All Questions
Tagged with limits-colimits higher-category-theory
15
questions
1
vote
1
answer
195
views
Computing the homotopy limit of a constant diagram.
Let $X$ be a nice space, and view it as an $\infty$-groupoid via its singular simplicial set. Consider the constant functor $\mathbb{S}$ valued functor to Spectra, mapping all simplices to the sphere ...
- 6,900
1
vote
0
answers
59
views
Sheaves valued in a $k$-category
Let $\mathcal{C}$ be a $k$-category which we regard as an $\infty$-category whose objects all happens to be $k$-truncated.
A sheaf $F$ valued in $\mathcal{C}$, in the $\infty$-categorical sense, is a ...
- 1,613
1
vote
0
answers
43
views
The boundary of an open set as the homotopy limit of the open set minus compact subsets.
Let $M$ be a manifold and $U \subseteq M$ be a relative compact open set of $M$. I run into an equivalence
$$\partial U \cong \operatorname{holim}_{K \subseteq U} U \setminus K $$
where the inverse ...
- 1,613
4
votes
1
answer
170
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Limits in the localization of a category of fibrant objects
Suppose we have category $\mathcal{C}$ which has the structure of a category of fibrant objects, and suppose we have a functor $F:I\to \mathcal{C}$ with a limit $\lim F$ in $\mathcal{C}$.
If we have ...
- 1,388
6
votes
1
answer
729
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Possible adjoint to Yoneda embedding and Repeated Yoneda embedding?
While thinking about Yoneda embedding, I came up with following two questions (I should apologies, if those are too vague):
Does the Yoneda embedding $y :\mathcal{C}\to\mathbf{Set}^{\mathcal{C}^{\...
- 18.4k
2
votes
1
answer
345
views
Homotopy coherence
$\require{AMScd}$
I am trying to get an understanding of the meaning of homotopy coherence - in order to understand homotopy limits and colimits - in the category $\mathbf{Top}$. Often when I see this ...
- 3,326
4
votes
1
answer
454
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coequalizer of simplicial sets
This is the statement whose first line of proof I am confused. This is on page 10, of Goerss, Jardine's Simplicial Homotopy Theory.
(i) How does one prove the "presentation" of $\partial \Delta^n$?
...
- 9,618
2
votes
1
answer
58
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VAR, Algebra and local presentability
Here1, on the page 282, I would like to understand why precisely Examples of $k$-ary operations are all $k-\mathrm{lim}$, BUT $k-\textrm{colim}$ must be filtered. Where have we used that $k$ is ...
- 3,978
4
votes
1
answer
350
views
The one-arrow category as a weighted limit in Cat
Many categories can be defined as weighted limits or colimits in the 2-category of categories Cat. For example the category 1 (one object with its identity) is the terminal object of Cat, the category ...
- 1,548
2
votes
1
answer
107
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Examples of (co)complete 2-categories that aren't (co)complete as a 1-category
Title says it all. Are there examples of 2-categories that are (co)complete (with 2-(co)limits) such that their underlying 1-categories aren't (co)complete as a 1-category?
- 695
0
votes
1
answer
470
views
Weighted limits in the $Cat$-category of categories
What is a weighted limit in the $Cat$-category of categories, functors and natural transformations?
I can find the general definition of a weighted limit for enriched categories in Kelly's book or ...
- 1,548
2
votes
1
answer
271
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How to define a weighted cone?
Let $F : I \to C$ be a diagram in $C$, and $N$ an object of $C$. A cone from $N$ to $F$ is a family of morphism $P_X : N \to F(X)$ such that for every morphism $f : i1 \to i2$ in $I$, $F(f) \circ P_X =...
- 1,548
4
votes
1
answer
1k
views
Construction of 2-limits in 2-categories
Limits in a category can be built as a combination of the basic limits that are products and equalizers. Is there a similar construction for 2-limits in a 2-category? If yes, is it from products and ...
- 1,548
2
votes
0
answers
49
views
Why is the category of n-categries cocomplete?
I am studying a variant of the notion of (strict) n-categories, and I would like to show that this makes a cocomplete category. For that I planned on adapting the proof of the cocompleteness of n-...
- 343
7
votes
0
answers
251
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Why are split coequalizers "contractible"?
In the book Toposes, Triples and Theories by Barr and Wells, the authors define a contractible coequalizer (elsewhere known as a split coequalizer) to be a commutative diagram:
$A \rightrightarrows_{d^...
- 1,861