Questions tagged [limits-and-colimits]

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

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Directed colimit of fully faithful functors

Suppose that for every $n\in\mathbb{N}$ we have a category $\mathcal{C}_n$ and a fully faithful functor $F_n:\mathcal{C}_n\hookrightarrow \mathcal{C}_{n+1}$. My question is whether fully faithful ...
MikeTrooper's user avatar
-1 votes
1 answer
132 views

Categories that admit all finite products but not all finite coproducts

What are examples for categories that admit all finite products but not all finite coproducts? (See also this question: Categories that admit all products but not all coproducts .)
Yilmaz Caddesi's user avatar
5 votes
1 answer
116 views

Co/limits and 2-co/limits of categories in the $\infty$- and $(\infty,2)$-category of $\infty$-categories

Recently, in a conversation with Gabriel, the following question came up: Question. Do co/limits of categories taken in the $\infty$-category of $\infty$-categories agree with the usual co/limits ...
Emily's user avatar
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2 votes
0 answers
81 views

Weighted limits and co-Yoneda

Is there a good reference that discusses weighted limits through the lens of the co-Yoneda embedding? Recall that the limit of a functor $F:\mathcal{C}\to{\bf Set}$ is canonically given by the set $${...
Alec Rhea's user avatar
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12 votes
1 answer
177 views

Large V-categories admitting the construction of V-presheaves

By a result of Foltz, and Freyd and Street, a category $C$ is essentially small (i.e. equivalent to a small category) if and only if both $C$ and $[C^{\text{op}}, \mathrm{Set}]$ are locally small. I ...
varkor's user avatar
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2 votes
0 answers
64 views

Colimits from van Kampen cocones

Let $\mathcal{C}$ be a category with pullbacks, $\mathcal{J}$ a small category, $F : \mathcal{J} \to \mathcal{C}$ a diagram and $\kappa : F \Rightarrow \Delta X$ a cocone in $\mathcal{C}$. Let $\...
Naïm Favier's user avatar
3 votes
1 answer
139 views

Kernels and cokernels in a quotient of an abelian category

I am trying to understand the construction of the quotient of an abelian category called the Serre quotient or Gabriel quotient. From the description here: https://en.wikipedia.org/wiki/...
Ji Woong Park's user avatar
7 votes
1 answer
199 views

Compact objects in slice categories of finitely presentable categories

Given a locally finitely presentable category $\mathscr C$ and an object $X \in \mathscr C$, it is not so hard to show that a morphism $(X \to Y)$ is compact in $\mathscr C_{X/}$ if it can be obtained ...
R. van Dobben de Bruyn's user avatar
5 votes
1 answer
327 views

Day convolution and sheafification

$\DeclareMathOperator\Psh{Psh}\DeclareMathOperator\Sh{Sh}\newcommand\copower{\mathrm{copower}}$I was looking through Bodil Biering's thesis On the Logic of Bunched Implications - and its relation to ...
Anthony D'Arienzo's user avatar
6 votes
1 answer
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Does the 2-category of double categories and vertical transformations have flexible limits?

Consider the 2-category of pseudo-double categories (with the weak composition in the horizontal direction and the strict composition in the vertical direction), strong double functors, and vertical ...
David Jaz Myers's user avatar
7 votes
1 answer
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Can every weighted colimit in a $\mathbf{Pos}$-enriched category be rephrased as a conical colimit?

For ordinary category theory, we have the following fact. A weighted colimit of a functor can always be equivalently expressed as a colimit of a different functor. Specifically, the weighted colimit ...
Nick Hu's user avatar
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1 answer
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Comparing stabilization of stable category modulo injectives and a Verdier localization

Let $\mathcal A$ be an abelian category with enough injectives. Let $\mathcal I$ be the collection of injective objects. Let $\mathcal A/\mathcal I$ be the quotient category whose objects are same as ...
Snake Eyes's user avatar
3 votes
1 answer
108 views

