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Questions tagged [limits-and-colimits]

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

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On intersection of kernels in enveloping algebras

Thinking about the following: if $k$ is a field and $S,T\subseteq A$ an inclusion of $k$-subalgebras, and denote $A^e=A\otimes_k A$ the enveloping algebra associated to $A$. Name $K(B)=\langle b\...
Vincenzo Di Bartolo's user avatar
3 votes
0 answers
42 views

From pushout in rings to pushout and pullback in bimodules (Mayer Vietoris presentations)

In a paper ("Mayer vietoris presentations over colimits of rings", W. Dicks) one finds this reasoning at the beginning. I am currently trying to prove the following higher dimensional ...
Vincenzo Di Bartolo's user avatar
1 vote
0 answers
45 views

contravariant finiteness and limit closure: is there dual to a result of Crawley-Boevey?

Let $\mathcal A$ be a locally finitely presented category. Theorem 4.2 of https://doi.org/10.1080/00927879408824927 says that given a full additive subcategory $\mathcal D$ of finitely presented ...
Alex's user avatar
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4 votes
1 answer
126 views

Cocompletion without cocontinuous functors

The forgetful functor from the 2-category $\mathsf{Cats}^{\mathrm{loc.small}}_{\mathrm{cocomp}}$ of locally small cocomplete categories and cocontinuous functors to the 2-category $\mathsf{Cats}^{\...
Emily's user avatar
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5 votes
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115 views

Compact objects in categories of categories

I am interested in the compact objects of various categories of categories. For example, $Cat^{small}$ is presentable and has compact objects that are retracts of finite colimits of $\Delta^n$, the $n$...
user39598's user avatar
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0 answers
110 views

Extremally disconnected sets as building blocks for compact Hausdorff spaces

Is every compact Hausdorff space the filtered colimit of compact extremally disconnected spaces?
Peter Kropholler's user avatar
3 votes
2 answers
257 views

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
5 votes
1 answer
408 views

Does coproduct preserve cohomology in differential graded algebra category

Consider two cochain DGA (differential graded algebras) named $A$ and $B$. By "coproduct" of two DGA I mean the category theory coproduct, not the coalgebra coproduct. It is defined in "...
wer's user avatar
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-1 votes
1 answer
150 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
136 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
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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
200 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
77 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
161 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
8 votes
1 answer
218 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
354 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
144 views

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
227 views

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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5 votes
1 answer
226 views

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
113 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
190 views

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
211 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
402 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
0 answers
129 views

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
  • 8,915
5 votes
1 answer
286 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
  • 2,043
2 votes
1 answer
164 views

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
  • 355
6 votes
1 answer
438 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
396 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
  • 1,133
10 votes
1 answer
392 views

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
  • 14.9k
1 vote
1 answer
217 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
  • 231
3 votes
1 answer
137 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
94 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
  • 231
6 votes
1 answer
212 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
451 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
  • 11.3k
9 votes
0 answers
124 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
  • 8,915
2 votes
0 answers
125 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
  • 9,047
2 votes
1 answer
163 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
136 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
444 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
648 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
147 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
  • 549
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
  • 8,915
8 votes
2 answers
273 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
366 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
  • 433
10 votes
0 answers
211 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
  • 11.3k
6 votes
1 answer
259 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
204 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
3 votes
2 answers
946 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

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