Questions tagged [limits-and-colimits]

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

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Why do filtered colimits commute with finite limits?

It's not hard to show that this is true in the category Set, and proofs have been written down in many places. But all the ones I know are a bit fiddly. Question 1: is there a soft proof of this fact?...
Steve Lack's user avatar
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37 votes
7 answers
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Limits in category theory and analysis

Is it possible to regard limits in analysis (say, of real sequences or more generally nets in topological spaces) as limits in category theory? Is there some formal connection? Edit ('13): Perhaps it ...
Martin Brandenburg's user avatar
33 votes
5 answers
3k 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. ...
26 votes
2 answers
2k views

Is every commutative ring a limit of noetherian rings?

Edit of Feb. 14, 2019. After Laurent Moret-Bailly's accepted answer, only Questions 4 and 5 remain open. I don't care that much about Question 4, but I'm very curious about Question 5, which is Do ...
Pierre-Yves Gaillard's user avatar
25 votes
1 answer
704 views

Finite limits in the category of smooth manifolds?

The category of smooth manifolds does not have all finite limits. However, it does have some limits: it has finite products, it has splittings of idempotents, and it has certain other limits if we ...
Mike Shulman's user avatar
21 votes
4 answers
2k views

Conceptual reason that monadic functors create limits?

Let $U: Alg_T \to C$ be the forgetful functor from the category of algebras of $T: C \to C$ ($T$ could be a monad; I'm happy to think about the simpler case where $T$ is just an endofunctor or pointed ...
Tim Campion's user avatar
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20 votes
2 answers
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Is every compact topological ring a profinite ring?

There are a lot of compact (Hausdorff) groups, whereas every compact field is finite. What about rings? Is there a classification theorem for compact rings? If you take a cofiltered limit of finite ...
Gene S. Kopp's user avatar
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19 votes
4 answers
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Categorical description of the restricted product (Adeles)

Background on the Adèles The Adèles $\mathbb{A}_K$ of a number field or function field $K$ are defined as a restricted product of the complete local fields $K_\nu$, where $\nu$ ranges over all places ...
Konrad Voelkel's user avatar
18 votes
1 answer
3k views

Coend computation

Let $F:A^{\mbox{op}} \to \mbox{Set}$ and define $G_a:A\times A^{\mbox{op}} \to \mbox{Set}$ $G_a(b,c) = \mbox{hom}(a,b) \times F(c)$. I think the coend of $G_a$, $\int^AG_a$, ought to be $F(a)$--...
Mike Stay's user avatar
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17 votes
2 answers
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Exactness of filtered colimits

Are filtered colimits exact in all abelian categories? In Set, filtered colimits commute with finite limits. The proof carries over to categories sufficiently like Set (i.e. where you can chase ...
Martin Orr's user avatar
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16 votes
10 answers
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References for homotopy colimit

(1) What are some good references for homotopy colimits? (2) Where can I find a reference for the following concrete construction of a homotopy colimit? Start with a partial ordering, which I will ...
Kevin Walker's user avatar
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16 votes
4 answers
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Is there a tricategory of bicategories and biprofunctors?

Background There is a bicategory where the objects are categories, the 1-morphisms are profunctors, and the 2-morphisms are morphisms of profunctors. The non-obvious part of this assertion is that ...
Evan Jenkins's user avatar
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16 votes
3 answers
911 views

Cofinality for coends?

Recall that a functor $I \xrightarrow u J$ is cofinal if it has the property that for any functor $J \xrightarrow F C$, we have that $\varinjlim F \cong \varinjlim Fu$ via the canonical map, either ...
Tim Campion's user avatar
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15 votes
2 answers
626 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 ...
varkor's user avatar
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15 votes
2 answers
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Monoidal functors $\mathcal C \to [\mathcal D,\mathcal V]$ are monoidal functors $\mathcal C \otimes \mathcal D \to \mathcal V$?

