Questions tagged [limits-and-colimits]

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

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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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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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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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4 votes
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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
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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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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
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2 answers
338 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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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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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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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
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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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1 answer
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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
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1 answer
433 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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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
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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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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
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0 answers
123 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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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
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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
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2 answers
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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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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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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
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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
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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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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
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6 votes
1 answer
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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
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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
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2 answers
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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
182 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
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6 votes
1 answer
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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
366 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
83 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
130 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
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12 votes
0 answers
243 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
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4 votes
2 answers
250 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
3 votes
1 answer
215 views

Question about the proof of Kerodon tag 030V (Proposition 7.3.7.1)

$\require{AMScd}$ Related to this, I have a question about the proof given in Kerodon of the following result: Proposition 7.3.7.1: Let $C$ be an $\infty$-category, let $\bar{F} : C^\rhd \to D$ be a ...
daniel gratzer's user avatar
2 votes
0 answers
174 views

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 ...
Eric's user avatar
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1 vote
1 answer
405 views

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\...
Samuel Adrian Antz's user avatar
3 votes
1 answer
270 views

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 ...
FDR's user avatar
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3 votes
0 answers
122 views

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 ...
Alec Rhea's user avatar
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5 votes
0 answers
268 views

$V$-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 ...
Jxt921's user avatar
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7 votes
1 answer
395 views

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. ...
Alec Rhea's user avatar
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3 votes
0 answers
78 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 $P_n \colon \mathsf{C}^{op} \times \mathsf{C} \to \mathsf{Set}$ ...
Mario Román's user avatar
4 votes
1 answer
234 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 (strict) 2-pushouts in the 2-category 2-Cat ...
varkor's user avatar
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5 votes
0 answers
136 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}:\...
Jonas Linssen's user avatar
2 votes
0 answers
86 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 ...
postdoc's user avatar
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2 votes
0 answers
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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 ...
kevkev1695's user avatar
6 votes
1 answer
225 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 ...
Ivan Di Liberti's user avatar
10 votes
1 answer
300 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 $\mathsf{C}$ ...
Emilio Minichiello's user avatar
0 votes
1 answer
171 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 ...
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