All Questions
Tagged with limits-and-colimits higher-category-theory
35 questions
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
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
5
votes
0
answers
154
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 $n$...
- 521
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
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
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
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
3
votes
1
answer
147
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 ...
- 81
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 $\mathbf{Set}$ ...
2
votes
1
answer
706
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\...
5
votes
1
answer
311
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 ...
- 10.6k
3
votes
0
answers
180
views
For which categories $D$ is a $D^{\vartriangleleft\vartriangleright}$-shaped diagram in a stable $\infty$-category a limit iff it is a colimit?
Throughout, I'll omit the "$\infty$" from the term "$\infty$-category".
It is well-known (and sometimes even included in the definition, although not by Lurie) that pushouts and ...
- 728
7
votes
1
answer
306
views
Filtered 2-colimits commute with finite 2-limits
Is there an explicit proof anywhere in the literature that filtered 2-colimits commute with finite 2-limits (all meant in the weak bicategorical sense) in the 2-category of groupoids? I have only ...
- 66.7k
4
votes
1
answer
286
views
Explicit description of a pullback of $(2,1)$-categories
In the 1-category of 2-categories, with objects being categories enriched over Cat, and morphisms being 2-functors, is there an explicit way to describe a pullback of two functors $G:E\to D$ and $F:C\...
- 85
15
votes
0
answers
332
views
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 ...
- 63.9k
1
vote
0
answers
71
views
Gluing categorical limit over subgraphs
Let $C$ be a category, and $\Gamma$ a graph in $C$. Under good conditions it makes sense to talk about the limit $\lim \Gamma$ of $\Gamma$ in $C$.
Suppose $\Gamma$ is the union of two subgraphs $\...
- 5,230
9
votes
3
answers
911
views
Decomposing a (co)limit by decomposing the indexing diagram
Let $D: I \to \mathcal C$ be a diagram, and suppose we have a colimit decomposition $I = \varinjlim_{j \in J} I_j$ in $Cat$. Then under certain conditions, we can decompose the colimit of $D$ as $\...
- 63.9k
12
votes
2
answers
564
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 ...
- 898
8
votes
2
answers
472
views
Classification of absolute 2-limits?
Let $\mathcal V$ be a good enriching category. Recall that an enriched limit weight $\phi: D \to \mathcal V$ is called absolute if $\phi$-weighted limits are preserved by any $\mathcal V$-enriched ...
- 63.9k
16
votes
3
answers
1k
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 ...
- 63.9k
5
votes
1
answer
339
views
Diagonal of a diagram of codescent objects
Given the following diagram in a $2$-category, in which squares of the same "type" commute, where each column and each row is a strong codescent diagram (Edit: it should be reflexive as well), is ...
- 63.1k
11
votes
1
answer
518
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}^\...
- 10.1k
3
votes
0
answers
82
views
Does a cartesian transformation induce a cartesian transformation on absolute limit cones?
Fix a category or $\infty$-category $C$ with all small limits.
We call a natural transformation $\alpha\colon f\to g$ between two functors $f,g\colon K\to C$ Cartesian, if for every arrow $k\colon a\...
- 725
11
votes
1
answer
1k
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) ...
- 335
4
votes
1
answer
443
views
Do coproducts of infinity-groupoids commute with pullbacks?
As stated in this question, coproducts commute with pullbacks in the category of sets.
Let $Grpd_{\infty}$ denote the $\infty$-category of $\infty$-groupoids. Do coproducts commute with pullbacks in $...
- 439
2
votes
0
answers
200
views
Connected families of objects in $(\infty,1)$-categories?
Background
Let $C$ be an (extensive) category. An object $X\in C$ is called connected if the functor $Hom(X,-):C\rightarrow Set$ preserves coproducts.
Given a category $C$, one can consider the ...
- 439
2
votes
1
answer
206
views
Appropriate morphisms and 2-morphisms in Ind(C)
As I was trying to understand the category $Ind(C)$ of diagrams of the form $I \to C$, where $I$ is a small filtered $(0,1)$-category, I wondered whether it is possible to define morphisms directly, ...
- 1,813
1
vote
0
answers
415
views
Continuity of Kan extension along the Yoneda embedding
Let $\mathcal{C}$ be a category and $h_-: \mathcal{C} \to \mathrm{Set}^{\mathcal{C}^{op}}$ be the Yoneda embedding. Let $\mathcal{A}$ be a cocomplete category and $F: \mathcal{C} \to \mathcal{A}$ a ...
- 805
5
votes
1
answer
724
views
Which E_∞-spaces are homotopy colimits of k-truncated E_∞-spaces?
This question is closely related to my previous question about modules over truncated sphere spectra, in particular, it has the same motivation.
Recall that every space (or ∞-groupoid) can be ...
- 37.8k
6
votes
1
answer
244
views
2-completeness analog of completeness theorem
It's not hard to see that a category is finitely complete if it has finite products and equalizers. In short, this is because one can write all limits as iterations of these two "operations".
I ...
- 4,842
16
votes
4
answers
1k
views
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 ...
- 7,237