Questions tagged [limits-and-colimits]

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

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3
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0answers
52 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 $P_n \colon \mathsf{C}^{op} \times \mathsf{C} \to \mathsf{Set}$ ...
4
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106 views

Are locally fully faithful 2-functors closed under 2-pushout in 2-Cat?

It is known that fully faithful functors are closed under pushouts in Cat (e.g. Lemma 4.9 of this paper). Are locally fully faithful 2-functors closed under 2-pushouts in the 2-category 2-Cat of 2-...
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97 views

Are weighted limits terminal in a category of cones?

Consider a Benabou-cosmos $(\mathcal{V},\otimes,J)$, $\mathcal{V}$-categories $\mathcal{I},\mathcal{C}$ and $\mathcal{V}$-functors $\mathcal{W}:\mathcal{I} \rightarrow \mathcal{V}$ and $\mathcal{D}:\...
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51 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 ...
2
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86 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 ...
7
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1answer
203 views

Stability properties of essential geometric morphisms

Notation. $\mathsf{Topoi}$ is the bicategory of topoi, geometric morphisms and natural transformations between left adjoints. $\mathsf{Topoi}_{\text{ess}}$ is the bicategory of topoi, essential ...
10
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1answer
185 views

Weak descent and effective equivalence relations

I want to prove that weak descent of a $1$-category implies the classical Giraud axioms. More precisely, let $\mathsf{C}$ be a cocomplete, finitely complete $1$-category. We say that $\mathsf{C}$ ...
0
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1answer
145 views

The direct limit of invertible functions on a variety

(I asked this question a couple of days back on Stackexchange but with no success, it seems elementary, but I am struggling to go about attempting it.) Let $X$ be a smooth geometrically integral ...
7
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1answer
244 views

Finite coproducts commute with which limits in Set?

It is well known that (small) coproducts commute with connected limits in $\mathbf{Set}$. With which class of limits do finite coproducts commute? Ideally, we should furthermore like to know whether ...
5
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87 views

Applications of $FP_\infty$ groups preserving direct systems

In [1], the author proves that given a group $G$ and a directed system $(M_\lambda)_\lambda$ of $G$-modules, the induced maps $$\varinjlim H^k(G,M_\lambda) \to H^k(G,\varinjlim M_\lambda)$$ are ...
13
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2answers
408 views

Original reference for categories of presheaves as free cocompletions of small categories

It is well known that, for a small category $\mathbf A$, the category $\widehat{\mathbf A} = [\mathbf A^\circ, \mathbf{Set}]$ of presheaves on $\mathbf A$ together with the Yoneda embedding $\mathbf A ...
10
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2answers
489 views

Do colimits of manifolds coincide with underlying colimits as topological spaces?

Categories of manifolds (possibly with extra structure) tend not to have all colimits. Other questions have addressed when colimits of manifolds exist. I'd like to know what we can say in general ...
3
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147 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 ...
3
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0answers
84 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 ...
2
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61 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 ...
5
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1answer
174 views

Dense subcategory of measurable spaces

Recall the notion of a dense subcategory $\mathcal{D}$ of a category $\mathcal{C}$. It means that the restricted Yoneda functor $\mathcal{C} \to \mathrm{Hom}(\mathcal{D}^{op},\mathbf{Set})$, $A \...
8
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2answers
266 views

Strongly compact categories (reference request)

The notion of a "compact category" was introduced by Isbell$\color{red}{^{1,2}}$. A locally small category $\mathcal{C}$ is called compact when every functor $\mathcal{C} \to \mathcal{D}$ ...
3
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1answer
98 views

Examples of (co)lax idempotent pseudocomonads on Cat

A lax idempotent pseudomonad, also called a KZ doctrine or KZ monad, is a pseudomonad $(T, \mu, \eta)$ with the property that $T \eta \dashv \mu \dashv \eta T$. Lax idempotent pseudomonads were ...
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1answer
226 views

Limit along the category of all algebraic curves over a field

Let $k$ be algebraically closed field of charactersistic zero and $\mathcal C$ be the category of irreducible smooth projective curves over $k$ and non-constant maps between them. I have a functor $F\...
7
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79 views

Original reference for the Fam construction

For a category $\mathbf C$, the category of families of $\mathbf C$, denoted $\mathrm{Fam}(\mathbf C)$ is the free coproduct completion of $\mathbf C$. Explicitly, the objects of $\mathbf C$ are given ...
2
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3answers
117 views

Conditions for exact projective limits for some Mittag-Leffler systems?

