All Questions
Tagged with colimits or limits-and-colimits
347 questions
6
votes
1
answer
433
views
Equivalence relations in arbitrary categories
Let $C$ be a category and $A\in\mathrm{ob}(C)$. A relation is a subobject $q:Q\hookrightarrow A^{\times 2}$ and the quotient is defined as the coequalizer
$$A/Q:=\mathrm{coeq}\left(Q\stackrel{q}{\...
- 42.4k
17
votes
10
answers
3k
views
References for homotopy colimit
(1) What are some good references for homotopy colimits?
(2) Where can I find a reference for the following concrete construction of a homotopy colimit? Start with a partial ordering, which I will ...
- 6,023
6
votes
1
answer
685
views
Basic example of a formal affine scheme, functorial point of view
$\let\opn=\operatorname$For my BA thesis I have to describe formal groups from the functorial point of view. I am hence reading Strickland - Formal Schemes and Formal Groups, which is apparently the ...
- 11.8k
4
votes
1
answer
206
views
Algorithmically deciding existence of finite limits in a category
Given $\Sigma$ a consistent finite first order theory in vocabulary $L$, one can consider the category of its models $\mathcal{M}(\Sigma)$, its objects are the models of $\Sigma$ and arrows are ...
- 5,031
3
votes
1
answer
138
views
Two directed colimits of same spaces with different inclusions
For any natural number $n$, let $i_{n},j_{n}:X_{n}\rightarrow X_{n+1}$ be a pair of monomorphisms of simplcial sets.
Define $$X=\operatorname*{colim}_n \{\cdots X_n \rightarrow_{i_n} X_{n+1}\cdots \}...
- 2,152
0
votes
1
answer
203
views
Filtered colimit of a topological space
Suppose that $X$ is a space filtered by closed subspaces $X_{1}\subset X_{2}\subset \dots$.
As topological space $X=\operatorname{colim}_{n}X_{n}$.
We define $Y_{n}=X_{n+1}/X_{n}$, and consider the ...
- 4,713
6
votes
0
answers
630
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
$$\...
5
votes
1
answer
279
views
Inductive limit commutes with topological tensor product
Consider $H \left(U \right); U \subset \mathbb{C}$ - space of holomorphic functions with compact-open topology. In this topology, this space is Montel, nuclear and Frechet. I want to take the ...
- 799
2
votes
0
answers
60
views
direct limit in locally convex modules and continuous map
Let we have short exact sequences of LCM over LC algebra $A$ with continuous linear maps
$$
0\to B_j\;{\xrightarrow {\ f_j\ }}\;C_j\;{\xrightarrow {\ g_j\ }}\;D_j\to 0.
$$
We can take inductive limit (...
- 4,729
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
7
votes
0
answers
417
views
When do Kan extensions preserve colimits?
Assume that we have a pair of functors $Y:A \to B$ and $F:A \to C$ where $A$ is an essentially small category, $B,C$ are cocomplete categories and $Y,F$ preserve colimits. Assume also that for some ...
- 718
6
votes
1
answer
219
views
Is the projection onto the regular image an epimorphism?
Let $f:X\to Y$ be a morphism in a category $\mathcal{C}$.
Let $m:I\hookrightarrow Y$ be the regular image of $f$. This means that $f$ can be written as $f=m\circ e$, with $m$ regular mono (i.e. being ...
- 2,821
11
votes
1
answer
518
views
Weighted (co)limits as adjunctions
It's well known that a category $\mathcal{C}$ having (conical) limits/colimits of shape $\mathcal{D}$ is equivalent to the diagonal functor $\Delta^\mathcal{D}_\mathcal{C}:\mathcal{C}\to\mathcal{C}^\...
- 10.1k
27
votes
2
answers
2k
views
Is every commutative ring a limit of noetherian rings?
Edit of Feb. 14, 2019. After Laurent Moret-Bailly's accepted answer, only Questions 4 and 5 remain open. I don't care that much about Question 4, but I'm very curious about Question 5, which is
Do ...
- 19.6k
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
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
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
1
vote
2
answers
691
views
Yoneda Embedding and pull back
Given a manifold $M$ we have a geometric stack associated to it namely $\underline{M}$ whose objects are smooth maps to $M$. For the sake of consistency I am writing $BM$ for $\underline{M}$.
Given a ...
- 6,023
15
votes
2
answers
697
views
Monoidal functors $\mathcal C \to [\mathcal D,\mathcal V]$ are monoidal functors $\mathcal C \otimes \mathcal D \to \mathcal V$?
It is well known (e.g., Reference for "lax monoidal functors" = "monoids under Day convolution" ) that if $\mathcal C$ is a monoidal $\mathcal V$-enriched category, then a monoid ...
