All Questions
Tagged with limits-and-colimits infinity-categories
12 questions
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
6
votes
1
answer
315
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, ...
- 2,303
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
3
votes
1
answer
276
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 ...
- 1,312
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
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
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 ...
- 1,082
9
votes
1
answer
609
views
Higher-dimensional version of the "Magic Cube Lemma" for homotopy pushouts/pullbacks
The "Magic Cube Lemma" is a surprising (to me) relationship between (homotopy) pushouts and (homotopy) pullbacks of spaces:
Consider a cubical diagram $I^3\to \mathcal{S}$ in the $\infty$-category of ...
- 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
2
votes
0
answers
134
views
Colimits of small categories in the category of big categories
This is a question regarding set-theoretic issues in higher category theory. Let $\text{Cat}_{\infty}^{big}(\omega)$ be the $\infty$-category of (not necessarily small) $\infty$-categories with ...
- 21
5
votes
0
answers
448
views
Examples of nonstable ∞-categories in which sifted colimits commute with finite limits
What are some natural examples (if any) of nonstable ∞-categories in which finite limits commute with sifted colimits (or rather just colimits over Δ^op)?
Stable ∞-categories do satisfy this property,...
- 37.8k
6
votes
1
answer
613
views
Does Ind-completion commute with finite limits?
The broad and vague question is in the title. The more precise question is:
Say $\{\mathcal{C}_i\}$ is a finite diagram of (essentially small) stable $\infty$-categories and exact functors with ...
- 61