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

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**1**answer

121 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?

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votes

**1**answer

96 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}...

**3**

votes

**1**answer

152 views

### Definition of dense functors

Definition. A functor $F:\mathsf C\rightarrow \mathsf D$ is dense if every $D\in \mathsf D$ is the vertex of the following colimit $$\varinjlim \left(F\downarrow D\rightarrow\mathsf C\rightarrow \...

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**0**answers

43 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 ...

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votes

**0**answers

131 views

### Yoneda embedding and Horn sentences

The following is taken from Borceux and Bourn's Mal'cev, Protomodular, Homological, and Semi-Abelian Categories.
Metatheorem 0.1.3. Let $\mathcal P$ be a statement of the form $\varphi\implies \...

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votes

**0**answers

157 views

### $\mathbb{Z}_2$ as a colimit of $\mathbb{Z}^*$

Consider the multiplicative monoid $(\mathbb{Z}^*,\cdot)$ of nonzero integers. By multiplication, $\mathbb{Z}_2$ is a right (or left) $\mathbb{Z}^*$-set. According to the co-Yoneda lemma $\mathbb{Z}_2$...

**3**

votes

**1**answer

55 views

### Dual of colimit in $\text{Ban}_1$

I learned in J. Castillo's Hitchhiker guide to categorical Banach space theory that, by a theorem of Semadeni and Zidenberg, limits and colimits exist in the category $\text{Ban}_1$ of Banach spaces ...

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votes

**1**answer

142 views

### Explicit calculations of small homotopy limits of CDGAs

I would like to carry out explicit calculations of homotopy limits of certain simple diagrams of CDGAS. My set-up is the following : I have a finite graded poset $R$ with minimal element $0$ and a ...

**8**

votes

**1**answer

163 views

### Intuition for density comonad in relation to lifting problems

In Emily Riehl's Categorical Homotopy Theory, there is a section on Garner's Small Object Argument which I'm trying and failing to understand. Originally I followed most of Garner's paper, using the ...

**2**

votes

**1**answer

138 views

### Appropriate morphisms and 2-morphisms in Ind(C)

As I was trying to understand the category $Ind(C)$ of diagrams of the form $I \to C$, where $I$ is a small filtered $(0,1)$-category, I wondered whether it is possible to define morphisms directly, ...

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**0**answers

130 views

### Topological localization of (infinite) inverse limits

The classical localization of topological spaces at a given set of primes $\mathcal{P}$ is a functor $\mathcal{T}\xrightarrow{(-)_{(\mathcal{P)}}}\mathcal{T}$ from a suitable category of topological ...

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**1**answer

189 views

### How “nondegenerate” are amalgamated free products of C*-algebras?

In the following, I assume all algebras are unital. Let $A$ and $B$ be C*-algebras that each contain (isomorphic copies of) a common C*-subalgebra $C$. Let $A *_C B$ denote the amalgamated free ...

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votes

**1**answer

213 views

### colimits in Cat via coproducts and coequalizers

I am attempting to do a calculation of a colimit in $Cat$, the category of small categories. To this end, people have suggested that I do this by calculating coproducts and using coequalizers. I ...

**2**

votes

**1**answer

80 views

### What do you get when you apply a universal cocone to a colimit functor

Any colimit can be represented as a functor $F$ left adjoint to a particular diagonal functor $\Delta: C \rightarrow C^J$. The unit of this adjunction is the natural transformation $\eta_K: K \...

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votes

**1**answer

133 views

### What is an example of a colimit-dense generator which is not dense?

An object $G$ of a category $\mathcal{C}$ is a dense generator if every object $X$ is the colimit of the canonical diagram of copies of $G$ mapping to $X$. (This canonical diagram is indexed by the ...

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votes

**1**answer

298 views

### Representable functors and direct limits

Let $\mathcal{F}:\mathrm{Sch}/S \to \{\mathrm{Sets}\}$ be a representable functor. Denote by $X$ the scheme representing $\mathcal{F}$. The question is whether the natural tranformation $\mathcal{F}(-)...

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**0**answers

127 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 $n>0$, ...

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**0**answers

100 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 $\mathrm{Spec}(R)$. ...

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**0**answers

197 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 ...

