5
votes
0answers
37 views

A theorem of Kan regarding fibrant replacement

Recall that there is an adjunction $$\mathrm{Sd} \dashv \mathrm{Ex} : \mathbf{sSet} \to \mathbf{sSet}$$ where $\mathrm{Sd} (\Delta^n)$ is the first barycentric subdivision of $\Delta^n$. There is a ...
0
votes
1answer
26 views

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 ...
2
votes
1answer
36 views

Product of simplicial complexes?

Given two abstract simplicial complexes $\mathcal{K}$ and $\mathcal{L}$, what is the definition of their product $\mathcal{K} \times \mathcal{L}$, as another abstract simplicial complex? Basically I'm ...
5
votes
1answer
56 views

Cohomological Whitehead theorem

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 ...
3
votes
1answer
47 views

Does $\pi_0$ preserve infinite products?

The functor $\pi_0\colon \operatorname{sSets}\to \operatorname{Sets}$ has value on a simplicial set $X$ the coequalizer of the two boundary morphisms $d_0,d_1\colon X_1\to X_0$. It is easy to see ...
3
votes
1answer
56 views

n-truncated simplicial set

It might be a trivial question. So, I apologise in advance. Let $ \Delta ^{op}_n $ be the full subcategory of $ \Delta ^{op} $ such that the set of objects of $ \Delta ^{op}_n $ is $ \left\{ 0, ...
4
votes
1answer
87 views

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 ...
4
votes
1answer
77 views

Model structure on sSet

Which is the model structure on $ \text{sSet} $ (category of simplicial sets) that makes $\text{sSet}$ Quillen equivalent to the category $ \text{Cat} $ (of small categories) by the adjunction ...
1
vote
1answer
52 views

Simplicial n-sphere as a coequalizer? (trivial question)

I know that it is a trivial question. And it can be found in any text on simplicial sets. (So, sorry about that). But I'm studying the Jardine's Book on Simplicial Homotopy Theory (Jardine and ...
5
votes
1answer
69 views

Fat geometric realization weakly equivalent to the usual one

Let $X$ be a simplicial set. Recall that we can associate to it a certain topological space, the geometric realization, given by $ | X | = \int ^{n \in \Delta} X_{n} \times \Delta _{n} \simeq ...
4
votes
1answer
156 views

What are simplicial topological spaces intuitively?

(NOTE: I reposted the question to MO. Please answer there.) I tend to imagine simplicial objects in a category as some kind of "topological objects", with a notion of homotopy. Simplicial sets are ...
5
votes
0answers
118 views

Characterising singular homology among a more general class of cosimplicial spaces

Is there a way to characterise (up to isomorphism) the simplicial spaces $F: \Delta \to \underline{\text{Top}}$ with $F( \underline{n}) \subset \mathbb{R}^{n+1}$ compact and $F(\underline{0})$ a ...
3
votes
1answer
73 views

Cech cohomology and cohomology of a category : a cluster of questions.

I apologize in advance : what follows is a bit of a mess. Also, I think it might be a big tautology, but i don't see it yet. My question is about the rapport of Cech cohomology and cohomology of a ...
2
votes
1answer
42 views

Understanding Quillens Theorem A

Let me restate the theorem: Let $F\colon\mathcal{C}\to\mathcal{D}$ be a functor. If $F\downarrow x$ is contractible for every $x\in\operatorname{Ob}(\mathcal{D})$, then $F$ is a homotopy ...
5
votes
1answer
117 views

Homology other than singular?

Usually, one defines $n$-th homology functor on topological spaces as the composite functor $$ \mathbf{Top} \to [\Delta^\mathrm{op},\mathbf{Set}] \to [\Delta^\mathrm{op},R\!-\!\mathbf{Mod}] \overset C ...
2
votes
1answer
74 views

Show that two different embeddings of the figure-eight in the torus are not homotopic

Note, we can express the torus $|X.| \cong T$ as a square with edges denoted by $e$ and $f$, the diagonal by $g$, and faces $T_1$ and $T_2$, and a single vertex $v$, with appropriate identifications. ...
6
votes
0answers
100 views

Playing with the torus and semisimplicial sets (prove that $\phi$ and $\psi$ are not homotopic)

Recall that we can express the torus $|X.| \cong T$ as a square with edges $e$ and $f$, diagonal $g$, faces $T_1$ and $T_2$, and a single vertex $v$, and appropriate identifications. Let $Y.$ be the ...
1
vote
1answer
80 views

Show that any continuous map $f: X \rightarrow Y$ induces a map of semisimplicial sets $Sing(X). \rightarrow Sing(Y).$

I want to show that any continuous map $f: X \rightarrow Y$ induces a map of semisimplicial sets $Sing(X). \rightarrow Sing(Y).$, but I'm confused about how to do so. I guess the main confusion is ...
2
votes
1answer
86 views

What is the cone over a simplicial set?

At the moment I'm reading through Edward B. Curtis, 'Simplicial Homotopy Theory' (Advances in Mathematics 6, 107-209 (1971)) in order to learn about simplicial sets and I run into a problem where the ...
0
votes
0answers
26 views

simplicial complex bijection [duplicate]

Given two compact Hausdorff spaces $X$ and $Y$ and $h \colon X \to Y$ a homeomorphism, how can I prove that $h_{\mathfrak{A}} : N(\mathfrak{A}) \to N(h(\mathfrak{A}))$ is a bijection where ...
4
votes
2answers
170 views

'homotopy' between morphisms of a 'topological' or 'algebraic' category (Stanley-Reisner ring)

In what follows, a homotopy is a congruence $\simeq$ on a given category. Given such a homotopy, objects $X$ and $Y$ of the given category are homotopy equivalent when there exist morphisms ...
1
vote
2answers
70 views

horn of a simplex

I'm reading one book of simplicial homotopy. It's just amazing. But I am stuck at the very beginning of the book. He let a simplicial set be a contravariant functor between the category $ \Delta $ of ...
8
votes
2answers
263 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 ...
0
votes
1answer
36 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
40 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 ...
5
votes
1answer
126 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 ...
0
votes
0answers
41 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
125 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 ...
1
vote
0answers
68 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
71 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 ...
4
votes
0answers
187 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 ...
4
votes
2answers
115 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 ...
8
votes
1answer
978 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
78 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
2answers
77 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
vote
0answers
60 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 ...
2
votes
1answer
215 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
3answers
219 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 ...
1
vote
1answer
67 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
137 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
1answer
88 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 ...
2
votes
0answers
104 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 ...
3
votes
1answer
132 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 ...
4
votes
1answer
530 views

Barycentric subdivisions of simplices yield a simplicial complex

The following interesting result (in particular parts (b) and (d)) is stated either as a obvious fact or as an exercise in several books on algebraic topology: The barycenter $b_\sigma$ of an ...
8
votes
3answers
386 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 ...
2
votes
1answer
107 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 ...
1
vote
1answer
72 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 ...
2
votes
1answer
117 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. ...
4
votes
1answer
404 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 ...
5
votes
2answers
214 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 ...