The simplicial-stuff tag has no wiki summary.
8
votes
1answer
241 views
How does hocolim relate to Hom?
In a usual category $\mathcal{C}$ one can pull the colim out of the Hom like ...
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 ...
7
votes
1answer
326 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 ...
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.
...
5
votes
1answer
110 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 ...
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 ...
5
votes
0answers
102 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 ...
4
votes
2answers
179 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])$, ...
4
votes
1answer
51 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 ...
3
votes
2answers
154 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 ...
3
votes
1answer
250 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 ...
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
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 ...
3
votes
0answers
30 views
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 ...
3
votes
0answers
39 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 ...
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 ...
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 ...
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 ...
3
votes
0answers
104 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 ...
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
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:
...
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 := ...
2
votes
2answers
67 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 ...
2
votes
1answer
121 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
...
2
votes
2answers
120 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$. ...
2
votes
1answer
99 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 ...
2
votes
2answers
55 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$ ...
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 ...
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 ...
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
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
...
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 ...
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
183 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 ...
2
votes
0answers
74 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) ...
1
vote
1answer
104 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. ...
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 ...
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 ...
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
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 ...
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 ...
1
vote
1answer
54 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 ...
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 ...
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 ...
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
0answers
141 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
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 ...
1
vote
0answers
90 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 ...
1
vote
0answers
50 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 ...
