# Questions tagged [simplicial-stuff]

For questions about simplicial sets (functors from simplex category to sets), simplicial (co)algebras and simplicial objects in other categories; geometric realization, model structures, Dold-Kan correspondence etc. Please do not use for questions about geometry of simplices nor about triangulations.

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### Non simplicial model categories

What are some interesting/notable examples of non simplicial model categories? What are some notable examples of model categories which cannot be seen to be Quillen equivalent to a simplicial one? ...
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### Locally finite abstract simplicial complex has only finite simplices?

I usually think of a locally finite abstract simplicial complex as having that any simplex has only finitely many higher dimensional simplices containing it. Namely if $\sigma$ is a $d$-simplex, then ...
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### A simplicial structure on symmetric groups

Do the symmetric groups admit a simplicial structure? By this, I mean a functor $X: \Delta^{op} \to \text{Sets}$ such that $X(n) = S_n$. More explicitly, one has to find functions (not necessarily ...
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### How to divide a unit space into many simplices?

I'm sorry, it may be simple and stupid but I didn't find any relative solutions on the Internet. Given the unit hypercube $C$ in the Euclidean space $R^n$, how to divide (or we can say "partition", I ...
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### How is a simplicial complex completely determined by a simplicial set?

I was wondering how a simplicial complex is completely detremined by a simplicial set. I say let $K$ be a simplicial complex. Then $K=(V,S)$ where $V$ is a set and $S$ is a collection of finite non-...
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### Dyadic division of a simplex - is it possible in dimension > 2?

Assume T is a triangle. We can join the middle points of the sides of T, and this form a division of T into 4 triangles, homothetic to the original triangle, and with sides equal to sides(T)/2. Let me ...
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### Introduction to $E_\infty$-algebras

To give an idea on how limited my background is: a few days ago, I have heard the term operad for the first time. I am trying to understand a text about dg-, $A_\infty$- and $E_\infty$-algebras in the ...
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### Proving that for every element $x\in X_n$ there is a unique $m$, surjection $f\in\Delta(n, m)$, and nondegenerate $y\in X_m$ such that $x=f^*(y)$.

Let $X$ be a simplicial set, and $|X|=X\otimes_\Delta\Delta^{\bullet}$ its geometric realization. We say $x\in X_n$ is degenerate if there is a surjection $f\in\Delta(n,m)$ with $m<n$ and $y\in X_m$...
I am trying to read Goerss and Jardine's book (Simplicial Homotopy Theory) and in the proof of Theorem 7.10 (in chapter 1), they claim that there is a simplicial homotopy $\Delta^n\times\Delta^1\to\... 0answers 25 views ### A lifting problem in$\infty$-categories Let$\mathcal{C}$be an$\infty$-category and consider the outer horn inclusion$\Lambda_3 \subset \Delta$. Given a diagram$f:\Lambda_3 \to \mathcal{C}$, if the image of$\Delta^{\{2,3\}}$... 1answer 44 views ### Why two simplices become equal in the colimit The following is an image of a proof found in Hovey's Model Categories, for which my question concerns just the second paragraph. I do not see the justification for the statement that we can find$\...
The following is an image of a proof from Hovey's Model Categories: How exactly do we know that $s\restriction_{\partial{\Delta[n]}}$ factors through $X_n$? Since $\partial{\Delta[n]}$ has only ...