# Tagged Questions

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-Kahn correspondence etc. Please do not use for questions about geometry of simplices nor about ...

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### The line between the centers of two k-simplices crosses through the centers of lower dimensional simplices

"The line between the centers of two k-simplices crosses through the centers of lower dimensional simplices." Is this always true? The barycentric centers are equidistant from the vertices. k=2 So ...
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### Explanation of notation for face and degeneracy

The (co)face are historically denoted by $d$, while the (co)degeneracy maps are denoted by $s$. Why is this, instead of the obvious $f$ and $d$? In the simplicial category $\Delta$, you also realize ...
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Let $X$ and $Y$ be CW complexes (resp. Kan complexes) and let $f : X \to Y$ be a continuous map (resp. morphism of simplicial sets). The following seems to be a folklore result: Theorem. The ...
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### Existence and homotopies of embeddings between simplicial complexes

Let $K$ and $L$ be simplicial complexes, $m=\dim K$, and $h:|K|\rightarrow |L|$ be a continuous map. Show that $h$ is homotopic to a map carrying $K$ into $L^{(m)}$, the $m$-skeleton of $L$. I'm ...
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### Two definitions of equivariant sheaves

Let $G$ be a topological group. Here are two definitions of $G$-equivariant sheaves on a $G$-space $X$. (a) Define an $G$-equivariant sheaf by a sheaf $F$ (étalé space) equipped with a $G$-action ...
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### Is a simplicial subcomplex of a contractible simplicial CW-complex also contractible?

Let $Y,X$ be simplical CW-complexes (By that I mean functors $\Delta^{op}\to CW$. In other words, for each natural number $n$ there exist CW-complexes $X_n$ and $Y_n$ and there are face and degeneracy ...
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### A question about partitioning the unit cube into simplexes

Let $C$ be the unit cube in $\mathbb{R}^n$. I want to simplicize $C$ (i.e. partition it into simplexes), but I have to follow these two rules: I can't add any vertices to $C$. The vertices of each ...
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### What is a good way to simplicize the integer lattice?

I have a function $f$ defined on the positive integer lattice in $\mathbb{R}^n$ (i.e. the vectors in $\mathbb{R}^n$ whose coordinates are all nonnegative integers). I want to extend the domain of $f$ ...
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### Random mixing of the space of triangulations of a surface

Summary: How quickly does the edge-flip random walk in the space of triangulations of a closed, connected, orientable surface converge to the uniform distribution over all triangulations? Let $M$ be ...