All Questions
Tagged with simplicial-stuff limits-colimits
13 questions
2
votes
0
answers
104
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What morphism is sent to a monomorphism by the left Kan extension ${\rm Lan}_{\Delta}\colon{\bf Set^\Delta\to\bf Set^{\hat\Delta}}$ along Yoneda?
For any small category $C$, let us write $\hat{C} = \mathbf{Set}_C$ for the presheaf category $\mathbf{Set}^{C^{\mathrm{op}}}$, and $y=y_C\colon C\to \mathbf{Set}_C$ for the Yoneda embedding. Consider ...
- 152
2
votes
1
answer
46
views
Reference request for realizing a simplicial set as the homotopy colimit of its simplices
I know that
$$X\simeq hocolim_{Simp(X)}\Delta^n,$$
where $Simp(X)$ is the category of simplices of $X$, I know this for example because of proposition 7.5 of the nLab's page for homotopy limits. ...
- 1,579
3
votes
1
answer
267
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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 ...
- 6,958
3
votes
1
answer
208
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Difficulty using the (co)limit formulae to construct the $n$-(co)skeleton left and right Kan extensions for truncated simplicial objects
Tl;Dr - I’m struggling to show that the $n$-skeleton is a Kan extension, from the basic limit formula (this should be possible, as it was “left to the reader” in my book). I’m also struggling to even ...
- 44.7k
0
votes
0
answers
84
views
Calculating Colimits in The category of Simplicial Sets
I am struggling to understand how to take colimits in the category of simplicial sets. To make my notation clear, I will use $\delta_i : [n-1]\to[n]$ to denote the $i$-th face map (it picks out the ...
- 1,965
2
votes
1
answer
251
views
On a definition of Spivak's fuzzy set
In the paper "Metric Realization of Fuzzy Simplicial Sets" of David Spivak it takes $I=(0,1]$ as poset and consider it as a category. He gives it a Grothendieck topology induce it from ...
- 2,004
1
vote
1
answer
64
views
Homotopy cofinality of $\Delta^{op}$ in $\Delta^{op}\times \Delta^{op}$
There is the usual diagonal inclusion $i:\Delta^{op}\to\Delta^{op}\times \Delta^{op}$ which is easily seen to be cofinal in the $1$-categorical sense, and so one can compute colimits on $\Delta^{op}\...
- 44.4k
4
votes
1
answer
480
views
Do colimits/limits exist in category of enriched categories?
This question may be too general. I am interested in references or proofs for special cases. I follow the definition in Chapter I of Kelly's Enriched Category.
Let $V$ be a monoidal category theory. ...
- 1,037
4
votes
1
answer
478
views
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 $\partial \Delta^n$?
...
- 9,766
3
votes
1
answer
313
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Which simplicial sets are filtered colimits of standard simplices?
The question is all in the title : every simplicial set is a colimit of the standard simplices $\Delta^n$, but I'm wondering which ones are filtered or directed colimits of these, if there's a nice ...
- 44.4k
0
votes
1
answer
95
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How to write the prism as a colimit of simplices (in the category of simplicial sets)?
It is intuitively clear that a prism $\Delta ^n\times \Delta ^1$ can be triangulated. If I am interpreting this correctly, it should be possible to give a formula for the prism as a colimit (in the ...
- 14.2k
2
votes
1
answer
145
views
For which families of subsets is the colimit of the Čech nerve realized by the union?
Let $(U_i\subset U)$ be a family of subspaces of a topological space $U$. Consider the Čech nerve of this family, given by the simplicial object furnished by taking iterated kernel pairs of the ...
- 14.2k
6
votes
1
answer
262
views
Question about limit of cosimplicial diagram associated with a sheaf
Let $F$ be a presheaf of sets on the usual topological site. Let $\left\{U_i\rightarrow U\right\}$ be covering of $U$. On the second page of Hypercovers and Simplicial Presheaves the authors write the ...
- 14.2k