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5
votes
1answer
108 views
What is the homotopy colimit of the Cech nerve as a bi-simplical set?
Let $f:Y_\bullet\to X_\bullet$ be an epimorphism of simplicial sets and define the bi-simplicial set
$$
F_{\bullet\bullet}=\ldots Y\times_X Y\times_X Y\underset{\to}{\underset{\to}{\to}}Y\times_X Y ...
1
vote
0answers
24 views
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 ...
2
votes
2answers
53 views
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$ ...
3
votes
1answer
38 views
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 ...
0
votes
1answer
33 views
Paths between 0-cells in a classifying space. II
Let $\mathcal{C}$ be a small category and $X,Y$ objects within.
If $X$ and $Y$ (as $0$-cells) lie within the same path-component in $B\mathcal{C}$, can one say anything in general on how $X$ and ...
1
vote
1answer
37 views
Paths between 0-cells in a classifying space.
Let $\mathcal{C}$ be a small category. Giving a morphism $u: X \to Y$ in $\mathcal{C}$ is equivalent to giving a functor $f: \{0 < 1 \} \to \mathcal{C}$ (with $f(0) = X$, $f(1) = Y$, and $f(0 \to ...
2
votes
1answer
77 views
What exactly is the CW complex structure on a geometric realisation?
This is likely a silly question.
Definitions:
$\bullet$ $\Delta_n = \{ (t_0, \dots, t_n) \: | \: 0 \leq t_i \leq 1, \sum_i t_i = 1 \}$
$\bullet$ Given $f: \underline{m} \to \underline{n}$ in ...
3
votes
0answers
37 views
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 ...
0
votes
0answers
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 ...
5
votes
0answers
100 views
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 ...
2
votes
1answer
37 views
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 ...
4
votes
1answer
50 views
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 ...
1
vote
1answer
34 views
Regular triangulation of compact oriented hyperbolic space
Is there a good way of explicitly constructing a regular triangulation of a compact orientable hyperbolic 2-manifold, ideally with any desired vertex degree $\ge 7$? I only need the topology, not any ...
2
votes
1answer
65 views
adjunction relation
Assume I have simplicial sets $X:\Delta^{op}\rightarrow Set$ and $Y:\Delta^{op}\rightarrow Set$, then I can form the simplicial set
$\operatorname{Hom}(X,Y)_n := ...
1
vote
0answers
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 ...
1
vote
1answer
45 views
Does the suspension functor preserve fibrations?
Let $X_{\bullet}$ be a simplicial set and let $\Sigma X_{\bullet}$ denote its simplicial suspension. If $X_{\bullet} \to Y_{\bullet}$ is a fibration, then is $\Sigma X_{\bullet} \to \Sigma ...
3
votes
0answers
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 ...
3
votes
0answers
51 views
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 ...
7
votes
1answer
317 views
Difference between simplicial and singular homology?
I am having some difficulties understanding the difference between simplicial and singular homology. I am aware of the fact that they are isomorphic, i.e. the homology groups are in fact the same (and ...
2
votes
0answers
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|} ...
2
votes
2answers
66 views
The empty set in homotopy theoretic terms (as a simplicial set/top. space)
I am currently confused about the empty set in terms of its path components and how this fits into the Quillen adjunction between topological spaces and simplicial sets. Probably, one of my ...
3
votes
2answers
148 views
Kan fibrations and surjectivity
I have a basic question on the usual model structure on simplicial sets.
What is the relation between being a Kan (trivial maybe ?) fibration and
surjectivity ?
Surjectivity here means either ...
3
votes
0answers
52 views
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 ...
1
vote
0answers
140 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*$): \Delta \rightarrow$ ...
1
vote
1answer
62 views
Decomposition of simplicial G-sets as a colimits of its simplicial G-subsets
Every simplicial set is the colimit of its finite simplicial subsets. Suppose $G$ be a finite discrete set.
Is every simplicial $G$-set a colimit of its finite simplicial $G$-subsets? I'm ...
0
votes
1answer
59 views
How to show that $\Delta[n]$ isn't Kan fibrant…?
This is the problem: I have to prove that $\Delta[n]$ isn't Kan fibrant for n >=2. Does anyone how idea how to do it?
3
votes
0answers
67 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 ...
0
votes
0answers
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 ...
2
votes
1answer
39 views
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
...
1
vote
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 ...
2
votes
0answers
60 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 ...
1
vote
1answer
70 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 ...
2
votes
2answers
63 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:
...
1
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0answers
58 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 ...
5
votes
2answers
117 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.
...
3
votes
2answers
153 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 ...
2
votes
1answer
119 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
...
5
votes
2answers
194 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 ...
0
votes
0answers
45 views
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 ...
2
votes
3answers
153 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 ...
2
votes
0answers
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 ...
1
vote
1answer
45 views
Is the Image of a Simplicial Complex under a Simplicial Map again a Simplicial Complex?
I think this is trivially true, but just wanted confirmation. A simple yes or no would be great! Thanks.
1
vote
0answers
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 ...
0
votes
0answers
83 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 ...
0
votes
1answer
116 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 ...
7
votes
3answers
278 views
What is combinatorial homotopy theory?
Edit: After a discussion with t.b. we agreed that this question aims to a different answer from this one, for more information you can read the comment below.
Many times I've heard people ...
0
votes
1answer
54 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 ...
0
votes
1answer
76 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 ...
2
votes
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 ...
2
votes
0answers
103 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 ...