**4**

votes

**1**answer

165 views

### Relation between dual of nuclear space $(\substack{\text{lim} \\ \leftarrow i} H_i)'$ and $\substack{\text{colim} \\ i \rightarrow } H_i$

Let $\substack{\text{lim} \\ \leftarrow i} H_i$ be a nuclear space, considered as the limit of the codirected diagram $$... \to H_2 \to H_1 \to H_0,$$
with $f_{ji}:H_i \to H_j$ being the trace class ...

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vote

**0**answers

121 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 ...

**4**

votes

**1**answer

632 views

### Direct limit of compact topological spaces

Let $\{X_n\}_{n\in \mathbb{N}}$ be a direct system of compact topological spaces, meaning that we have morphisms $f_i\colon X_i \to X_{i+1}$ with the necessary compatibility conditions. Is there any ...

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votes

**0**answers

261 views

### Is it possible to assume that an étale neighborhood is connected?

I am new to étale topology (though I've seen Grothendieck's sites before).
Let $S:=\mathcal{O}^\textrm{sh}_{X,x}$ be the strict local ring of a point $x$ of a scheme $X=\operatorname{Spec}R$ (over a ...

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vote

**1**answer

102 views

### Groups arising as direct limits of a stationary system of primitive matrices over the integers

I am interested in the kinds of groups of the form $\displaystyle\lim_{\longrightarrow}(\mathbf{Z}^k,M)$ where $M$ is a primitive (some power of $M$ has strictly positive components) $k\times k$ ...

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**0**answers

102 views

### Contractibility of a poset-indexed colimit

Let $(X,\leq)$ be a poset with distinguished element $p$, and let $P'$ be the poset of "finite chains which weakly descend to $p$" given by all $\sigma = (x_0 \geq x_1 \geq \cdots \geq x_k \geq p)$ ...

**1**

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**0**answers

96 views

### Pre-Order induced by continuous functions

I'm an newbie in category theory, but I want use it to solve a pre-order question I encountered in my research:
Let $X$ be a convex&compact subset of $\mathbb{R}^n$. $f,g: X \rightarrow [0,1]$ ...

**4**

votes

**2**answers

267 views

### Canonical presentation of pro-modules over pro-rings

Let $A = (\dotsc \twoheadrightarrow A_2 \twoheadrightarrow A_1 \twoheadrightarrow A_0)$ be a (commutative) pro-ring with surjective transition maps. Consider the category $\mathcal{M} := \varprojlim_i ...

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**1**answer

438 views

### Do direct limits (filtered colimits) commute with pullbacks, in C*-algebras?

I asked this question already on math.stackexchange, but maybe it is also useful to ask this here, since it was not answered there.
Suppose we have three directed sequences of $C^*$-algebras, say $(...

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votes

**3**answers

404 views

### Maximum cardinality of a filtered limit of finite sets

Let $(I,<)$ be a directed, partially ordered set. Consider an inverse system $(S_i)_{i \in I}$ of finite sets, i. e. a functor $S:I^{op}\to \mathbf{FinSet}$. What is the maximum possible ...

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votes

**0**answers

161 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 ...

**0**

votes

**1**answer

169 views

### Direct limit of primitive integral matrices

I'm interested in computing the direct limit of an arbitrary $2\times 2$ primitive matrix over $\mathbf{Z}$. That is for a fixed primitive matrix $M$, the colimit $\displaystyle\lim_{\stackrel{\...

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**0**answers

144 views

### Continuity of Kan extension along the Yoneda embedding

Let $\mathcal{C}$ be a category and $h_-: \mathcal{C} \to \mathrm{Set}^{\mathcal{C}^{op}}$ be the Yoneda embedding. Let $\mathcal{A}$ be a cocomplete category and $F: \mathcal{C} \to \mathcal{A}$ a ...

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votes

**2**answers

395 views

### Unexpected interaction between limits and colimits

Let $D$ be a limit-complete category. My vague question is: given two diagrams in $D$, what comparisons between them induce a morphism of their limits? I'm especially interested in the case that the ...

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vote

**4**answers

324 views

### Homology of infinite intersection

If $X_1\supseteq X_2\supseteq \ldots$ is a sequence of "nice" compact spaces, I would like to know whether the natural map from $H_*(\cap X_i)$ to the inverse limit $\lim \, H_*(X_i)$ is surjective. ...