It is well known (e.g., Reference for "lax monoidal functors" = "monoids under Day convolution" ) that if $\mathcal C$ is a monoidal $\mathcal V$-enriched category, then a monoid ...
John Gowers's user avatar
14 votes
2 answers
745 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}...
Martin Brandenburg's user avatar
14 votes
2 answers
1k views

Colimits of manifolds

This question tells us that in general colimits do not exist in the category of manifolds. However, this negative answer is not very satisfying. A manifold can be considered as a colimit of its altas....
user avatar
14 votes
1 answer
1k views

What is a (partial) left adjoint of the Yoneda embedding called?

It is a fairly special property for the Yoneda embedding $A \hookrightarrow \mathcal{P}A$ of a category to have a left adjoint defined everywhere (this happens just when $A$ is total). However, a ...
Mike Shulman's user avatar
14 votes
1 answer
583 views

How "nondegenerate" are amalgamated free products of C*-algebras?

In the following, I assume all algebras are unital. Let $A$ and $B$ be C*-algebras that each contain (isomorphic copies of) a common C*-subalgebra $C$. Let $A *_C B$ denote the amalgamated free ...
Manny Reyes's user avatar
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14 votes
0 answers
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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 ...
Tim Campion's user avatar
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14 votes
0 answers
842 views

Grothendieck construction and coends

In category theory, both the Grothendieck construction and coends are represented by a sort of "integral sign", respectively: $$ \int F $$ for a functor $F:C\to\mathbf{Cat}$, and: $$ \int^x G(x,x) $$ ...
geodude's user avatar
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12 votes
5 answers
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Motivation of filtered colimits

I am trying to move in categorical algebra beyond the basics. A Lawvere theory L is a small category with finite products. (I know that there also is a functor $(skeleton(FinSet))^{op}\to L$, which ...
beroal's user avatar
  • 520
12 votes
1 answer
2k views

Is every ring the direct limit of Noetherian rings?

Are there any examples of commutative rings that do not occur as direct limits of Noetherian rings?
Achilleas K's user avatar
12 votes
2 answers
678 views

Defining abstract varieties and their morphisms over a finitely generated subfield of the base field

Let $k$ be an algebraically closed field. By a finitely generated subfield of $k$ I mean a subfield $k_0\subset k$ that is finitely generated over the prime subfield of $k$ (that is, over $\mathbb Q$ ...
Mikhail Borovoi's user avatar
12 votes
1 answer
2k views

Adjoint Functors as Initial Objects of Some Category

Just as universal arrows can be characterized as initial objects of some appropriate comma category, and (co)limits can be characterized as (initial) terminal objects of the appropriate (co)cone ...
Dmitry V's user avatar
  • 433
12 votes
2 answers
508 views

Does forgetting colimits preserve colimits?

For each regular cardinal $\kappa$ let $\operatorname{Cat}_{\kappa}$ be the $(2,1)$-category of small categories with $\kappa$-small colimits, and functors that preserve those colimits. For each pair ...
G. Stefanich's user avatar
12 votes
1 answer
435 views

About pointwise Kan extension

Suppose that you want to look at the left Kan extension of a functor $F : \mathcal{C} \to \mathcal{A}$ along a functor $K : \mathcal{C} \to \mathcal{B}$. It is widely known that if the colimit of the ...
L.Guetta's user avatar
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12 votes
1 answer
178 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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11 votes
5 answers
708 views

Colimits in the category of (not necessarily locally convex) topological vector spaces

Do colimits in the category of (not necessarily locally convex) topological vector spaces (over R, C, respectively) exist in general? If no, is there a well-known condition of when they exist? If ...
Junekey Jeon's user avatar
11 votes
2 answers
494 views

How to interpret topologically that the equalizer in Groupoids of ${\rm id}, {\rm id}: BG \rightrightarrows BG$ is $G/G$ (adjoint action)?

Let $G$ be a (discrete) group, and $1/G$ the corresponding groupoid with one object. Consider the diagram in (the 2-category) Groupoids with one vertex, labeled $1/G$, the one arrow from that vertex ...
Theo Johnson-Freyd's user avatar
11 votes
2 answers
3k views

Is Sheafification Functor Exact?

I know that sheafification functor from the category of abelian presheaves on $C$ to the category of abelian sheaves on $C$. Here, $C$ is a category with Grothendieck pretopology. My question is: ...
Hiro's user avatar
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11 votes
1 answer
1k views

Do direct limits (filtered colimits) commute with pullbacks, in C*-algebras?

