The simplicial-stuff tag has no wiki summary.
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Pseudomanifolds
An $n$-dimensional (closed) pseudomanifold is a finite simplicial complex $X$ such that
(i) every simplex is a face of an $n$-simplex
(ii) every $(n-1)$-simplex is a face of exactly two ...
2
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1answer
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Inner product between certain vectors on a simplex.
For $n\geq 2$, let $\Delta^n$ be a regular $n$-dimensional simplex in $\mathbb{R}^n$ centered at the origin ${\bf 0}$ and inscribed in the unit sphere $S^{n-1}$.
Let ${\bf v}_0,{\bf v}_1,\ldots,{\bf ...
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Colimits of cosimplicial rings
The forgetful functor from the category of rings to the category of sets does not preserve arbitrary colimits. But it does preserve filtered colimits. For example, the colimit of a sequence of rings
...
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1answer
52 views
Boundary of a simplicial set in terms of a coequalizer
I am trying to understand why we have a coequalizer
$\sqcup_{0 \leq i < j \leq n} |\Delta^{n-2}| \rightrightarrows \sqcup_{0 \leq i \leq n} |\Delta^{n-1}| \rightarrow |\partial \Delta^n|$. What ...
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Inverse functor in proof of Dold Kan Correspondence
I´m looking at the proof of the Dold-Kan correspondence. Let $SA$ be the category of simplicial objects in an abelian category $A$ and $CH_{\ge0}(A) $ the category of non-negative chain complexes in ...
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For which coverings by “geometrically nice” sets does the nerve admit “Vietoris-Rips-like” approximations?
It is well known that the nerve (or Čech complex) of a covering by metric balls is nicely approximated by the Vietoris-Rips complex. Being a flag complex on its 1-simplices, the latter is ...
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150 views
Algebraic Morse theory
In 2005, prof. Emil Skoldberg developed a theory, similar to Forman's Discrete Morse Theory, but suited for arbitrary based chain complexes, in his Morse Theory from an algebraic viewpoint. I'm going ...
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Hypervolume under the square of an n-simplex
I posted this question a while ago, but since I didn't have much luck I though I'd reformulate it and try again.
Question: What is the general form of the equation that gives the hypervolume under ...
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What is an “absolute, equational pushout”?
I was leafing through Joyal and Tierney's Notes on simplicial homotopy theory.
In the first few lines of the section on the skeleton of a simplicial set they display the simplicial identities as a ...
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66 views
What is a copresheaf on a “precategory”?
Let $\mathcal{S}$ be a category with pullbacks. A precategory $\mathbb{C}$ in the sense of Borceux and Janelidze [Galois Theories, §7.2] comprises three objects $C_0, C_1, C_2$ in $\mathcal{S}$ and ...
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64 views
root mean square distance between two simplices
As the title says, I want to compute the root mean square distance between two n-dimensional simplices. Say I have two surfaces $S$ and $S'$, the mean error is
$$
d_m(S,S') = \frac{1}{|S|} ...
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Ways to decompose a torus for finite element method so that each cell contains a complete revolution of the major radius
I've got a finite element problem involving paths around the interior of a torus. For this particular problem I think I could make things more computationally efficient if each cell in the mesh made ...
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0answers
76 views
left inverse to trivial fibration is trivial cofibration
It is claimed that in the model category of simplicial sets (with usual model structure), a trivial fibration $X \to Y$ has a section, which is a trivial cofibration.
Now, I see that there is a ...
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102 views
How does a simplicial map induce a map on chain complexes
I am working on the following exercise: Show that a simplicial map induces a map on chain complexes (hence maps on homology between simplicial complexes).
Here is what I have so far...
Let $K, L$ be ...
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182 views
does it make sense to tensor a simplicial set with a simplicial ring?
Given a simplicial commutative ring $A$ and a simplicial set $K$ does $K \otimes A$ make sense as a (commutative) simplicial ring?
I'm asking as I've seen the expression $S^n \otimes A$ written down ...
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73 views
Local injective model structure for simplicial presheaves
The category of simplicial presheaves on a small Grothendieck site $\mathcal{C}$ can be given a model structure by defining weak equivalences and cofibrations sectionwise. It's called the (global) ...
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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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50 views
geometric realization of a free simplicial group
Call a simplicial group $G_{\bullet}$ free if for each $n$, the group $G_n$ is a free group.
How does the geometric realization of $G_{\bullet}$ look like?
Can its nondegenerate and ...
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139 views
Understanding the definition and notation of geometric realization
I had trouble making sense of the definition of geometric realization of a simplicial set. Let $\Delta^n$ be the standard n-simplex defined as the functor $\hom_\Delta(-,$*n*$): \Delta \rightarrow$ ...
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Can one define “simplicial” EM spaces?
Let $\Delta$ be the well-known simplicial category, and denote with $\widehat\Delta_\bf C$ the category of simplicial objects in $\bf C$ ($\widehat\Delta$ for short).
Given $\mathcal ...
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89 views
The Fundamental Group of a Polyhedron Depends Only on its $2$-Skeleton
Does anyone have a good quick proof of this using the Simplicial Approximation Theorem? I'm aware that it comes out as a corollary when considering edge paths and the edge group, but this seems like ...
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Valuations, simplicial complexes and arithmetics
I am currently working on an Introduction to geometric probability (Klein, Rota, 1997). This book is very stimulating, and I find myself toying with the subject, and a note in particular (pp 95-97 for ...
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34 views
An variation of “Lk” in simplicial complexes
Let $\mathcal S$ be a simplicial complex. For a simplex $S \in \mathcal S$, we have the closure, the star and the link
$\operatorname{Cl} S := \{ F \in \mathcal S : F \leq S \}$
$\operatorname{St} S ...
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46 views
Teminology for a “supercomplex”
If $\Delta_{1}$ and $\Delta_{2}$ are abstract simplicial complexes and $\Delta_{1}\subseteq \Delta_{2}$, then we say that $\Delta_{1}$ is a subcomplex of $\Delta_{2}$. Is there a terminology for ...
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What is a category of simplicial complexes where collapsibility is well-defined?
Given this discussion http://mathoverflow.net/questions/87557/a-simplicial-complex-which-is-not-collapsible-but-whose-barycentric-subdivision, it seems that the notion of collapsibility is not ...
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Generalizing Euler's Polyhedral Formula for Graph Embeddings in Higher Dimensions.
In the plane, Euler's Polyhedral formula tells us that $V - E + F = \chi$, where for graph embeddings we have that $\chi = 1$. Alternatively, we can think of a graph embedding as a simplicial ...