- 383
5
votes
2
answers
474
views
Limits, colimits and universes
For many purposes in category theory, we consider limit and colimits of diagrams $F\colon\mathsf{J\to C}$ where $\mathsf{J}$ is small category, that is, a category where the classes of objects and ...
- 10.1k
2
votes
1
answer
130
views
Reflexive coequalizer and singular functor.
Does the singular functor $sing: Top\rightarrow SimpSet$ form the category of topological spaces to simplicial sets commutes with reflexive coequalizers?
Recall that the singular functor $sing$ is ...
- 3,411
11
votes
1
answer
1k
views
Functoriality of (co)limits in $\infty$-categories
I have some questions about the functoriality of (co)limits in $\infty$-categories, say in the framework of Lurie's Higher Topos Theory.
From the general stuff about Kan-extensions (HTT 4.3.2.6) ...
- 3,411
5
votes
0
answers
80
views
Is there an analogue of final functors for genuine 2-categorical limits
A functor $I\to J$ of $1$-categories is called final, if each undercategory $(j, I)$ is connected.
More generally, for $(\infty,1)$-categories there is an analogous notion where one requires the ...
- 725
12
votes
2
answers
706
views
Defining abstract varieties and their morphisms over a finitely generated subfield of the base field
Let $k$ be an algebraically closed field.
By a finitely generated subfield of $k$ I mean a subfield $k_0\subset k$ that is finitely generated over the prime subfield of $k$ (that is, over $\mathbb Q$ ...
- 101
5
votes
1
answer
279
views
Behaviour of direct limit with matrices
I am trying to understand direct limits in the category of $C^*$-algebras by self reading. My last question was also related to direct limits. Here is another of my doubts:
Let $(A_n,f_n)$ be a ...
- 3,768
2
votes
2
answers
723
views
Behaviour of Direct limit with quotient and double dual
I am trying to understand direct limit in category of $C^*$ algebras.
Is it well known that direct limit behaves well with double dual and quotient of $C^*$ algebras?
Any references or ideas?
P....
- 42.8k
7
votes
1
answer
366
views
Are sifted (2,1)-colimits of fully faithful functors again fully faithful? (And a de-categorified variant)
1) Suppose that I have a sifted diagram of categories $\mathcal{C}_i$, another of the same shape $\mathcal{D}_i$, and that I have a system $F_i:\mathcal{C}_i\to\mathcal{D}_i$ commuting with the ...
- 6,829
8
votes
1
answer
1k
views
Surjectivity of a map on inverse limits
(The following is crossposted from Math.SE, where the question did not receive any answers.)
I am looking for a proof of the following lemma from P. Gabriel's Des catégories abéliennes (Chap. IV, §3, ...
- 16.4k
4
votes
1
answer
338
views
Chow group over function field and algebraic equivalence
It is known that for smooth projective varieties $X,Y$ over $k=\bar k,$ $$CH^d(X_{k(Y)})=\varinjlim_{U\subset Y\ open}CH^d(X\times_k U)$$
I was wondering whether there was such an equality with ...
- 532
1
vote
0
answers
132
views
When is a nested sequence of closed sets a colimit?
Let $X$ denote a topological space and $X_0\subset X_1\subset \ldots\subset X$ a nested sequence of closed subsets of $X$ such that $$ \bigcup_i X_i =X$$
It is easy to see that in the general case $X$...
- 1,174
2
votes
3
answers
235
views
Example of an $\omega_1$ decreasing chain of dense semicontinua?
In his well-known paper Bellamy constructs an indecomposable continua with exactly two composants. The setup is as follows:
We have an inverse-system $\{X(\alpha); f^\alpha_\beta: \beta,\alpha < \...
- 4,786
9
votes
2
answers
839
views
Calculating limits progressively
Consider the problem of finding the limit of the following diagram:
$$ \require{AMScd} \begin{CD}
& & & & E
\\ & & & & @VVV
\\ && C @>>> D
\\ & &...
- 66.8k
7
votes
0
answers
219
views
Pushout of Nisnevich sheaves
Let us consider the projective line $\mathbb{P}^1$ over a field $k$ and take the following open embeddings
$$j_{\epsilon}\colon \mathbb{P}^1\setminus\{0,\infty\} \to \mathbb{P}^1\setminus\{\epsilon\}$$...
- 303
9
votes
1
answer
657
views
Must an inverse limit of simply connected groups be simply connected?
While the fundamental group $\pi_1$ preserves products, it is not true in general that an inverse limit of simply connected topological spaces is simply connected. I would like to know if similar ...
