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Questions tagged [limits-and-colimits]

For questions on limits and colimts in the sense of category theory, and related notions.

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2answers
320 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 ...
2
votes
1answer
58 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 ...
9
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1answer
234 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) ...
4
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0answers
53 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 ...
4
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2answers
284 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 ...
4
votes
1answer
133 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 ...
12
votes
2answers
521 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 $\mathbb Q$ ...
2
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2answers
137 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....
8
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1answer
143 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 ...
5
votes
1answer
176 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, ...
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0answers
73 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 $X$...
2
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3answers
206 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 < \...
9
votes
2answers
545 views

Calculating limits progressively

Consider the problem of finding the limit of the following diagram: $$ \require{AMScd} \begin{CD} & & & & E \\ & & & & @VVV \\ && C @>>> D \\ & &...
7
votes
0answers
117 views

Pushout of Nisnevich sheaves

Let us consider the projective line $\mathbb{P}^1$ over a field $k$ and take the following open embeddings $$j_{\epsilon}\colon \mathbb{P}^1\setminus\{0,\infty\} \to \mathbb{P}^1\setminus\{\epsilon\}$$...
3
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0answers
233 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: $$ \int^x G(x,x) $$ ...
7
votes
1answer
165 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 ...
10
votes
1answer
233 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 ...
7
votes
1answer
333 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 ...
3
votes
1answer
91 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 ...
7
votes
0answers
243 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 ...
4
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0answers
123 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 ...
7
votes
0answers
172 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 \...
1
vote
1answer
96 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 \, \...
1
vote
0answers
102 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 ...
2
votes
0answers
163 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}^+(\...
2
votes
1answer
74 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 ...
2
votes
0answers
97 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
vote
1answer
72 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
votes
1answer
249 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 $...
3
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1answer
144 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 \\ \...
1
vote
1answer
317 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 ...
2
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0answers
162 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 ...
7
votes
2answers
591 views

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, ...
1
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1answer
252 views

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 \...
6
votes
1answer
366 views

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 ...
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vote
0answers
79 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 ...
3
votes
0answers
101 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 $\...
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0answers
67 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 arrow from ...
6
votes
2answers
301 views

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 $\...
2
votes
1answer
114 views

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 ...
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0answers
69 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 $CMon(C)$ (...
5
votes
1answer
147 views

(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 ...
2
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0answers
148 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 $$...
5
votes
0answers
355 views

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 ...
3
votes
2answers
336 views

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 ...
5
votes
1answer
103 views

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 ...
4
votes
1answer
175 views

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 ...
5
votes
0answers
171 views

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,...
6
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1answer
228 views

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?
0
votes
1answer
249 views

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}...