All Questions
Tagged with colimits or limits-and-colimits
347 questions
6
votes
1
answer
4k
views
Which limits does group cohomology commute with?
For a discrete group G, if $M$ is a direct/inverse limit of $M_i$, is $H^i(G, M)$ the direct/inverse limit of the $H^i(G, M_i)$? Of course, cohomology commutes with finite direct sums, but how about ...
- 135
1
vote
1
answer
665
views
Colimit notation
In Wikipedia's article on Kan extensions 1, in the view of Kan extensions as colimits, I am confused about the notation: $(Lan_F X)(b) = \varinjlim_{f:Fa \to b} X(a)$.
Wikipedia says that the colimit ...
- 61
9
votes
1
answer
950
views
The crude monadicity theorem
In order to test the monadicity of a functor, there is a precise monadicity theorem (PM) as well as a crude monadicity theorem (CM), see the nlab. In CM, the forgetful functor should create reflexive ...
- 63.1k
3
votes
0
answers
867
views
The inductive and projective limits of compact Hausdorff topological groups
Are there conditions known under which the inductive or projective limit of a family of compact Hausdorff topological groups is compact? (For instance, such a result is valid for the projective limit ...
- 5,407
6
votes
2
answers
222
views
Local finality condition (for re-indexing parameterized colimits)
I'm in need of a condition that is analogous to the "finality" condition in the following lemma:
Lemma: A functor $F\colon A\to B$ is final if and only if for any functor $x\colon B\to Set$, the ...
- 8,659
12
votes
1
answer
2k
views
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 ...
- 433
16
votes
4
answers
4k
views
When do Kan extensions preserve limits/colimits?
I'm guessing the answer to this question is well-known:
Suppose that $Y:C \to P$ and $F:C \to D$ are functors with $D$ cocomplete. Then one can define the point-wise Kan extension $\mathbf{Lan}_Y\...
- 15.5k
10
votes
2
answers
863
views
Fubini theorem for hocolim
I wanted to ask the following question,
Suppose $\mathbf{M}$ a cofibrantly generated model category and $I,~J$ two small categories. Suppose that $F:J\rightarrow \mathbf{M}^{\mathrm{I}}$ is a functor. ...
- 1,974
6
votes
0
answers
629
views
Inverse limit of graded rings
Let $(I,\le)$ be a directed set and let $(\rho^{\beta\alpha}: R^\beta \to R^\alpha)_{\alpha \le \beta}$ be an $I$-directed system of $\mathbb{Z}$-graded rings whose multiplication is denoted by
$$\...
- 16.2k
0
votes
0
answers
221
views
how many ways can an algebra be a weighted colimit of free algebras?
For a given weight $W : \mathcal{S}^{op} \to \mathcal{V}$ and diagram $D : \mathcal{S} \to \mathcal{A}$, the weighted colimit is an object $W \cdot D$ together with an isomorphism
$$\mathcal{A}(W\cdot ...
- 3,393
14
votes
1
answer
2k
views
What is a (partial) left adjoint of the Yoneda embedding called?
It is a fairly special property for the Yoneda embedding $A \hookrightarrow \mathcal{P}A$ of a category to have a left adjoint defined everywhere (this happens just when $A$ is total). However, a ...
- 66.7k
21
votes
4
answers
3k
views
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 ...
- 1,926
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 ...
- 1,213
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:
...
- 945
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 ...
8
votes
3
answers
2k
views
Is the category of small categories locally presentable?
I was wondering whether the various model structures on the category of small categories are combinatorial. I think that the ones I know are at least cofibrantly generated. In order to be ...
- 15.2k
10
votes
1
answer
454
views
Given a small category with some colimits, can the rest of the colimits be added?
Let $\mathcal{A}$ be a small category with some ( maybe no) colimits. What I would like to be able to do is add the rest of the colimits in a universal way. The Yoneda lemma will not work, since this ...
- 319
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 ...
- 4,842
5
votes
2
answers
630
views
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
2
votes
0
answers
562
views
Direct Limits and Limits of Nets
A net is a function from a directed set into a topological space, and it is said to converge to a point if certain conditions are satisfied. Similarly, a direct system is a function from a directed ...
- 15.4k
6
votes
2
answers
926
views
Is the category of toposes cocomplete ?
Hello.
[Edits between brackets.]
Does the [1-]category of [elementary] toposes [with logical morphisms] admit any [1-]colimits ?
[By colimit I mean initial object in the category of outgoing ...
- 61
3
votes
1
answer
233
views
Aspherical amalgamations without injective maps
The situation I find myself in is as follows: I have a CW complex $X$ which is covered by two subcomplexes $A$ and $B$ and I know that $A$, $B$ and $A \cap B$ are connected and aspherical. The term ...
- 1,680
3
votes
1
answer
392
views
Can cones (toric monoids) be built as colimits of their faces?
