All Questions
Tagged with colimits or limits-and-colimits
347 questions
1
vote
1
answer
606
views
Does the inverse limit of complexes with bounded cohomology have a bounded cohomology?
Let $A$ be a ring (commutative and noetherian if it helps).
Suppose we are given an inverse system $M_i$ of complexes of $A$-modules (where $i$ is a natural number),
and integers $a<b$
such that ...
CommunityBot
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9
votes
2
answers
2k
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Is there a useful limit or co-limit of a diagram that has only a single object?
I'm starting to study category theory kind of informally and everytime I read about the definitions of limits and co-limits, the first three examples are always the same:
terminal/initial objects,
...
CommunityBot
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3
votes
1
answer
673
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Can comma categories of small categories be understood as limits/colimits in $\textbf{Cat}$?
Let $F: C \to D$ be a functor of small categories. One can form the comma categories $F/$ and $/F$ with objects
\begin{align*}
(c,d,\phi) && \phi: F(c) \to d \\
(c,d,\psi) && \psi: d \...
- 63.9k
7
votes
1
answer
861
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Continuity of Alexander-Spanier cohomology
Suppose that a paracompact space $X$ is the inverse limit of paracompact
spaces $X_{i}$ (that is $X=\varprojlim X_{i}$) and $H^{\ast }$ is Alexander-Spanier
cohomology with closed supports. Then the ...
- 17.4k
1
vote
0
answers
102
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Hom-set of ind-objects of the same shape
The usual definition of hom-sets between ind-objects in a category $C$ is:
$$
\operatorname{ind-}C(F,G) := \lim{}_{a\in A} \operatorname{colim}_{b\in B} \operatorname{Hom}(Fa,Gb)\;.
$$
where $F:A\to ...
- 2,129
3
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0
answers
139
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Colimits of algebras of an endofunctor
I try to understand a proof in Adamek-Rosicky's book "Locally presentable and accessible categories", Cambridge University Press 1994. In Corollary 2.75 (p. 121) it is proven that the category $\...
- 5,482
6
votes
2
answers
659
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Limit of a sequence of locally presentable categories
Let $\dotsc \to \mathcal{C}_2 \xrightarrow{F_1} \mathcal{C}_1 \xrightarrow{F_0}\mathcal{C}_0$ be a sequence of cocontinuous functors between locally presentable categories. Consider the limit $\...
- 63.9k
2
votes
1
answer
187
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On countable homotopy colimits in (the derived categories of) AB3 abelian categories
If $h_i:A_i\to A_{i+1}$ is a countable chain of morphisms in an abelian category $A$ that is AB3 then one can consider the (Bökstedt-Neeman) homotopy colimit of $A_i$ in $D^b(A)$. This is a two-term ...
- 16.9k
1
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0
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77
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When is the 2-category of Commutative Monoids (Co)complete?
Let $C$ be a strict 2-category which is bicomplete (has all 2-(co)limits). Assume further that $C$ is symmetric monoidal. Denote by $CMon(C)$ its 2-category of commutative monoids. When is $CMon(C)$ (...
- 913
5
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1
answer
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(Co)completness of the 2-category of ``2-Rings"
Let $2Ring$ denote the 2-category of cocomplete categories with monoidal structures that preserve colimits in each argument. The morphisms are cocontinuous and strong monoidal functors (which are ...
- 54.6k
2
votes
0
answers
184
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Infinite iterates of the contravariant hom endofunctors on sets
My recent answer to Is it possible to define higher cardinal arithmetics (about defining infinite tetrations) requires something I don't know. Here is the simplest case.
Take a set $S$ and consider
$$...
CommunityBot
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6
votes
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
4
votes
2
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545
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Where does the name "filtered colimit" come from?
There are a lot of articles which explain what filtered colimits are (e.g. https://ncatlab.org/nlab/show/filtered+limit), but I couldn't find why they are named "filtered colimits".
It doesn't look ...
- 18.4k
5
votes
1
answer
152
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Finite well-completeness and the small object argument?
I'm reading a few papers on reflective factorization systems and I've just noticed they're all mentioning a procedure which seems very similar to the small object argument.
First of all, some ...
- 66.8k
4
votes
1
answer
316
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Forgetful Functor $MonCat\rightarrow Cat$ preserves filtered colimits?
Did not get an answer from the Stack Exchange.
Let $MonCat$ and $Cat$ denote the 2-categories of monoidal categories with strict monoidal functors and small categories, respectively.
There is a ...
- 66.8k
5
votes
0
answers
448
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Examples of nonstable ∞-categories in which sifted colimits commute with finite limits
What are some natural examples (if any) of nonstable ∞-categories in which finite limits commute with sifted colimits (or rather just colimits over Δ^op)?
Stable ∞-categories do satisfy this property,...
- 37.8k
11
votes
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-...
