All Questions
Tagged with limits-and-colimits ct.category-theory
221 questions
6
votes
0
answers
152
views
Covering categories with posets
Let $C$ be a small (1-)category.
There is always a poset $D$ and a functor $p : D \to C$ such that:
$p$ is surjective on objects, i.e. for every $c$ in $C$ there is a $d$ in $D$ such that $p (d) = c$,...
- 15.9k
5
votes
1
answer
198
views
Density formula in small presheaves
I've been trying to write down a proof of Di Liberti's Kan lemma fortissimo on the existence of left Kan extensions, given as Lemma 3.3 in the nLab entry on Kan extensions. Let $F : \mathcal{A} \to \...
- 51
2
votes
1
answer
74
views
Conditions for partially applied induced product functor to preserve colimits
Let $\mathcal{C}$ have products, $A, B \in \mathcal{C}_0$, ${\times}\colon \mathcal{C}^2\to \mathcal{C}$ be the functor that sends objects to their product.
Then the induced product ${\boxtimes}\colon ...
- 25
1
vote
2
answers
214
views
Presentable categories as colimits of finitely presentable categories
I am trying to understand the relationship betweeen compactly generated presentable categories, also called finitely presentable categories, and general presentable categories (which I have less ...
- 521
5
votes
1
answer
156
views
The inverse limit of a sequence of ring surjections commutes with taking difference subsets of the respective units & gluing in some primes?
Define $R_n := \Bbb{Z}/p_n\#$ the ring of integers modulo primorial $p_n\# = p_n p_{n-1} \cdots p_1$. Let $U_n$ denote the group of units modulo $p_n\#$ in these rings.
Then if $f_{n,n+1}: \Bbb{Z}/p_{...
- 260
11
votes
1
answer
451
views
Is every petite category essentially small?
A locally small category $\mathscr C$ is called petite if, for every functor $F : \mathscr C \to \mathscr D$ with locally small codomain, and for every object $D \in \mathscr D$, the presheaf $\...
- 10.6k
1
vote
1
answer
234
views
Notion of $\kappa$-sifted categories?
Let $\kappa$ be a regular cardinal. It seems reasonable to introduce the following definition:
Definition. A simplicial set $K$ is $\kappa$-sifted if, for every set $E$ with $\lvert E\rvert<\kappa$...
- 2,806
8
votes
1
answer
390
views
Pushouts vs contractible colimits
Suppose that $C$ has all weakly contractible colimits, i.e. colimits of functors $F: I \rightarrow C$ where the geometric realization $|I|$ is weakly contractible. Then $C$ has pushouts and filtered ...
- 521
3
votes
0
answers
55
views
Universal property of 2-presheaves and pseudo/lax/colax natural transformations
For each small 2-category $\mathscr K$, the 2-category $[\mathscr K^\circ, \mathrm{Cat}]$ of 2-functors and 2-natural transformations has a universal property: it is the free cocompletion of $\mathscr ...
- 10.6k
5
votes
0
answers
89
views
Free cocompletion of a 2-category under pseudo colimits, lax colimits, and colax colimits
Let $\mathscr K$ be a small 2-category. It follows from $\mathrm{Cat}$-enriched category theory that the free cocompletion of $\mathscr K$ under strict 2-colimits of 2-functors is given by the 2-...
- 10.6k
6
votes
1
answer
281
views
Are adjoints closed under pushouts?
The category $PrL$ of locally presentable categories has all colimits. In particular, if
$A_1 \leftarrow A_0 \rightarrow A_2$ is a diagram of presentable categories, with left adjoint functors between ...
- 521
4
votes
1
answer
140
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}^{\...
- 11.8k
4
votes
2
answers
314
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 ...
- 119
5
votes
1
answer
436
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 "...
- 159
-1
votes
1
answer
178
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 .)
5
votes
1
answer
158
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 ...
- 11.8k
2
votes
0
answers
100
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 $${...
- 10.1k
13
votes
1
answer
224
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 ...
- 10.6k
1
vote
0
answers
88
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 $\...
- 453
3
votes
1
answer
223
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/...
- 323
8
votes
1
answer
253
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 ...
- 28.7k
5
votes
1
answer
411
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 ...
6
votes
1
answer
155
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 ...
7
votes
1
answer
253
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 ...
- 173
8
votes
1
answer
258
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. ...
- 2,624
4
votes
0
answers
131
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 ...
- 1,949
9
votes
0
answers
103
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 ...
- 10.6k
2
votes
1
answer
168
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 ...
- 355
6
votes
1
answer
480
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 ...
- 165
6
votes
2
answers
425
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$ ...
- 1,173
10
votes
1
answer
498
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 ...
- 15.9k
1
vote
1
answer
232
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 \...
- 231
2
votes
1
answer
95
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^...
- 231
5
votes
1
answer
465
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 ...
- 11.8k
9
votes
0
answers
129
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 ...
- 10.6k
2
votes
0
answers
162
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 ...
- 10.1k
2
votes
1
answer
180
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 ...
- 505
6
votes
0
answers
142
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_{...
- 42.4k
9
votes
1
answer
734
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 ...
- 295
0
votes
2
answers
156
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 ...
- 639
9
votes
0
answers
103
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 ...
- 10.6k
8
votes
2
answers
306
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 ...
- 167
9
votes
1
answer
389
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 ...
- 285
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 ...
- 433
10
votes
0
answers
228
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}{\...
- 11.8k
6
votes
1
answer
267
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-...
- 775
5
votes
1
answer
244
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 ...
- 309
3
votes
2
answers
987
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 $\mathbf{Set}$ ...
3
votes
1
answer
512
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 ...
- 119
10
votes
0
answers
145
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(\...
- 1,001