**22**

votes

**1**answer

297 views

### Finite limits in the category of smooth manifolds?

The category of smooth manifolds does not have all finite limits. However, it does have some limits: it has finite products, it has splittings of idempotents, and it has certain other limits if we ...

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votes

**0**answers

363 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. ...

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votes

**1**answer

584 views

### What are the uses of Limits and Colimits of Category Theory in every day problems? [closed]

I am interested in knowing how we can use the concepts of Limits and Colimits in modeling problems in every day life? Could anyone provide (Software) engineering examples, perhaps? Or describe ...

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votes

**1**answer

271 views

### Reference request: colimits of locally presentable categories

Consider the 2-category of locally presentable categories, cocontinuous functors, and natural transformations. I believe that this 2-category is 2-cocomplete in the sense of containing all small 2-...

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votes

**1**answer

323 views

### Is there a practical criterion to determine whether the limit of a diagram of real chain complexes is also a homotopy limit?

Consider a diagram D: I→ChR of real connective chain complexes.
In the example I have in mind all chain complexes are concentrated in some fixed degree n.
There is a canonical map lim D → holim D ...

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votes

**0**answers

205 views

### Ends and Coends - Analogues for higher arity - Horn Filling

Consider the setting of categories enriched over a suitable monoidal category $\mathbb V$.
We define $$\mathrm{Dist}(X,Y):=\mathbb V−\mathrm{Cat}(X^ \mathrm{op}⊗Y,\mathbb V).$$
Recall the definition ...

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votes

**1**answer

254 views

### Limits in span categories

What are the limits in the span categories? and what is known about them in the literature?

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**1**answer

240 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) ...

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votes

**1**answer

274 views

### Does Ind-completion commute with finite limits?

The broad and vague question is in the title. The more precise question is:
Say $\{\mathcal{C}_i\}$ is a finite diagram of (essentially small) stable $\infty$-categories and exact functors with ...

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votes

**1**answer

112 views

### Co-completeness of differential stacks?

I once heard a rumour that various nice categories of stacks were co-complete. Gepner and Henriques, working from the groupoids point of view, give a construction [link] of 2-colimits of topological ...

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votes

**2**answers

350 views

### grothendieck construction for profunctors

Given categories $X$ and $Y$ and a strong functor
$$D:X^{op}\times Y\to Cat$$
we can of course build the oplax colimit
$$\mathrm{colim}^{oplax}_{X^{op}\times Y}D$$
via the usual (covariant) ...

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vote

**0**answers

140 views

### Composition of Cat-valued distributors - compatible with grothendieck construction?

Let $C$ be a category and $F\in[C^{op}, Cat]$ be a strong functor.
(1) There are functors
$$hom_C(c',c)\times F(c)\to F(c').$$
(2) The grothendieck construction gives a 2-equvalence
$$\int_C: [C^{...

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votes

**2**answers

299 views

### Projective limit construction of a semigroup

Let $\tilde{\mathbb N}$ be the Abelian semigroup (under addition) given by $\mathbb N\cup\{0,\infty\}$, and let $S_n$ be the Abelian monoid $\tilde{\mathbb N}^{2^n}$ under point-wise addition. ...

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votes

**1**answer

492 views

### Which E_∞-spaces are homotopy colimits of k-truncated E_∞-spaces?

This question is closely related to my previous question about modules over truncated sphere spectra, in particular, it has the same motivation.
Recall that every space (or ∞-groupoid) can be ...

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votes

**0**answers

418 views

### Constructing pointwise Kan extensions as adjoints to some functor

Background
I'm working on formalizing some category theory in Coq at https://bitbucket.org/JasonGross/catdb. Currently, I'm in the process of formalizing pointwise Kan extensions. Partly because I'...

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votes

**1**answer

314 views

### Is the evaluation of polynomial functors appropriately continuous?

I'd like a nice proof of the following fact.
Let $C$ and $D$ be categories, and let $\mathbf{Cat}/(C\times D)$ be the usual (1-categorical) slice category whose objects are triples $(X,F\colon X\to C,...