# Questions tagged [limits-and-colimits]

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

277
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### Is the homotopy limit of derived schemes along affine maps a derived scheme?

The title question is true in the setting of ordinary limits and ordinary schemes; that is, given an inverse limit of schemes along affine maps, the limit still lives in the category of schemes.
I'd ...

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### Formula for the left adjoint of the nerve functor?

I recently stumbled upon a formula for the left adjoint of the nerve functor. Let $X$ and $Y$ be simplicial sets, then:
\begin{equation}
\mathbf{sSet}(X,Y)
\cong\mathbf{sSet}(\varinjlim_{\Delta^n\...

3
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1
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### Derived functors of inverse limit in abelian categories?

I have a finite poset $I$ and an inverse system $A: I^{op}\longrightarrow \mathscr C$ taking values in an abelian category $\mathscr C$.
I suppose that $\mathscr C$ has direct sums. Given that my ...

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### Weighted limit calculus

In coend calculus by Fosco Loregian, it is mentioned that Lawvere conjectured that co/ends constitute a categorification of logical calculus. In his own words:
The somewhat far-fetched conjecture ...

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### Is the structure presheaf of $\text{Spec}(A)$ a limit of presheaves?

In Shafarevich's Basic Algebraic Geometry, he defines the structure presheaf on $\text{Spec}(A)$ first by $\mathcal{O}(D(f))\cong A_f$ and then $\mathcal{O}(U)=\lim_{D(f)\subset U}\mathcal{O}(D(f))$, ...

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### $V$-cat and $V$-graph: coequalizers in the category of enriched functors

This question is regarding the 1974 JPAA paper $V$-cat and $V$-graph by Harvey Wolff.
To be precise, I don't understand a certain step in the proof of Corollary 2.9, which (the corollary) is crucial ...

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### A fibration equivalent to having a terminal object

It is well known that the codomain functor $$cod:\mathcal{C}^\to\to\mathcal{C}$$ from the arrow category of a category $\mathcal{C}$ to itself is a fibration iff $\mathcal{C}$ has binary pullbacks.
...

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### 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 $P_n \colon \mathsf{C}^{op} \times \mathsf{C} \to \mathsf{Set}$ ...

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### 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 (strict) 2-pushouts in the 2-category 2-Cat ...

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### 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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### 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 ...

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### 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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### 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 ...

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### 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 $\mathsf{C}$ ...

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1
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### 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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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 \...

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### 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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### 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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### 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\...

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### 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 ...

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### 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 \...

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### 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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1
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### 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 ...

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### 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, ...

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### 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. ...

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### 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 ...

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### 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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### 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\...

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### 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 ...

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1
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### 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
votes

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### 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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### 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 ...

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### 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 ...

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### 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 ...

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### 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 / ...

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### 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 ...

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### 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_{...

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### 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 $X$...

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### 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 ...

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### 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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### 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

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358
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### 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 ...

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votes

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### 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 $(...