Questions tagged [limits-and-colimits]
For questions on limits and colimts in the sense of category theory, and related notions.
333
questions
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0
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Homotopy colimits in subcategories of combinatorial model categories
We know that, in a combinatorial simplicial model category $\mathbf{M}$, we can find a regular cardinal $\lambda$ large enough that all $\lambda$-filtered homotopy colimits can be computed as strict ...
- 1,071
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votes
0
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140
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Do pseudo 2-limits commute?
It is a well-known fact that if $F:\mathcal{C}_1\times\mathcal{C}_2\rightarrow \mathcal{D}$ is a functor (between 1-categories), then $F$ has a limit if and only if $F:\mathcal{C}_1\rightarrow Fun(\...
- 901
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A right adjoint preserves Phi-colimits if and only if the left adjoint does what?
Let $\Phi$ be a class of categories (e.g. filtered categories), and consider an adjunction $L : \mathbf C \rightleftarrows \mathbf D : R$. A $\Phi$-colimit is a colimit whose diagram is in $\Phi$. We ...
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4
votes
2
answers
264
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Change of coordinates for coends
I recall that there was a theorem mimicking the change of variables' integral formula. Surprisingly, I can't find it on the Fosco Loregian book. The change of variables formula states that, if $f: E \...
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3
votes
1
answer
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Question about the proof of Kerodon tag 030V (Proposition 7.3.7.1)
$\require{AMScd}$
Related to this, I have a question about the proof given in Kerodon of the following result:
Proposition 7.3.7.1: Let $C$ be an $\infty$-category, let $\bar{F} : C^\rhd \to D$ be a ...
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2
votes
0
answers
178
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Is the homotopy limit of derived schemes along affine maps a derived scheme?
The title question is true in the setting of ordinary limits and ordinary schemes; that is, given an inverse limit of schemes along affine maps, the limit still lives in the category of schemes.
I'd ...
- 301
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1
answer
542
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Formula for the left adjoint of the nerve functor?
I recently stumbled upon a formula for the left adjoint of the nerve functor. Let $X$ and $Y$ be simplicial sets, then:
\begin{equation}
\mathbf{sSet}(X,Y)
\cong\mathbf{sSet}(\varinjlim_{\Delta^n\...
4
votes
1
answer
393
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Derived functors of inverse limit in abelian categories?
I have a finite poset $I$ and an inverse system $A: I^{op}\longrightarrow \mathscr C$ taking values in an abelian category $\mathscr C$.
I suppose that $\mathscr C$ has direct sums. Given that my ...
- 41
9
votes
1
answer
319
views
Algebraically-free monadicity theorem
The monadicity theorem characterises when a functor $u : \mathbf B \to \mathbf E$ is the forgetful functor from the category of algebras for some monad on $\mathbf E$ (up to an equivalence over $\...
- 8,755
3
votes
0
answers
144
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Weighted limit calculus
In coend calculus by Fosco Loregian, it is mentioned that Lawvere conjectured that co/ends constitute a categorification of logical calculus. In his own words:
The somewhat far-fetched conjecture ...
- 9,009
5
votes
0
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301
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$V$-cat and $V$-graph: coequalizers in the category of enriched functors
This question is regarding the 1974 JPAA paper $V$-cat and $V$-graph by Harvey Wolff.
To be precise, I don't understand a certain step in the proof of Corollary 2.9, which (the corollary) is crucial ...
- 1,095
7
votes
1
answer
408
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A fibration equivalent to having a terminal object
It is well known that the codomain functor $$cod:\mathcal{C}^\to\to\mathcal{C}$$ from the arrow category of a category $\mathcal{C}$ to itself is a fibration iff $\mathcal{C}$ has binary pullbacks.
...
- 9,009
3
votes
0
answers
81
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When is Tw(C) {ω}-filtered?
I am interested in categories $\mathsf{C}$ for which coends commute with $\omega$-chain limits. That is, given a chain of profunctors $P_n \colon \mathsf{C}^{op} \times \mathsf{C} \to \mathsf{Set}$ ...
- 305
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votes
1
answer
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Are locally fully faithful 2-functors closed under 2-pushout in 2-Cat?
It is known that fully faithful functors are closed under pushouts in Cat (e.g. Lemma 4.9 of this paper). Are locally fully faithful 2-functors closed under (strict) 2-pushouts in the 2-category 2-Cat ...
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143
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Are weighted limits terminal in a category of cones?
