All Questions
Tagged with colimits or limits-and-colimits
105 questions with no upvoted or accepted answers
4
votes
0
answers
212
views
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
3
votes
0
answers
55
views
Universal property of 2-presheaves and pseudo/lax/colax natural transformations
For each small 2-category $\mathscr K$, the 2-category $[\mathscr K^\circ, \mathrm{Cat}]$ of 2-functors and 2-natural transformations has a universal property: it is the free cocompletion of $\mathscr ...
- 10.6k
3
votes
0
answers
156
views
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 ...
- 10.1k
3
votes
0
answers
85
views
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 $P_n \colon \mathsf{C}^{op} \times \mathsf{C} \to \mathsf{Set}$ ...
- 326
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
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
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
3
votes
0
answers
82
views
Does a cartesian transformation induce a cartesian transformation on absolute limit cones?
Fix a category or $\infty$-category $C$ with all small limits.
We call a natural transformation $\alpha\colon f\to g$ between two functors $f,g\colon K\to C$ Cartesian, if for every arrow $k\colon a\...
- 725
3
votes
0
answers
139
views
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
3
votes
0
answers
593
views
Inverse limits of schemes and open subsets
Let $R$ be a discrete valuation ring, $\{A_i\}_{i \in I}$ be a direct system of $R$-algebras and $A$ the limit of the system. Let $X$ be a noetherian projective scheme over $\mathrm{Spec}(R)$. ...
- 2,022
3
votes
0
answers
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
3
votes
0
answers
361
views
Which reflexive coequalizer diagrams are projectively cofibrant?
Consider the walking reflexive pair category W,
which consists of two objects 0 and 1 and three generating
morphisms f: 0→1, g: 0→1, and h: 1→0
satisfying the relation fh=gh=id₁.
Consider the ...
- 37.8k
3
votes
0
answers
528
views
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. ...
- 1,271
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
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
2
votes
0
answers
100
views
Weighted limits and co-Yoneda
Is there a good reference that discusses weighted limits through the lens of the co-Yoneda embedding?
Recall that the limit of a functor $F:\mathcal{C}\to{\bf Set}$ is canonically given by the set $${...
- 10.1k
2
votes
0
answers
163
views
Initial cones, terminal cocones
We're all familiar with terminal cones/initial cocones in the form of limits/colimits.
What about initial cones and terminal cocones?
While writing an answer to a related question the concept ...
- 10.1k
2
votes
0
answers
189
views
Is the homotopy limit of derived schemes along affine maps a derived scheme?
The title question is true in the setting of ordinary limits and ordinary schemes; that is, given an inverse limit of schemes along affine maps, the limit still lives in the category of schemes.
I'd ...
- 301
2
votes
0
answers
108
views
Characterization of inverse limits of finite-dimensional convex cones
Consider a countable inverse system $C_1\substack{f_1 \\ \leftarrow} C_2 \substack{f_2 \\ \leftarrow} C_3 \substack{f_3 \\ \leftarrow} \ldots$ where the $C_i$ are finite-dimensional convex cones of ...
- 21
2
votes
0
answers
93
views
Is there a category of "chains of modules" that behaves well with taking direct limits?
I came up with the following definition of a category of certain "chains of modules" and want to know if this concept is already known and studied.
Let $R$ be ring. An object in our category ...
- 696
2
votes
0
answers
109
views
Cofinality for natural transformations
Given a diagram $D\colon\mathcal{C}\longrightarrow\mathcal{D}$, we say that a functor $J\colon\mathcal{I}\longrightarrow{C}$ is cofinal if we have a natural isomorphism
$$
\mathrm{colim}\left(\mathcal{...
- 11.8k
2
votes
0
answers
255
views
Ultralimit of metric spaces vs. inductive limits of underlying topological spaces
Let $\{(X_n,d_n)\}_{n =1}^{\infty}$ be a sequence of bounded metric spaces such that:
$X_n \subseteq X_{n+1}$ is a metric subspace of $X_n$.
Let $\omega$ denote a non-principal ultrafilter (i.e.: ...
- 5,405
2
votes
0
answers
116
views
Primitive ideals of inductive limits of $C^*$-algebras
I am trying to understand ideals of direct limits in the category of $C^{\ast}$-algebras.
Let $(A_n,f_n)$ be a direct sequence of $C^{\ast}$-algebras and let $I$ be a primitive (modular) ideal of ...
- 1,115
2
votes
0
answers
60
views
direct limit in locally convex modules and continuous map
Let we have short exact sequences of LCM over LC algebra $A$ with continuous linear maps
$$
0\to B_j\;{\xrightarrow {\ f_j\ }}\;C_j\;{\xrightarrow {\ g_j\ }}\;D_j\to 0.
$$
We can take inductive limit (...
- 171
2
votes
0
answers
163
views
Coend of full subcategory
$\require{AMScd}$Let $F:\mathcal{C}^{op}\times \mathcal{C} \to \mathcal{D}$ be a functor and $\mathcal{C}' \subseteq \mathcal{C}$ a full subcategory. Assume that the coends $C$ over $F$ and $C'$ over $...