I asked this question already on math.stackexchange, but maybe it is also useful to ask this here, since it was not answered there. Suppose we have three directed sequences of $C^*$-algebras, say $(...
Opluoos35's user avatar
  • 113
11 votes
2 answers
938 views

How to understand adjoint functors?

I asked this same question on MathUnderflow two weeks ago but didn't receive any answer. Now that I am thinking more, it feels like the most suitable place for this question is here. I have a good ...
Bumblebee's user avatar
  • 1,019
11 votes
1 answer
471 views

Weighted (co)limits as adjunctions

It's well known that a category $\mathcal{C}$ having (conical) limits/colimits of shape $\mathcal{D}$ is equivalent to the diagonal functor $\Delta^\mathcal{D}_\mathcal{C}:\mathcal{C}\to\mathcal{C}^\...
Alec Rhea's user avatar
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11 votes
1 answer
490 views

Is there a practical criterion to determine whether the limit of a diagram of real chain complexes is also a homotopy limit?

Consider a diagram D: I→ChR of real connective chain complexes. In the example I have in mind all chain complexes are concentrated in some fixed degree n. There is a canonical map lim D → holim D ...
Dmitri Pavlov's user avatar
11 votes
1 answer
522 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 $X$...
Vivek Shende's user avatar
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11 votes
0 answers
334 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
  • 8,675
10 votes
3 answers
1k views

Can filtered colimits be computed in the homotopy category?

For $\mathcal{S}$ the $(\infty,1)$-category of spaces its homotopy category $h\mathcal{S}$ does not have pushouts or pullbacks. Even if it does, they won't always agree with the (homotopy) pushouts or ...
Jan Steinebrunner's user avatar
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
2 answers
846 views

Fubini theorem for hocolim

I wanted to ask the following question, Suppose $\mathbf{M}$ a cofibrantly generated model category and $I,~J$ two small categories. Suppose that $F:J\rightarrow \mathbf{M}^{\mathrm{I}}$ is a functor. ...
Ilias A.'s user avatar
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10 votes
2 answers
649 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 ...
Alastair Grant-Stuart's user avatar
10 votes
1 answer
560 views

Explicit description of the oplax limit of a functor to Cat?

The nCatLab Grothendieck construction page gives an explicit description of the oplax colimit of any functor to Cat. Can someone give me a similarly explicit description (the objects and morphisms) ...
Jason Gross's user avatar
10 votes
1 answer
731 views

Decomposing a large colimit as a pushout of smaller colimits

I would like to find a reference in the literature for the following result. I have it on high authority that it isn't in 'Categories for the Working Mathematician' and I can't find it in Borceux's ...
James Griffin's user avatar
10 votes
1 answer
383 views

Sufficient sets of colimits in small categories

Let $C$ be a small category, and consider the class of diagrams $G:D\to C$, with $D$ a small category, that have colimits in $C$. This is a proper class even when $C$ is very small, e.g. whenever $D$ ...
Mike Shulman's user avatar
10 votes
1 answer
329 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
10 votes
1 answer
352 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
10 votes
1 answer
821 views

Functoriality of (co)limits in $\infty$-categories

I have some questions about the functoriality of (co)limits in $\infty$-categories, say in the framework of Lurie's Higher Topos Theory. From the general stuff about Kan-extensions (HTT 4.3.2.6) ...
Robin Stoll's user avatar
10 votes
1 answer
509 views

Is the evaluation of polynomial functors appropriately continuous?

I'd like a nice proof of the following fact. Let $C$ and $D$ be categories, and let $\mathbf{Cat}/(C\times D)$ be the usual (1-categorical) slice category whose objects are triples $(X,F\colon X\to C,...
David Spivak's user avatar
  • 8,549
10 votes
0 answers
138 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
9 votes
2 answers
2k views

Is there a useful limit or co-limit of a diagram that has only a single object?

I'm starting to study category theory kind of informally and everytime I read about the definitions of limits and co-limits, the first three examples are always the same: terminal/initial objects, ...
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