All Questions
Tagged with colimits or limits-and-colimits
347 questions
7
votes
1
answer
697
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 ...
- 63.9k
10
votes
2
answers
690
views
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 ...
- 63.9k
5
votes
0
answers
146
views
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 ...
8
votes
2
answers
339
views
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
14
votes
2
answers
761
views
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}...
- 21.3k
3
votes
1
answer
145
views
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 ...
- 66.7k
3
votes
0
answers
180
views
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
1
vote
0
answers
90
views
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 ...
3
votes
0
answers
109
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 ...
- 10.6k
5
votes
1
answer
217
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 \...
- 37.8k
4
votes
0
answers
244
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 \...
- 1,361
2
votes
1
answer
240
views
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\...
CommunityBot
- 1
2
votes
3
answers
224
views
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 \...
CommunityBot
- 1
8
votes
0
answers
165
views
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
5
votes
1
answer
256
views
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, ...
- 42.4k
3
votes
1
answer
285
views
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 ...
- 16.4k
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 ...
- 35.4k
6
votes
0
answers
291
views
When is every element of a coend of abelian groups contained in one of the summands?
Let $I$ be a small category and let $D : I^{\mathrm{op}} \times I \to \mathsf{Ab}$ be a functor. The coend
$$\int^{i \in I} D(i,i)$$
can be constructed as the direct sum $\bigoplus_{i \in I} D(i,i)$ ...
- 63.1k
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?...
- 66.7k
7
votes
1
answer
306
views
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 ...
- 10.6k
4
votes
1
answer
286
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\...
- 66.7k
4
votes
1
answer
132
views
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 ...
- 15.9k
8
votes
1
answer
341
views
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 ...
- 15.8k
7
votes
1
answer
615
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 ("...
- 2,607
2
votes
1
answer
232
views
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 ...
- 15.8k
2
votes
1
answer
211
views
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_{...
CommunityBot
- 1
15
votes
0
answers
332
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 ...
- 63.9k
9
votes
1
answer
405
views
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 ...
- 25.2k
6
votes
0
answers
139
views
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 / ...
- 10.6k
7
votes
1
answer
236
views
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 ...
- 5,429
11
votes
1
answer
555
views
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 $X$...
- 3,938
10
votes
0
answers
446
views
Milnor's universal bundle as a colimit?
I have had Milnor's construction of the classifying space of a topological group explained to me on multiple occasions, and seen it described briefly in various places. But only now am I reading the ...
- 1,225
3
votes
1
answer
390
views
Filtered colim of F-groups
A group G is said to have a property F if there exists a finite aspherical CW-complex of which it is the fundamental group (according to wikipedia).
question: is there a full characterization of ...
- 3,730
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. ...
- 63.9k
10
votes
1
answer
581
views
Explicit description of the oplax limit of a functor to Cat?
The nCatLab Grothendieck construction page gives an explicit description of the oplax colimit of any functor to Cat. Can someone give me a similarly explicit description (the objects and morphisms) ...
- 3,249
5
votes
1
answer
317
views
Is Cauchy completion the largest extension with the same free cocompletion?
EDIT Title has been edited.
Let $C$ be a category, and $$\hat{C} = [C^{op}, (Set)]$$ be its free cocompletion. Despite its name, the free cocompletion of free cocompletion is not equivalent to the ...
- 5,230
1
vote
0
answers
71
views
Gluing categorical limit over subgraphs
Let $C$ be a category, and $\Gamma$ a graph in $C$. Under good conditions it makes sense to talk about the limit $\lim \Gamma$ of $\Gamma$ in $C$.
Suppose $\Gamma$ is the union of two subgraphs $\...
- 5,230
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 ...
- 364
5
votes
1
answer
245
views
Colimits of short exact sequences of C*-algebras
Assume I have an inductive system of short exact sequences of $C^{\ast}$-algebras (i.e., short exact sequences $0 \to A_n \to B_n \to C_n \to 0$ together with transformations from the $n$-th to the $(...
- 2,263
1
vote
0
answers
81
views
Examples of spaces which have explicit expression as colimits in $\mathrm{Top}$
$\DeclareMathOperator\Ball{Ball}$Question: What "well-known" spaces can be explicitly written down in the form $\bigcup_k \phi_k C(K_n,\mathbb{R}^m)$; where $K_n$ is a non-empty compact ...
- 5,405
5
votes
0
answers
368
views
Continuity property for Čech cohomology
Suppose we have an inverse system of compact Hausdorff spaces $\lbrace X_i , \varphi_{ij} \rbrace_{i\in I}$ and that each space has a presheaf $\Gamma_i$ assigned to it in such a way that $\Gamma_i(\...
- 63.9k
7
votes
2
answers
812
views
Do filtered colimits commute with finite limits in the category of pointed sets?
It seems to be the case that filtered colimits commute with finite limits in the category Set (for instance, this is shown in Why do filtered colimits commute with finite limits?), but does the same ...
- 6,162
5
votes
0
answers
220
views
Pushout of $C^*$-algebras using generalised morphisms
There is a known construction of pushout of $C^*$-algebras, or rather, the amalgamated free product, which is universal for commutative squares of $*$-homomorphisms. Jensen and Thomsen in their book ...
- 35.4k
7
votes
0
answers
266
views
Relation between two limit presentations of Eilenberg--Moore objects
Let $\mathbb{T}=({\cal T}\colon C\to C,\mu,\eta)$ be a monad (in the
$2$-category $\mathsf{Cat}$), which we view as a $2$-functor
$\mathbb{T}\colon\mathsf{B}\Delta_{\mathrm{a}}\to\mathsf{Cat}$ (where
$...
- 446
10
votes
1
answer
333
views
2-monads for categories with a class of (co)limits
This question concerns the strictness of (co)completions, at various levels of generality.
In Blackwell–Kelly–Power's Two-dimensional monad theory, the authors state
For instance, the 2-category $\...
- 940
9
votes
3
answers
911
views
Decomposing a (co)limit by decomposing the indexing diagram
Let $D: I \to \mathcal C$ be a diagram, and suppose we have a colimit decomposition $I = \varinjlim_{j \in J} I_j$ in $Cat$. Then under certain conditions, we can decompose the colimit of $D$ as $\...
- 13.5k
3
votes
0
answers
163
views
Classifying spaces of amalgamated topological monoids
Let $\mathsf{Top}_*$ be the category of well-based spaces and $\mathsf{TopMon}$ the category of topological monoids. Recall the James construction $\mathcal{J}:\mathsf{Top}_*\to \mathsf{TopMon}$ which ...
- 1,623
7
votes
2
answers
3k
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Does a fully faithful functor always preserve limits and colimits?
I read on this n-lab page that a fully faithful functor $F: C\to D$ reflects all limits and colimits by the universal property.
On the other hand, I think a fully faithful functor does not always ...
- 476
2
votes
0
answers
109
views
Cofinality for natural transformations
Given a diagram $D\colon\mathcal{C}\longrightarrow\mathcal{D}$, we say that a functor $J\colon\mathcal{I}\longrightarrow{C}$ is cofinal if we have a natural isomorphism
$$
\mathrm{colim}\left(\mathcal{...
- 11.8k
1
vote
1
answer
303
views
Comparison of product topology and colimit topology in sequence spaces
In Munkres Theorem 20.4 it is shown that the (relative) uniform topology induced by:
$$
d(x,y)\triangleq \sup_{n \in \mathbb{N}} d(x_n,y_n)
$$
is strictly finer than the product topology on $\prod_{n \...
CommunityBot
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