colimit?
Let's consider a diagram $\Phi: \lambda \to \mathcal{T}_*$
$$
X_0 \to X_1 \to \cdots \to X_\xi \to X_{\xi+1} \to \cdots
$$
of pointed spaces,
indexed by some ordinal $\lambda$, in which each $X_\xi$ ...
- 12.5k
7
votes
1
answer
214
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Algebras for products or limits of monads
If a category $C$ has limits of a certain type, then the category of monads on $C$ has the same type of limits, and these limits are computed "levelwise" (i.e. are preserved by the forgetful ...
- 42.4k
9
votes
2
answers
988
views
Reference for homotopy colimit = total complex
I'm looking for a reference for the following fact:
take a simplicial chain complex $ X:\Delta^{op}\to Ch_{\geq 0}(\mathcal A)$ for $\mathcal A$ a nice abelian category (say, cocomplete with enough ...
- 15.8k
0
votes
1
answer
177
views
Convergence in LB-spaces
Edit:
Let $X$ be a strict LB-space described by $\lim X_n$ and suppose that $\{x_n\}_{n \in \mathbb{N}}$ converges in $X$. I'm looking for a reference showing that $x_n$ must converge in some $X_N$.
- 5,405
5
votes
1
answer
339
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Diagonal of a diagram of codescent objects
Given the following diagram in a $2$-category, in which squares of the same "type" commute, where each column and each row is a strong codescent diagram (Edit: it should be reflexive as well), is ...
- 63.1k
7
votes
4
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1k
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Existence of homotopy limits and colimits in model categories
I am not an expert, thus I apologize if my question is very naive. Let $\mathsf{M}$ be a model category (I do not assume any functoriality on the factorization),
Q1. Is there a reference where it is ...
- 9,111
3
votes
1
answer
138
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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 \}...
- 71
4
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0
answers
212
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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 ...
- 16.9k
6
votes
1
answer
219
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 ...
- 2,129
15
votes
2
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696
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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 ...
- 907
12
votes
2
answers
706
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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 $\mathbb Q$ ...
- 14.1k
8
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1
answer
1k
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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, ...
- 1,042
6
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0
answers
812
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Limit of metric spaces
Let $\{X_n\}_{n\in \mathbb{N}}$ be a collection of T2 topological spaces, with maps $f_n\colon X_n \to X_{n+1}$. These maps are continuous and open. Let $X$ be the direct limit of this system.
Assume ...
- 2,384
0
votes
1
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959
views
Representable functors and direct limits
Let $\mathcal{F}:\mathrm{Sch}/S \to \{\mathrm{Sets}\}$ be a representable functor. Denote by $X$ the scheme representing $\mathcal{F}$. The question is whether the natural tranformation $\mathcal{F}(-)...
- 1,981
2
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0
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160
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Universal property of limits of invertible sheaves
Let $R$ be a discrete valuation ring, $m$ the maximal ideal and $f:X \to \mathrm{Spec}(R)$ be a flat, proper morphism of relative dimension $1$. Assume further that $X$ is regular. For any $n>0$, ...
- 2,022
3
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0
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385
views
How does one compute a colimit of monoidal categories?
The question is in the title. I'm also happy to get answers about (your favorite adjective) monoidal categories.
Here's a guess:
In order to compute a colimit of monoids we can push everything down ...
- 41
7
votes
1
answer
2k
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Direct limit of compact topological spaces
Let $\{X_n\}_{n\in \mathbb{N}}$ be a direct system of compact topological spaces, meaning that we have morphisms $f_i\colon X_i \to X_{i+1}$ with the necessary compatibility conditions. Is there any ...
- 2,384
11
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1
answer
1k
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Reference request: colimits of locally presentable categories
Consider the 2-category of locally presentable categories, cocontinuous functors, and natural transformations. I believe that this 2-category is 2-cocomplete in the sense of containing all small 2-...
- 54.6k
6
votes
1
answer
474
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Limits in span categories
What are the limits in the span categories? and what is known about them in the literature?
- 61
5
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2
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630
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Where can I find an explicit description of the pseudocolimit of a small pseudofunctor to Cat?
Given a functor from a small category to $Set$, we can describe the colimit set as a quotient of the disjoint union of image sets by an equivalence relation arising from morphisms in the source ...
- 45.7k
8
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1
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606
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Comparing colimits in schemes with colimits in sheaves of sets
Suppose I have a diagram of schemes, and I know that the colimit exists in the category of schemes. How does this colimit compare with the colimit of the corresponding sheaves (I'm being nonspecific ...
- 5,548
12
votes
5
answers
5k
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Motivation of filtered colimits
I am trying to move in categorical algebra beyond the basics. A Lawvere theory L is a small category with finite products. (I know that there also is a functor $(skeleton(FinSet))^{op}\to L$, which ...
- 530
17
votes
10
answers
3k
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References for homotopy colimit
(1) What are some good references for homotopy colimits?
(2) Where can I find a reference for the following concrete construction of a homotopy colimit? Start with a partial ordering, which I will ...
- 12.8k