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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115 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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106 views

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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1answer
208 views

manifold as simplicial complex

I want to know the topological relation between a manifold and a simplicial complex. I know that a simplicial complex cannot be a manifold since its a union of simplices which are manifolds of ...
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2answers
112 views

Equivalent definition for a collection of simplices to be a simplicial complex

I am reading the following lemma from Munkres' Elements of Algebraic Topology: Lemma 2.1 A collection $K$ of simplices is a simplicial complex if and only if the follow hold: ...
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64 views

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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2answers
135 views

Are bounded open regions in $\mathbb{R}^n$ determined by their boundary?

Let $U$ and $V$ be two bounded open regions in $\mathbb{R}^n$, and let us further assume that their topological boundaries are nice enough that they are homeomorphic to finite simplicial complexes. ...
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2answers
243 views

Triangulate rectangular parallelepiped in $\mathbb{R}^{n}$

I need to triangulate the n-dimensioned rectangular parallelepiped in $\mathbb{R}^{n}$ into a set of $n$-simplices. Could you suggest me any known algorithm for that or maybe an extension of Delaunay ...
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1answer
293 views

Comparison between various types of cell complexes

There are the following (and more) types of geometric cell complexes: 1) The geometric realization of a simplicial set 2) CW-complexes 3) The geometric realization of an abstract simplicial complex ...
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234 views

Excision via simplicial sets

Let $X$ be a topological space, covered by a collection of open sets $\{U_\alpha\}$ (or more generally a cover by subspaces whose interiors cover $X$). Consider the singular simplicial set ...
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3answers
315 views

can the statement “a simplicial set is the nerve of a category if and only if it satisfies a horn-filling condition” be tweaked for groupoids?

For some reason I convinced myself that a simplicial set (or maybe I mean directly Kan complex) is homotopy equivalent to the nerve of a groupoid if and only if it has no higher homotopy groups. Is ...
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238 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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187 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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190 views

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 ...
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1answer
195 views

Maximum number of simplexes given n-element point sets in the plane

Does anyone know if it has been proved what the maximum number of simplexes occurring in the plane is for a given value of $n$ points? I am interested in this question in relation to packing problems ...
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1answer
108 views

Simplicial Complexes - the Closure of the Star is a Cone on the Link (proof?)

I'm trying to prove that $\overline{st_K(x)}$ is a cone on $lk_K(x)$, but can't seem to get anywhere! I know how to construct a topological cone given a space $X$. However I don't know any way to ...
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1answer
158 views

simplex and power set

I read the following: Let $M$ be a set. The simplex on $M$ is the set of all subsets of $M$; we denote this by $\Delta_M$. We will sometimes refer to the elements of $M$ as vertices of $\Delta_M$. A ...
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157 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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2answers
237 views

The subcomplex of degenerate simplices has trivial homology

Let $A_\bullet$ be a (non-augmented) simplicial object in an abelian category, with face maps $d_i : A_n \to A_{n-1}$ and degeneracy maps $s_i : A_n \to A_{n+1}$, $0 \le i \le n$, for each $n \ge 0$. ...
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1answer
131 views

Non-Kan Fibrations

In the definition of a Kan fibration (on nlab), i.e. for a map $\pi:Y\to X$ of simplicial sets the inclusion of any horn into $Y$ always lifts to an inclusion of the filled in horn if that filled in ...
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74 views

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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2answers
296 views

Passing pullbacks through adjunction

I'm having trouble following the proof of Proposition I.5.2 in Goerss-Jardine (Simplicial Homotopy Theory). After establishing the adjunction $\hat\Delta(X\times K,Y) \simeq \hat\Delta(K,[X,Y])$, ...
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1answer
126 views

Adjointness of the corresponding simplicial functors associated to an adjoint pair between categories

Suppose you have an adjoint pair of functors $F:C\to D$ and $G:D\to C$. Let ${\mathrm{Simp}}C$ and ${\mathrm{Simp}}D$ be the category of simplicial objects in the categories $C$ and $D$ respectively. ...
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1answer
597 views

Does the homology, homotopy, and geometric realization functors of a simplicial group preserve colimits?

Given a based simplicial group, you can find its reduced homology with coefficients in a field, homotopy, and geometric realization. These are functors. If I have a free product of based simplicial ...
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1answer
299 views

How does hocolim relate to Hom?

In a usual category $\mathcal{C}$ one can pull the colim out of the Hom like ...
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94 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) ...