Vanishing of self-hom in Spanier–Whitehead stabilization category

$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\SW{SW}$Let $R$ be a commutative Noetherian ring. For $R$-modules $M,N$, let $\mathcal I_R(M,N)$ be the collection of all $f\in \text{Hom}_R(M,N)$ ...
Snake Eyes's user avatar
8 votes
1 answer
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Sequential colimit of iterated quotients of Cauchy sequences

We work in constructive mathematics. The sets and functions in the foundations form a Grothendieck topos, which means that all colimits exist, and in particular, that all sequential colimits exist. ...
Madeleine Birchfield's user avatar
5 votes
0 answers
201 views

Duality and compactness for pro vector spaces

I have a somewhat basic question which I haven't been able to piece together from the literature. Background. We work over a field $\bf{k}$. Consider the category, $\bf{Pro}_{k}$, of pro- vector ...
E.B.'s user avatar
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9 votes
2 answers
393 views

How do these definitions of factorization algebra compare?

Question Several sources define (homotopy) factorization algebras in a seemingly different manner (I am looking at [CG], [Gi], and [CFM].) I wish to know how they compare with each other. I apologize ...
Ken's user avatar
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4 votes
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Weakening of terminal object in a category

I’ve come across a category $\mathcal{C}$ recently with an object $T$ such that any other object $X$ has a map $f:X\rightarrow T$, and for any two maps $f,g:X\rightarrow T$, there exists a (not ...
Chris H's user avatar
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8 votes
0 answers
94 views

Adjoining a morphism to a finitely complete category

Let $\mathscr C$ be a finitely complete category. Let $x, y$ be objects of $\mathscr C$. We can describe the universal property of freely adjoining a morphism $x \to y$ to $\mathscr C$: it comprises a ...
varkor's user avatar
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5 votes
1 answer
278 views

Commuting homotopy colimits and arbitrary products in spaces

Let $X : D \rightarrow Spc$ be a diagram with values in the $\infty$-category of spaces and $I$ some (discrete) set, not necessarily finite. ($D$ can be a 1-category if that makes statements easier, ...
Georg Lehner's user avatar
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2 votes
1 answer
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Inclusion of $1$-presheaves into $\infty$-presheaves preserves pushouts?

Let $\mathcal{R}$ be a $1$-category. Assume that one has a pushout of representable $1$-presheaves $\mathrm{y} A \cup_{\mathrm{y} B} \mathrm{y} C$ in $\mathsf{PSh}(\mathcal{R})$. Under which ...
HDB's user avatar
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6 votes
1 answer
424 views

Why isn't pullback-stability defined for individual colimits but for colimits with the same shape?

Circumstances: I'm studying Grothendieck's Galois Theory and recently encountered a proposition that discussed the stability of coproducts under pullback. And I found the page pullback-stable colimit ...
Linuxmetel's user avatar
6 votes
2 answers
390 views

Has the "Lambek embedding" into the category of (co)product-preserving presheaves been studied very much?

The Lambek embedding is a particular embedding which is similar to the Yoneda embedding. Suppose we have any category $C$. Recall that a presheaf on $C$ is defined as a contravariant functor from $C$ ...
Tanner Swett's user avatar
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10 votes
1 answer
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What does it mean for a category to be generated under (some) colimits?

This is going to be a long post, so let me state my question first and then explain what I am interested in by way of examples. Question. Is there any literature studying notions of generation under ...
Zhen Lin's user avatar
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1 vote
1 answer
211 views

Pointwise Kan extensions VS weighted limits

$\newcommand{\Dist}{\operatorname{Dist}} \newcommand{\Ran}{\operatorname{Ran}} \newcommand{\Lim}{\operatorname{Lim}}$ TLDR Given a pointwise kan extension, how can we go from $$ \Dist(B, C)(\phi_c \...
nicolas's user avatar
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3 votes
1 answer
136 views

Does the forgetful functor from a pointed $\left(\infty, 1\right)$-category only create weakly contractible colimits?