Suppose $L$ is a lattice (free abelian group) and $\sigma$ is a (pointed) spanning rational cone in $L\otimes\mathbb Q$. Then $M=L\cap \sigma$ is a monoid with $M^{gp}=L$. A monoid of this form is ...
62
votes
3
answers
9k
views
Why do filtered colimits commute with finite limits?
It's not hard to show that this is true in the category Set, and proofs have been written down in many places. But all the ones I know are a bit fiddly.
Question 1: is there a soft proof of this fact?...
- 3,141
8
votes
1
answer
606
views
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
views
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
11
votes
2
answers
511
views
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
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 ...
- 2,200
12
votes
1
answer
2k
views
Is every ring the direct limit of Noetherian rings?
Are there any examples of commutative rings that do not occur as direct limits of Noetherian rings?
- 441
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 ...
- 3,249
4
votes
3
answers
1k
views
Unions of sets exist? [closed]
Hello,
Probably this questions is very stupid, but anyway: It usually said that the category of sets is cocomplete, in particular meaning that we have disjoint unions of arbitrary families of sets, ...
- 5,562
9
votes
1
answer
814
views
How does Berger-Moerdijk's relative Boardman-Vogt work?
In "The Boardman-Vogt resolution of operads in monoidal model categories," the authors construct factorizations of sufficiently nice operad maps $P\to Q$ into a cofibration followed by a weak ...
- 4,588
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
...
- 3,249
14
votes
2
answers
2k
views
Colimits of manifolds
This question tells us that in general colimits do not exist in the category of manifolds.
However, this negative answer is not very satisfying. A manifold can be considered as a colimit of its altas....
user2529
16
votes
4
answers
1k
views
Is there a tricategory of bicategories and biprofunctors?
Background
There is a bicategory where the objects are categories, the 1-morphisms are profunctors, and the 2-morphisms are morphisms of profunctors. The non-obvious part of this assertion is that ...
- 7,237
10
votes
1
answer
828
views
Decomposing a large colimit as a pushout of smaller colimits
I would like to find a reference in the literature for the following result. I have it on high authority that it isn't in 'Categories for the Working Mathematician' and I can't find it in Borceux's ...
- 1,680
3
votes
0
answers
356
views
Colimit of an etale diagram of schemes
It is known that the category of schemes is not cocomplete (e.g. see this question: Colimits of schemes). However, do diagrams of schemes for which every morphism is etale have colimits? More ...
- 15.5k
5
votes
1
answer
918
views
colimits of spectral sequences
I'm looking for some references about colimits of spectral sequences.
More precisely: let $X : I \longrightarrow \cal{C}$ be a functor from a filtered category $I$ to the category of double cochain ...
- 1,975
1
vote
2
answers
708
views
Tot and colimits
This must be a well-known exercise with spectral sequences, but I don't know a reference for it. I'm trying to figure out when does $Tot$ commute with colimits.
More precisely, let $X$ be a double ...
- 1,975
3
votes
2
answers
2k
views
Simple examples of homotopy colimits
I am following the explicit construction of homotopy colimits as described by Dugger in the paper: "Primer on homotopy colimits", which can be found here: http://www.uoregon.edu/~ddugger/hocolim.pdf
...
- 53
9
votes
2
answers
1k
views
Coend computation continued
This is a follow-up question to this coend computation. Here's an attempt at a slightly simpler computation:
$\int^{a \in A} \mbox{hom}_A(a,a)$
This should be similar to the trace operator. In ...
- 1,532
18
votes
1
answer
4k
views
Coend computation
Let
$F:A^{\mbox{op}} \to \mbox{Set}$
and define
$G_a:A\times A^{\mbox{op}} \to \mbox{Set}$
$G_a(b,c) = \mbox{hom}(a,b) \times F(c)$.
I think the coend of $G_a$,
$\int^AG_a$,
ought to be $F(a)$--...
- 1,532
38
votes
7
answers
12k
views
Limits in category theory and analysis
Is it possible to regard limits in analysis (say, of real sequences or more generally nets in topological spaces) as limits in category theory? Is there some formal connection?
Edit ('13): Perhaps it ...
- 63.1k
5
votes
2
answers
6k
views
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
...
- 5,515
17
votes
2
answers
5k
views
Exactness of filtered colimits
Are filtered colimits exact in all abelian categories?
In Set, filtered colimits commute with finite limits. The proof carries over to categories sufficiently like Set (i.e. where you can chase ...
- 1,500
17
votes
10
answers
3k
views
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
7
votes
2
answers
2k
views
Is there a category in which finite limits and directed colimits *don't* commute
Andrew Critch asks at the 20-questions seminar:
In Set and AbGrp (the categories of sets and abelian groups, respectively), finite limits commute with directed colimits. As an example, if you're ...
- 1,059