Consider a Benabou-cosmos $(\mathcal{V},\otimes,J)$, $\mathcal{V}$-categories $\mathcal{I},\mathcal{C}$ and $\mathcal{V}$-functors $\mathcal{W}:\mathcal{I} \rightarrow \mathcal{V}$ and $\mathcal{D}:\...
- 225
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votes
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answers
99
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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
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votes
0
answers
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Is there a category of "chains of modules" that behaves well with taking direct limits?
I came up with the following definition of a category of certain "chains of modules" and want to know if this concept is already known and studied.
Let $R$ be ring. An object in our category ...
- 1,023
6
votes
1
answer
236
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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 ...
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10
votes
1
answer
332
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Weak descent and effective equivalence relations
I want to prove that weak descent of a $1$-category implies the classical Giraud axioms.
More precisely, let $\mathsf{C}$ be a cocomplete, finitely complete $1$-category. We say that $\mathsf{C}$ ...
0
votes
1
answer
174
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The direct limit of invertible functions on a variety
(I asked this question a couple of days back on Stackexchange but with no success, it seems elementary, but I am struggling to go about attempting it.)
Let $X$ be a smooth geometrically integral ...
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7
votes
1
answer
588
views
Finite coproducts commute with which limits in Set?
It is well known that (small) coproducts commute with connected limits in $\mathbf{Set}$. With which class of limits do finite coproducts commute?
Ideally, we should furthermore like to know whether ...
- 8,755
5
votes
0
answers
125
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Applications of $FP_\infty$ groups preserving direct systems
In [1], the author proves that given a group $G$ and a directed system $(M_\lambda)_\lambda$ of $G$-modules, the induced maps $$\varinjlim H^k(G,M_\lambda) \to H^k(G,\varinjlim M_\lambda)$$ are ...
15
votes
2
answers
633
views
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 ...
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10
votes
2
answers
651
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Do colimits of manifolds coincide with underlying colimits as topological spaces?
Categories of manifolds (possibly with extra structure) tend not to have all colimits.
Other questions have addressed when colimits of manifolds exist.
I'd like to know what we can say in general ...
3
votes
0
answers
177
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For which categories $D$ is a $D^{\vartriangleleft\vartriangleright}$-shaped diagram in a stable $\infty$-category a limit iff it is a colimit?
Throughout, I'll omit the "$\infty$" from the term "$\infty$-category".
It is well-known (and sometimes even included in the definition, although not by Lurie) that pushouts and ...
- 728
3
votes
0
answers
108
views
Density with respect to a family of diagrams, versus a class of weights
In Theorem 5.19 of Kelly's Basic Concepts of Enriched Category Theory, it is proven that a fully faithful functor $K \colon \mathcal A \to \mathcal C$ is dense if and only if $\mathcal C$ is the ...
- 8,755
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vote
0
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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 ...
5
votes
1
answer
202
views
Dense subcategory of measurable spaces
Recall the notion of a dense subcategory $\mathcal{D}$ of a category $\mathcal{C}$. It means that the restricted Yoneda functor $\mathcal{C} \to \mathrm{Hom}(\mathcal{D}^{op},\mathbf{Set})$, $A \...
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8
votes
2
answers
316
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Strongly compact categories (reference request)
The notion of a "compact category" was introduced by Isbell$\color{red}{^{1,2}}$. A locally small category $\mathcal{C}$ is called compact when every functor $\mathcal{C} \to \mathcal{D}$ ...
- 61.5k
3
votes
1
answer
141
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Examples of (co)lax idempotent pseudocomonads on Cat
A lax idempotent pseudomonad, also called a KZ doctrine or KZ monad, is a pseudomonad $(T, \mu, \eta)$ with the property that $T \eta \dashv \mu \dashv \eta T$. Lax idempotent pseudomonads were ...
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votes
1
answer
239
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Limit along the category of all algebraic curves over a field
Let $k$ be algebraically closed field of charactersistic zero and $\mathcal C$ be the category of irreducible smooth projective curves over $k$ and non-constant maps between them. I have a functor $F\...
user21167
8
votes
0
answers
150
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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 ...
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2
votes
3
answers
194
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Conditions for exact projective limits for some Mittag-Leffler systems?
Let $(M_i)_{i\in I}$ and $(N_i)_{i\in I}$ be Mittag-Leffler systems of $R$-modules. I have a map $(h_i)$ of projective systems such that every $h_i$ is surjective. I search for conditions for $\lim \...
4
votes
0
answers
243
views
Free vector space on a filtered limit
$\DeclareMathOperator\Set{Set}\DeclareMathOperator\Vect{Vect}\DeclareMathOperator\Coalg{Coalg}\DeclareMathOperator\ProVect{ProVect}\DeclareMathOperator\prolim{prolim}
$Let $K$ be a field and $F: \Set \...