Let $(M_i)_{i\in I}$ and $(N_i)_{i\in I}$ be Mittag-Leffler systems of $R$-modules. I have a map $(h_i)$ of projective systems such that every $h_i$ is surjective. I search for conditions for $\lim \...
4
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0answers
203 views

Free vector space on a filtered limit

$\DeclareMathOperator\Set{Set}\DeclareMathOperator\Vect{Vect}\DeclareMathOperator\Coalg{Coalg}\DeclareMathOperator\ProVect{ProVect}\DeclareMathOperator\prolim{prolim} $Let $K$ be a field and $F: \Set \...
2
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1answer
74 views

Semi-norms on LCS inductive limit of Banach Spaces

Let $(E_n,i_n)_{n\in\mathbb{N}}$ be an direct system of Banach spaces in the category of locally convex spaces (LCSs) with continuous linear maps and let $E_{\infty}$ by their inductive limit. What ...
4
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1answer
214 views

Example of a non-cocomplete model category of a realized limit sketch

Let $(\mathcal{E},\mathcal{S})$ be a realized limit sketch, i.e. a locally small category $\mathcal{E}$ with a class $\mathcal{S}$ of limit cones in it. It is not assumed that $\mathcal{E}$ is small, ...
30
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4answers
2k views

Shapes for category theory

Most texts on category theory define a (small) diagram in a category $\mathcal{A}$ as a functor $D : \mathcal{I} \to \mathcal{A}$ on a (small) category $\mathcal{I}$, called the shape of the diagram. ...
8
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1answer
193 views

Filtered 2-colimits commute with finite 2-limits

Is there an explicit proof anywhere in the literature that filtered 2-colimits commute with finite 2-limits (all meant in the weak bicategorical sense) in the 2-category of groupoids? I have only ...
12
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2answers
664 views

Is there a large colimit-sketch for topological spaces?

Question. Is there a large colimit-sketch $\mathcal{S}$ such that $\mathrm{Mod}(\mathcal{S}) \simeq \mathbf{Top}$? In other words, is there a category $\mathcal{E}$ with a class of cocones $\mathcal{S}...
3
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1answer
134 views

Explicit description of a pullback of $(2,1)$-categories

In the 1-category of 2-categories, with objects being categories enriched over Cat, and morphisms being 2-functors, is there an explicit way to describe a pullback of two functors $G:E\to D$ and $F:C\...
3
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1answer
90 views

Functoriality of weighted limits

Let $C$ be a complete category, let $I$ be a small category, let $F,G:I\to C$ be functors, and let $W,U:C\to\mathrm{Set}$ be also functors, which we view as "weights". The weighted limits ...
8
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1answer
220 views

Filling square to push-out in abelian category

Let $\mathcal{C}$ be an abelian category. In $\mathcal{C}$ we consider the diagram \begin{array}{ccc} A&&\\\ \downarrow&&\\\ C&\rightarrow&D \end{array} with arrows being ...
6
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1answer
468 views

Does Grothendieck's algebraization imply existence of colimits of schemes?

I am a little bit confused about two lemmas regarding Grothendieck's algebraization. Assume all algebras are defined over some field. Here is the short version of my question: Does Tag 09ZT ("...
2
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1answer
175 views

Morphisms from the empty diagram

Let $X$ be an object in a category, and let $D$ be the empty diagram in the same category (containing no objects, and therefore no morphisms). What should $\text{Hom}(D,X)$ be? The only reasonable ...
13
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0answers
189 views

Which limits distribute over which colimits in $Set$? How about in $Spaces$?

I've never really thought much about distributivity of limits and colimits -- I tend to think more about commutativity of limits and colimits. This question makes me want to change that. The question ...
8
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1answer
282 views

Commutation of limits and colimits: Is there a choice diagram?