- 1,813
2
votes
0
answers
52
views
Inductive limits of unitary groups and quantum mechanics
I'm curious if someone has seen concrete applications of $U(\infty)$ in quantum mechanics. Is it possible, for example, in some particular cases to write down the propagator as a limit of a sequence ...
- 445
2
votes
0
answers
134
views
Colimits of small categories in the category of big categories
This is a question regarding set-theoretic issues in higher category theory. Let $\text{Cat}_{\infty}^{big}(\omega)$ be the $\infty$-category of (not necessarily small) $\infty$-categories with ...
- 21
2
votes
0
answers
257
views
Why holim and not Rlim?
Let $\mathcal{A}$ be a Grothendieck category (I care mostly about modules over a ring). Let $\operatorname{Ch}^+(\mathcal{A})$ the category of bounded below cochain complexes, $\operatorname{D}^+(\...
- 3,079
2
votes
0
answers
133
views
Group on 2 generators and greedy relations that preserve exponential growth
I'm not sure if there's a specific question here, other than perhaps "is this object studided" / "is there a better way of looking at it"; if it's too vague for this forum I apologize.
First take the ...
- 1,203
2
votes
0
answers
200
views
Connected families of objects in $(\infty,1)$-categories?
Background
Let $C$ be an (extensive) category. An object $X\in C$ is called connected if the functor $Hom(X,-):C\rightarrow Set$ preserves coproducts.
Given a category $C$, one can consider the ...
- 439
2
votes
0
answers
184
views
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
$$...
- 18.4k
2
votes
0
answers
101
views
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 ...
- 173
2
votes
0
answers
160
views
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
2
votes
0
answers
564
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
1
vote
0
answers
54
views
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
1
vote
0
answers
127
views
Extremally disconnected sets as building blocks for compact Hausdorff spaces
Is every compact Hausdorff space the filtered colimit of compact extremally disconnected spaces?
- 1,638
1
vote
0
answers
88
views
Colimits from van Kampen cocones
Let $\mathcal{C}$ be a category with pullbacks, $\mathcal{J}$ a small category, $F : \mathcal{J} \to \mathcal{C}$ a diagram and $\kappa : F \Rightarrow \Delta X$ a cocone in $\mathcal{C}$. Let $\...
- 468
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 ...
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
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
1
vote
1
answer
379
views
Creating an inverse system which "stratifies density"
Setting:
Let $X'$ be a dense subset of an infinite-dimensional Fréchet space $X$ and suppose that $(X_n')_{n \in \mathbb{N}}$ is a nested sequence of non-empty subsets of $X'$ satisfying
$$
\bigcup_{n ...
- 5,405
1
vote
0
answers
67
views
Comparison of inductive limit topology with very-rapidly decaying non-convex $L^{!/2}$-space topology
This is related to these posts and here.
Let $L^1([n,n+1])$ denote the subspace of $L^p$-functions on $[0,\infty)$ essentially supported on $[-n,n]$. Denote the accelerated $\ell^1$-direct sum ...
- 5,405
1
vote
0
answers
61
views
Refinement: Can $L^1_{loc}$ be represented as colimit?
Let $\mu$ be a $\sigma$-finite measure on a measure space $(\mathbb{R}^d,\Sigma)$. Can $L^1_{\mu,loc}$ be represented as an injective-limit in the category of LCS (locally convex spaces and ...
- 5,405
1
vote
0
answers
222
views
Surjectivity of colimit maps for topological spaces
From this post and to (co)completness of the category Top of topological spaces and continuous functions we know that for any diagram $B_i$ and an object $A$ in Top, there are natural maps of sets:$\...
- 5,405
1
vote
0
answers
99
views
Gluing together dense subset of Projective Limit in $Ban_1$
Let $(X_n,\pi_n^{m})$ be a countable projective system in the category Ban$_1$ of Banach spaces and short linear maps (is (continuous) linear constructions). Then (co)-completeness of this category ...
- 5,405
1
vote
0
answers
213
views
Zero in colimit of sheaves category
This question is motivated by showing that the category $\mathbf{Sheaves} (X)$ from the open subset excluding the empty set category to the category of abelian group $\mathbf{Ab}$ has enough injective ...
- 1,168
1
vote
0
answers
132
views
When is a nested sequence of closed sets a colimit?
Let $X$ denote a topological space and $X_0\subset X_1\subset \ldots\subset X$ a nested sequence of closed subsets of $X$ such that $$ \bigcup_i X_i =X$$
It is easy to see that in the general case $X$...
- 1,174
1
vote
0
answers
102
views
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
1
vote
0
answers
195
views
Limit as a pushout
In Categories for Working Mathematician, Mac Lane describe a cartesian product as a limit for a functor $F$ from a discrete category $|J|$ : Any cone from an object $Z$ to $F$, is a collection of ...
- 231
1
vote
0
answers
77
views
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