CommunityBot
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8
votes
1
answer
492
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Completeness of 2-category of Monoidal Categories
Is the 2-category of monoidal categories complete? If not, can any conditions be imposed to satisfy completeness?
- 66.8k
0
votes
1
answer
631
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A map between direct limits
Let $C$ be a category which has all small colimits.
I have the following situation:
$\{A_i\}_{i \in I}$ and $\{B_j\}_{j \in J}$ are two directed systems in $C$,
with transition maps $\alpha_{i_1,i_2}...
- 63.9k
7
votes
1
answer
1k
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Definition of dense functors
Definition. A functor $F:\mathsf C\rightarrow \mathsf D$ is dense if every $D\in \mathsf D$ is the vertex of the following colimit $$\varinjlim \left(F\downarrow D\rightarrow\mathsf C\rightarrow \...
- 118k
2
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0
answers
101
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Reference for the existence of bicolimits in groupoids and categories?
I am looking for a reference of these, I would say, very well known facts. (strangely though finding a reference was bit trick for me).
Let $C$ be a category and $F:C\rightarrow Cat$ a 2-functor in ...
- 30.3k
8
votes
0
answers
191
views
Yoneda embedding and Horn sentences
The following is taken from Borceux and Bourn's Mal'cev, Protomodular, Homological, and Semi-Abelian Categories.
Metatheorem 0.1.3. Let $\mathcal P$ be a statement of the form $\varphi\implies \psi$, ...
CommunityBot
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9
votes
1
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344
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Explicit calculations of small homotopy limits of CDGAs
I would like to carry out explicit calculations of homotopy limits of certain simple diagrams of CDGAS. My set-up is the following : I have a finite graded poset $R$ with minimal element $0$ and a ...
- 30.3k
4
votes
1
answer
298
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Dual of colimit in $\text{Ban}_1$
I learned in J. Castillo's Hitchhiker guide to categorical Banach space theory that, by a theorem of Semadeni and Zidenberg, limits and colimits exist in the category $\text{Ban}_1$ of Banach spaces ...
- 25.8k
9
votes
1
answer
602
views
Intuition for density comonad in relation to lifting problems
In Emily Riehl's Categorical Homotopy Theory, there is a section on Garner's Small Object Argument which I'm trying and failing to understand. Originally I followed most of Garner's paper, using the ...
- 5,309
2
votes
1
answer
206
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Appropriate morphisms and 2-morphisms in Ind(C)
As I was trying to understand the category $Ind(C)$ of diagrams of the form $I \to C$, where $I$ is a small filtered $(0,1)$-category, I wondered whether it is possible to define morphisms directly, ...
- 15.5k
7
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0
answers
260
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Topological localization of (infinite) inverse limits
The classical localization of topological spaces at a given set of primes $\mathcal{P}$ is a functor $\mathcal{T}\xrightarrow{(-)_{(\mathcal{P)}}}\mathcal{T}$ from a suitable category of topological ...
- 19.6k
14
votes
1
answer
616
views
How "nondegenerate" are amalgamated free products of C*-algebras?
In the following, I assume all algebras are unital. Let $A$ and $B$ be C*-algebras that each contain (isomorphic copies of) a common C*-subalgebra $C$. Let $A *_C B$ denote the amalgamated free ...
- 3,984
20
votes
2
answers
2k
views
Is every compact topological ring a profinite ring?
There are a lot of compact (Hausdorff) groups, whereas every compact field is finite. What about rings? Is there a classification theorem for compact rings? If you take a cofiltered limit of finite ...
- 18.4k
6
votes
1
answer
244
views
2-completeness analog of completeness theorem
It's not hard to see that a category is finitely complete if it has finite products and equalizers. In short, this is because one can write all limits as iterations of these two "operations".
I ...
CommunityBot
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2
votes
1
answer
353
views
Pro-affine varieties over a local field
Let $K$ be a (perfect) local field, and let $S = \lim (\operatorname{Spec} A_i)_{i=0}^\infty$ be a pro-affine variety over $K$. This means that each $A_i$ is a finite type $K$-algebra and that the ...
- 4,746
6
votes
1
answer
674
views
permutation of projective limits with inductive limits
Hi everybody,
I have a lack of references concerning projective limits and injective limits. Up to my faults in Bourbaki there are only proj and inj limits indexed by a partially ordered set (not a ...
CommunityBot
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11
votes
2
answers
4k
views
Is Sheafification Functor Exact?
I know that sheafification functor from the category of abelian presheaves on $C$ to the category of abelian sheaves on $C$. Here, $C$ is a category with Grothendieck pretopology.
My question is:
...
CommunityBot
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7
votes
0
answers
639
views
Constructing pointwise Kan extensions as adjoints to some functor
Background
I'm working on formalizing some category theory in Coq at https://bitbucket.org/JasonGross/catdb. Currently, I'm in the process of formalizing pointwise Kan extensions. Partly because I'...