- 1,221
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votes
1
answer
218
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Semi-norms on LCS inductive limit of Banach Spaces
Let $(E_n,i_n)_{n\in\mathbb{N}}$ be an direct system of Banach spaces in the category of locally convex spaces (LCSs) with continuous linear maps and let $E_{\infty}$ by their inductive limit. What ...
- 4,969
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votes
1
answer
245
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Example of a non-cocomplete model category of a realized limit sketch
Let $(\mathcal{E},\mathcal{S})$ be a realized limit sketch, i.e. a locally small category $\mathcal{E}$ with a class $\mathcal{S}$ of limit cones in it. It is not assumed that $\mathcal{E}$ is small, ...
- 61.5k
33
votes
5
answers
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Shapes for category theory
Most texts on category theory define a (small) diagram in a category $\mathcal{A}$ as a functor $D : \mathcal{I} \to \mathcal{A}$ on a (small) category $\mathcal{I}$, called the shape of the diagram. ...
7
votes
1
answer
278
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Filtered 2-colimits commute with finite 2-limits
Is there an explicit proof anywhere in the literature that filtered 2-colimits commute with finite 2-limits (all meant in the weak bicategorical sense) in the 2-category of groupoids? I have only ...
- 65.1k
14
votes
2
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745
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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}...
- 61.5k
4
votes
1
answer
253
views
Explicit description of a pullback of $(2,1)$-categories
In the 1-category of 2-categories, with objects being categories enriched over Cat, and morphisms being 2-functors, is there an explicit way to describe a pullback of two functors $G:E\to D$ and $F:C\...
- 85
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votes
1
answer
129
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Functoriality of weighted limits
Let $C$ be a complete category, let $I$ be a small category, let $F,G:I\to C$ be functors, and let $W,U:C\to\mathrm{Set}$ be also functors, which we view as "weights".
The weighted limits ...
- 2,129
8
votes
1
answer
312
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Filling square to push-out in abelian category
Let $\mathcal{C}$ be an abelian category. In $\mathcal{C}$ we consider the diagram
\begin{array}{ccc}
A&&\\\
\downarrow&&\\\
C&\rightarrow&D
\end{array}
with arrows being ...
- 463
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votes
1
answer
595
views
Does Grothendieck's algebraization imply existence of colimits of schemes?
I am a little bit confused about two lemmas regarding Grothendieck's algebraization. Assume all algebras are defined over some field. Here is the short version of my question: Does Tag 09ZT ("...
- 5,851
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votes
1
answer
207
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Morphisms from the empty diagram
Let $X$ be an object in a category, and let $D$ be the empty diagram in the same category (containing no objects, and therefore no morphisms).
What should $\text{Hom}(D,X)$ be?
The only reasonable ...
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14
votes
0
answers
291
views
Which limits distribute over which colimits in $Set$? How about in $Spaces$?
I've never really thought much about distributivity of limits and colimits -- I tend to think more about commutativity of limits and colimits. This question makes me want to change that.
The question ...
- 61.5k
9
votes
1
answer
384
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Commutation of limits and colimits: Is there a choice diagram?
I was looking at this question about a "soft proof" of the fact that finite limits (shape $I$) commute with filtered colimits (shape $J$) in Set, using only the fact that the diagonal $J \to ...
- 500
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votes
0
answers
136
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Characterisation of essentially algebraic theories with a fixed set of sorts
It is well known (e.g. Palmgren–Vickers's Partial Horn logic and cartesian categories) that many-sorted essentially algebraic theories (equivalently partial Horn theories / quasi-equational theories / ...
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votes
1
answer
219
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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 ...
- 40.2k
2
votes
1
answer
210
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Define a sketch $s_{\mathbf{Grp}}$ such that $\mathbf{Grp}\backsimeq \mathbf{Mod}(s_{\mathbf{Grp}},\mathbf{Set})$
I have this MSE question with a two hundred bounty but even with the bounty this post got underviewed. So maybe here is a more suitable place to post it. The question follows:
(a) Define a sketch $s_{...
- 79
11
votes
1
answer
524
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Measure the failure of colimit to commute with taking free loops (or Hochschild homology)?
For a space1 $X$, let $\mathcal{L}X = \mathrm{Maps}(S^1, X)$ be the free loop space.
Inclusion of constant loops gives a natural map $X \to \mathcal{L}X$. This is not a homotopy equivalence unless $X$...
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