- 7,193
14
votes
0
answers
919
views
Grothendieck construction and coends
In category theory, both the Grothendieck construction and coends are represented by a sort of "integral sign", respectively:
$$
\int F
$$
for a functor $F:C\to\mathbf{Cat}$,
and:
$$
\int^x G(x,x)
$$
...
- 2,129
7
votes
1
answer
197
views
Homology of a limit of semidirect products
Suppose I have two families of groups $A_k$ and $B_k$ indexed by the natural numbers and suppose $B_k$ acts on $A_k$. Suppose there are groups homomorphisms $A_{k+1} \rtimes B_{k+1} \to A_k \rtimes ...
- 6,813
12
votes
1
answer
458
views
About pointwise Kan extension
Suppose that you want to look at the left Kan extension of a functor $F : \mathcal{C} \to \mathcal{A}$ along a functor $K : \mathcal{C} \to \mathcal{B}$. It is widely known that if the colimit of the ...
- 63.9k
3
votes
1
answer
152
views
Conditions on a diagram category such that constant limits always exist and is constant
I am looking for a good set of sufficient conditions to put on a category $C$ such that the following becomes true:
Given any $\infty$-category $S$ and a functor $f\colon N(C)\to S$ which is ...
- 725
7
votes
0
answers
555
views
maximal tensor product commutes with inductive limits
Let $(A_n, \phi_n)$ be an inductive system of $C^*$ algebras and let $B$ be an arbitary $C^*$ algebra.
I want to prove $(\varinjlim A_n)\otimes_{max} B \cong \varinjlim (A_n \otimes_{max} B)$. This ...
- 1,188
5
votes
0
answers
211
views
A strict directed colimit of Hausdorff locally-convex spaces that is not Hausdorff
We work in the category of locally-convex spaces (morphisms are the continuous linear maps). Let $\Lambda$ be a directed set, for every $\lambda \in \Lambda$ let $V_{\lambda}$ be a locally-convex ...
- 16.4k
8
votes
0
answers
291
views
Loop space functor and sequential colimits of inclusions
The question is about a fact that is mentioned as "evident" everywhere in the literature, so my guess is that some small detail is passing over my head. Here it is:
Let $X_0\hookrightarrow X_1 \...
- 6,051
1
vote
1
answer
161
views
terminology problem related to finitely generated objects
If $x$ is an object of a category $C$, one usually says that $x$ is if finite presentation (or compact) if for any direct filtered system ($y_i$) in $C$, the canonical map
$$f : \text{colim}_i \, \...
- 6,051
2
votes
0
answers
134
views
Colimits of small categories in the category of big categories
This is a question regarding set-theoretic issues in higher category theory. Let $\text{Cat}_{\infty}^{big}(\omega)$ be the $\infty$-category of (not necessarily small) $\infty$-categories with ...
- 21
2
votes
0
answers
257
views
Why holim and not Rlim?
Let $\mathcal{A}$ be a Grothendieck category (I care mostly about modules over a ring). Let $\operatorname{Ch}^+(\mathcal{A})$ the category of bounded below cochain complexes, $\operatorname{D}^+(\...
- 3,079
2
votes
1
answer
113
views
Lax co/limit as evaluation on terminal/initial
A quick question about lax co/limits.
Strictly, when $F : J\to \bf A$ is a diagram and $J$ has an initial object $\varnothing$, then $\varprojlim F \cong F(\varnothing)$; dually, if $\cal J$ has a ...
- 66.8k
4
votes
1
answer
443
views
Do coproducts of infinity-groupoids commute with pullbacks?
As stated in this question, coproducts commute with pullbacks in the category of sets.
Let $Grpd_{\infty}$ denote the $\infty$-category of $\infty$-groupoids. Do coproducts commute with pullbacks in $...
CommunityBot
- 1
2
votes
0
answers
133
views
Group on 2 generators and greedy relations that preserve exponential growth
I'm not sure if there's a specific question here, other than perhaps "is this object studided" / "is there a better way of looking at it"; if it's too vague for this forum I apologize.
First take the ...
- 1,203
1
vote
1
answer
134
views
Do "factoradic" lists form a finitary monad?
I'm trying to understand better what it means for a monad to be finitary. I know that Lawvere theories correspond to finitary monads, but I don't really understand the definition in terms of filtered ...
- 4,459
3
votes
1
answer
246
views
Localization of the pullback diagram
In the paper, Topologically Defined Classes of Commutative Rings, localization of the pullback diagram (with $v,$ surjective)
$$
\begin{array}
DD & \stackrel{v\ '}{\longrightarrow} & A \\
\...
- 3,041
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