Consider an $\left(\infty, 1\right)$-category $\mathcal{C}$ with a terminal object $1$. (I'm particularly interested in the case where $\mathcal{C}$ is a topos.) It is known that the forgetful functor ...
Perry Hart's user avatar
2 votes
1 answer
92 views

Weighted limits and Kan extension in Dist

(noting $\otimes$ for composition in distributors, $\phi_f : A \nrightarrow B = B(-,f=)$ and $\phi^f : B \nrightarrow A = B(f-,=)$ the embeddings of a functor $f:A\to B$ in $Dist$, and $Dist(A,B) = [B^...
nicolas's user avatar
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6 votes
1 answer
198 views

When does base-change in topological spaces preserve quotient maps?

The question when $(-) \times X$ preserves colimits in topological spaces is well-studied. Since it always preserves arbitrary coproducts (disjoint unions), one only has to show when it preserves ...
Lennart Meier's user avatar
5 votes
1 answer
447 views

Homotopy groups of categories of elements as higher colimits

Given a diagram of sets $D\colon\mathcal{C}\to\mathsf{Set}$, we have a bijection (Proof) $$\operatorname{colim}(D) \cong \pi_0 (\textstyle\int_\mathcal{C}D).$$ Is there any known application or ...
Emily's user avatar
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9 votes
0 answers
119 views

Is totality a (large) cocompleteness condition?

A locally small category $A$ is called total if its Yoneda embedding $A \to [A^\circ, \mathbf{Set}]$ has a left adjoint. Such categories are necessarily small-cocomplete (since the presheaf category ...
varkor's user avatar
  • 8,665
1 vote
0 answers
104 views

Initial cones, terminal cocones

We're all familiar with terminal cones/initial cocones in the form of limits/colimits. What about initial cones and terminal cocones? While writing an answer to a related question the concept ...
Alec Rhea's user avatar
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2 votes
1 answer
160 views

Non-cofiltered derived limits

As far as I know, the inverse limit and its derived functors can be defined even in case we are dealing with a functor $F: I \to A$ from a category $I$ that is not cofiltered. I would content myself ...
Matteo Casarosa's user avatar
6 votes
0 answers
134 views

Which Ends preserve filtered colimits?

Can we characterize entirely for which categories $C$ the end on $C$ preserves filtered colimits, in a sense that the natural map $$ \operatorname*{colim}_{i \in I} \int_{x \in C} A_i (x,x) \to \int_{...
Simon Henry's user avatar
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3 votes
1 answer
390 views

Homotopy colimit commutes with homotopy groups

I'm interested in something built upon the construction laid out in nlab article on Snaith's theorem Let $(E, \mu, \iota)$ be a ring spectrum. For $\beta \in \pi_n(E)$ an element of the $n$th stable ...
Excalibur's user avatar
  • 301
9 votes
1 answer
614 views

Does the category of locally compact Hausdorff spaces with proper maps have products?

nlab presents a proof that the category of locally compact Hausdorff spaces does not admit infinite products in general. In particular it shows that there is no infinite product of $\mathbb{R}$, since ...
Oddly Asymmetric's user avatar
0 votes
2 answers
144 views

Essentially zero inverse system of abelian groups

I am learning local cohomology from Hartshorne’s Local Cohomology book. My question is about the notion of essentially zero inverse system of abelian groups, which is defined to be an inverse system ...
Boris's user avatar
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9 votes
0 answers
99 views

Cocompleteness of enriched categories of algebras

A useful result due to Linton is that for a cocomplete category $C$ and monad $T$ on $C$, if the category of algebras $C^T$ admits reflexive coequalisers, then it is cocomplete (see here for a sketch ...
varkor's user avatar
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8 votes
2 answers
268 views

Reference for certain categorical limits

I would like to know if there is a special name for the following concept, papers that feature something similar or a general reference. Let $\mathcal{C}$ be a category and $\mathcal{D}$ a subcategory ...
Arturo's user avatar
  • 167
9 votes
1 answer
354 views

Is the coproduct $N=1+N$ universal?

Let $\mathcal{C}$ be a category with finite limits and a (parameterized) natural numbers object $(N,0,s)$. Let $1$ denote the terminal object of the category. It's easy to show that the following is a ...
Sambo's user avatar
  • 243
10 votes
1 answer
1k views

What's the intuition for weighted limits?

