# Questions tagged [limits-and-colimits]

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

**14**

votes

**2**answers

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

**1**answer

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**

votes

**1**answer

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**

votes

**0**answers

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**

votes

**2**answers

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

**1**answer

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

**2**answers

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**

votes

**2**answers

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**

votes

**1**answer

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

**1**answer

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, ...

**1**

vote

**0**answers

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**

votes

**3**answers

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

**2**answers

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

**0**answers

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**

votes

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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**

votes

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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**

votes

**1**answer

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

**1**answer

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**

votes

**0**answers

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

**2**answers

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**

vote

**1**answer

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

**1**answer

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 ...

**1**

vote

**0**answers

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

**0**answers

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 $\...

**1**

vote

**0**answers

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

**2**answers

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

**1**answer

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 ...

**1**

vote

**0**answers

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

**1**answer

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**

votes

**0**answers

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

**0**answers

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

**2**answers

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

**1**answer

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

**1**answer

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

**0**answers

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**

votes

**1**answer

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

**1**answer

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}...