I was looking at this question about a "soft proof" of the fact that finite limits (shape $I$) commute with filtered colimits (shape $J$) in Set, using only the fact that the diagonal $J \to ...
5
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0answers
102 views

Characterisation of essentially algebraic theories with a fixed set of sorts

It is well known (e.g. Palmgren–Vickers's Partial Horn logic and cartesian categories) that many-sorted essentially algebraic theories (equivalently partial Horn theories / quasi-equational theories / ...
7
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1answer
164 views

Free extension of algebra for an operad

I fix $C$ a symmetric monoidal model category (with a cofibrant unit if it helps). I'm assuming that it is closed, or at least that the tensor product commutes to colimits in each variable. If $X$ is ...
2
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1answer
206 views

Define a sketch $s_{\mathbf{Grp}}$ such that $\mathbf{Grp}\backsimeq \mathbf{Mod}(s_{\mathbf{Grp}},\mathbf{Set})$

I have this MSE question with a two hundred bounty but even with the bounty this post got underviewed. So maybe here is a more suitable place to post it. The question follows: (a) Define a sketch $s_{...
12
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1answer
371 views

Measure the failure of colimit to commute with taking free loops (or Hochschild homology)?

For a space1 $X$, let $\mathcal{L}X = \mathrm{Maps}(S^1, X)$ be the free loop space. Inclusion of constant loops gives a natural map $X \to \mathcal{L}X$. This is not a homotopy equivalence unless $X$...
9
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0answers
260 views

Milnor's universal bundle as a colimit?

I have had Milnor's construction of the classifying space of a topological group explained to me on multiple occasions, and seen it described briefly in various places. But only now am I reading the ...
3
votes
1answer
216 views

Is Cauchy completion the largest extension with the same free cocompletion?

EDIT Title has been edited. Let $C$ be a category, and $$\hat{C} = [C^{op}, (Set)]$$ be its free cocompletion. Despite its name, the free cocompletion of free cocompletion is not equivalent to the ...
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0answers
56 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 $\...
3
votes
1answer
353 views

Filtered colim of F-groups

A group G is said to have a property F if there exists a finite aspherical CW-complex of which it is the fundamental group (according to wikipedia). question: is there a full characterization of ...
5
votes
1answer
128 views

Colimits of short exact sequences of C*-algebras

Assume I have an inductive system of short exact sequences of $C^{\ast}$-algebras (i.e., short exact sequences $0 \to A_n \to B_n \to C_n \to 0$ together with transformations from the $n$-th to the $(...
1
vote
0answers
68 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 ...
4
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0answers
161 views

Continuity property for Čech cohomology

Suppose we have an inverse system of compact Hausdorff spaces $\lbrace X_i , \varphi_{ij} \rbrace_{i\in I}$ and that each space has a presheaf $\Gamma_i$ assigned to it in such a way that $\Gamma_i(\...
5
votes
2answers
492 views

Do filtered colimits commute with finite limits in the category of pointed sets?

It seems to be the case that filtered colimits commute with finite limits in the category Set (for instance, this is shown in Why do filtered colimits commute with finite limits?), but does the same ...
4
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0answers
166 views

Pushout of $C^*$-algebras using generalised morphisms

There is a known construction of pushout of $C^*$-algebras, or rather, the amalgamated free product, which is universal for commutative squares of $*$-homomorphisms. Jensen and Thomsen in their book ...
6
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0answers
158 views

Relation between two limit presentations of Eilenberg--Moore objects

Let $\mathbb{T}=({\cal T}\colon C\to C,\mu,\eta)$ be a monad (in the $2$-category $\mathsf{Cat}$), which we view as a $2$-functor $\mathbb{T}\colon\mathsf{B}\Delta_{\mathrm{a}}\to\mathsf{Cat}$ (where $...
7
votes
1answer
188 views

2-monads for categories with a class of (co)limits

This question concerns the strictness of (co)completions, at various levels of generality. In Blackwell–Kelly–Power's Two-dimensional monad theory, the authors state For instance, the 2-category $\...
9
votes
3answers
514 views

Decomposing a (co)limit by decomposing the indexing diagram

Let $D: I \to \mathcal C$ be a diagram, and suppose we have a colimit decomposition $I = \varinjlim_{j \in J} I_j$ in $Cat$. Then under certain conditions, we can decompose the colimit of $D$ as $\...

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