I am reading Fosco's Coend Calculus and Emily Riehl's Categorical Homotopy Theory, Riehl's book motivates it in the following way, Abstraction 1: Classical limits in terms of cones: Cones from an ...
MrPajeet's user avatar
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9 votes
0 answers
195 views

Natural cotransformations and "dual" co/limits

$\DeclareMathOperator{\id}{\mathrm{id}}\DeclareMathOperator{\Hom}{\mathrm{Hom}}\DeclareMathOperator{\UnCoNat}{\mathrm{UnCoNat}}\DeclareMathOperator{\UnNat}{\mathrm{UnNat}}\DeclareMathOperator{\CoNat}{\...
Emily's user avatar
  • 10k
6 votes
1 answer
249 views

Do the representations of a 2-functor naturally form a contractible 2-category?

In 1-category theory a representation of a 1-functor $F:C\to Set$ is a 0-cell $X$ in $C$ together with a universal element $u\in FX$ such that the transformation $C(X,-)\to F$ is an isomorphism (=a 1-...
Nico's user avatar
  • 775
5 votes
1 answer
201 views

limits and products stable $\infty$-category

In an abelian category $\mathcal{A}$, for a system $\{F_i,\phi_{ij}\}$ we have an exact sequence $0\to \lim F_i\to \prod F_i \to \prod F_i$ where the second map is given by $id-\prod\phi_{ij}$. Is ...
user197402's user avatar
2 votes
2 answers
936 views

Which functors preserve the number of connected components?

The categories $\mathbf{Top}$ of topological spaces, $\mathbf{sSet}$ of simplicial sets and $\mathbf{Cat}$ of small categories are all equipped with a functor $\pi_0$ into the category $\mathbf{Set}$ ...
Samuel Adrian Antz's user avatar
2 votes
1 answer
199 views

Surjection of a short exact sequence induced by spectral sequence (from paper of Schneider/Stuhler)

Let $K=\mathbb{Q}_p$ and $X$ a smooth separated rigid analytic variety over $K$ with coherent sheaf $\mathcal{F}$. Furthermore, $U \subset X$ is an open subvariety with admissible covering $$ \dotsb \...
KKD's user avatar
  • 463
6 votes
1 answer
329 views

Vanishing of higher limits

Let $I$ be a directed set and let $X_I$ be a corresponding inverse system of, say, (complex) vector spaces or abelian groups (in my case in general not finite-dimensional, resp. not finitely generated)...
AlexE's user avatar
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3 votes
1 answer
423 views

Proof without sieves: Equivalent grothendieck topologies have the same sheaves

I'm currently learning about sheaf theory with Angelo Vistoli’s 2007 Notes on Grothendieck topologies, fibered categories and descent theory. And in page 35, there is the following definition of a ...
Muster Maxfrau's user avatar
4 votes
0 answers
85 views

Homotopy colimits in subcategories of combinatorial model categories

We know that, in a combinatorial simplicial model category $\mathbf{M}$, we can find a regular cardinal $\lambda$ large enough that all $\lambda$-filtered homotopy colimits can be computed as strict ...
Giulio Lo Monaco's user avatar
10 votes
0 answers
137 views

Do pseudo 2-limits commute?

It is a well-known fact that if $F:\mathcal{C}_1\times\mathcal{C}_2\rightarrow \mathcal{D}$ is a functor (between 1-categories), then $F$ has a limit if and only if $F:\mathcal{C}_1\rightarrow Fun(\...
JeCl's user avatar
  • 901
11 votes
0 answers
325 views

A right adjoint preserves Phi-colimits if and only if the left adjoint does what?

Let $\Phi$ be a class of categories (e.g. filtered categories), and consider an adjunction $L : \mathbf C \rightleftarrows \mathbf D : R$. A $\Phi$-colimit is a colimit whose diagram is in $\Phi$. We ...
varkor's user avatar
  • 8,665
4 votes
2 answers
264 views

Change of coordinates for coends

I recall that there was a theorem mimicking the change of variables' integral formula. Surprisingly, I can't find it on the Fosco Loregian book. The change of variables formula states that, if $f: E \...
Andrea Marino's user avatar

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