CommunityBot
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3
votes
1
answer
4k
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colimits in Cat via coproducts and coequalizers
I am attempting to do a calculation of a colimit in $Cat$, the category of small categories. To this end, people have suggested that I do this by calculating coproducts and using coequalizers. I ...
- 624
11
votes
1
answer
502
views
Is there a practical criterion to determine whether the limit of a diagram of real chain complexes is also a homotopy limit?
Consider a diagram D: I→ChR of real connective chain complexes.
In the example I have in mind all chain complexes are concentrated in some fixed degree n.
There is a canonical map lim D → holim D ...
CommunityBot
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21
votes
4
answers
3k
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Categorical description of the restricted product (Adeles)
Background on the Adèles
The Adèles $\mathbb{A}_K$ of a number field or function field $K$ are defined as a restricted product of the complete local fields $K_\nu$, where $\nu$ ranges over all places ...
CommunityBot
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2
votes
2
answers
357
views
Projective limit construction of a semigroup
Let $\tilde{\mathbb N}$ be the Abelian semigroup (under addition) given by $\mathbb N\cup\{0,\infty\}$, and let $S_n$ be the Abelian monoid $\tilde{\mathbb N}^{2^n}$ under point-wise addition. ...
CommunityBot
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11
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2
answers
511
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How to interpret topologically that the equalizer in Groupoids of ${\rm id}, {\rm id}: BG \rightrightarrows BG$ is $G/G$ (adjoint action)?
Let $G$ be a (discrete) group, and $1/G$ the corresponding groupoid with one object. Consider the diagram in (the 2-category) Groupoids with one vertex, labeled $1/G$, the one arrow from that vertex ...
- 54.6k
3
votes
1
answer
124
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What do you get when you apply a universal cocone to a colimit functor
Any colimit can be represented as a functor $F$ left adjoint to a particular diagonal functor $\Delta: C \rightarrow C^J$. The unit of this adjunction is the natural transformation $\eta_K: K \...
- 66.8k
26
votes
1
answer
780
views
Finite limits in the category of smooth manifolds?
The category of smooth manifolds does not have all finite limits. However, it does have some limits: it has finite products, it has splittings of idempotents, and it has certain other limits if we ...
CommunityBot
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6
votes
1
answer
613
views
Does Ind-completion commute with finite limits?
The broad and vague question is in the title. The more precise question is:
Say $\{\mathcal{C}_i\}$ is a finite diagram of (essentially small) stable $\infty$-categories and exact functors with ...
CommunityBot
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10
votes
0
answers
650
views
(Co-)Limits and fibrations of DG-Categories?
First of all, let me see if I got the 1-categorical version right:
Let $\mathcal F:C\to Cat $ be a
(pseudo-) functor. The 2-colimit
$\mathrm{colim}_C\mathcal F$ is then
given by the Grothendieck
...
- 30.3k
6
votes
1
answer
475
views
Limits in span categories
What are the limits in the span categories? and what is known about them in the literature?
CommunityBot
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3
votes
4
answers
492
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Homology of infinite intersection
If $X_1\supseteq X_2\supseteq \ldots$ is a sequence of "nice" compact spaces, I would like to know whether the natural map from $H_*(\cap X_i)$ to the inverse limit $\lim \, H_*(X_i)$ is surjective. ...
CommunityBot
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12
votes
1
answer
2k
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Adjoint Functors as Initial Objects of Some Category
Just as universal arrows can be characterized as initial objects of some appropriate comma category, and (co)limits can be characterized as (initial) terminal objects of the appropriate (co)cone ...
CommunityBot
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3
votes
0
answers
528
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Homotopy category of groupoids
The nlab Ho(Cat) page says: morphisms in the homotopy category of groupoids $Ho(Gpd)$, have two equivalent description:
iso-classes of functors.
formally invert equivalence functors (i.e. ...
- 30.3k
5
votes
2
answers
6k
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On limits and Colimits
I want to ask a stupid question. Let $I$ be an infinite set and suppose $i$ belongs to $I$. I wonder whether following morphisms exist in general:
Hom($A$,colim $B_i) \to$ lim Hom($A,B_i$) and
...
- 30.3k
3
votes
1
answer
876
views
Strong colimits of categories.
Let $\mathcal C$ be a category and let $\mathcal F:\mathcal C\to\mathcal C\textrm{at}$ be a strong bifunctor. Given another category $\mathcal D$, let $\triangle_{\mathcal D}$ denote the constant ...
CommunityBot
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4
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1
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979
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What are the uses of Limits and Colimits of Category Theory in every day problems? [closed]
I am interested in knowing how we can use the concepts of Limits and Colimits in modeling problems in every day life? Could anyone provide (Software) engineering examples, perhaps? Or describe ...
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