Questions tagged [simplicial-stuff]

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-Kan correspondence etc. Please do not use for questions about geometry of simplices nor about triangulations.

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81 views

Derived functor of the realisation in a simplicial model category

Given a model category $C$, a simplicial symmetric monoidal model category $D$ (in the sense of Goerss-Jardine) and a left Quillen functor $F:C\to D$, define $|F|:C\to\mathsf{sSet}$ to be the functor $...
3
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1answer
268 views

Is the simplicial $n$-sphere a reduced Kan simplicial set?

The simplicial $n$-sphere is the unique simplicial set with one non-degenerate 0-simplex, one non-degenerate $n$-simplex, and no other non-degenerate simplices. The $n$-sphere is obtained from $\...
3
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1answer
149 views

Loop space has all abelian homotopy groups $i \geq 1$, done via simplicial sets

This is a proof from Goerss-Jardine (p.31): What do they mean by the multiplication $\star$? Should I think of $\pi_n(\Omega X, *)$ as $1$ simplices in $\mathsf{sSet}(\Delta^n, X)$? In order to do my ...
3
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1answer
319 views

Definition for simplicial complex

I'm confused between the definition of a simplex and a simplicial complex, as well as a face and a chain, and all of their various definitions of each. (Definition 0): A concrete simplicial complex ...
3
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1answer
423 views

Homology of simplicial set vs homology of its geometric realization?

Problem. Given a simplicial set $X$, we can associate to it a chain complex defined by $$C_n=\mathbb{Z}[X_n]$$(the free abelian group on the set $X_n$) with differentials $$d=\sum_i(-1)^id_i:C_n\to C_{...
3
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1answer
153 views

Degeneracies of simplex $y$ which appears as any face of some simplex $x$

Let $K$ be simplicial set and $d_i:K_n\rightarrow K_{n-1}$, $s_i: K_n\rightarrow K_{n+1}$ ($i = 0,...,n$) face and degeneracy maps respectively. Suppose we have some $x\in K_n$ with $d_0x = ... = ...
3
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1answer
485 views

cofaces and codegeneracies on Simplicial Sets

Let Δ be the category of finite ordinal numbers with order-preserving maps, i.e., Δ consists of objects strings A morphism f:n→[m] is an order-preserving function (a functor) and we can think of the ...
3
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1answer
910 views

Contradictory Orientations of Faces in Simplicial Complexes

From what I've read, an orientation of a simplex is an equivalence class of total orders on its vertices, where equivalence means up to even permutation. An orientation on an $n$-simplex induces an ...
3
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1answer
128 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 ...
3
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1answer
156 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. ...
3
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1answer
545 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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41 views

(Reference Request) Simplicial Complexes and Homology Book

A student has asked for references on simplicial complexes, and I remember a book but can't find its name. Pretty sure it was a yellow Springer book, but not sure what series - and I'm not even 100% ...
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46 views

Example of right homotopy not transitive in Kan-Quillen model structure

I am learning about abstract homotopy theory and know that when $Y$ is fibrant then right homotopy is an equivalence relation on the set of maps from $X$ to $Y$. When $Y$ isn’t, then transitivity ...
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60 views

How to divide a unit space into many simplices?

I'm sorry, it may be simple and stupid but I didn't find any relative solutions on the Internet. Given the unit hypercube $C$ in the Euclidean space $R^n$, how to divide (or we can say "partition", I ...
3
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91 views

Homology is homotopy invariant, fake proof with simplicial methods?

Here is a maybe false proof that I came up with that homology of topological spaces is homotopy invariant. I'm thinking that it is indeed fake because why hasn't anyone else come up with this much ...
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50 views

Generalization of singular homology of topological spaces to varieties over $k$.

To define the singular homology of topological spaces we define a sequence of topological spaces $\Delta^n$ and maps $r_i:\Delta^n \rightarrow \Delta^{n+1}$. I'm kinda skipping a lot of details ...
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0answers
41 views

Reproving a standard result about singular homology with methods from model categories

If $A \rightarrow X$ is a cofibration of topological spaces then the induced map $H_i(X,A)\rightarrow H_i(X/A,*)$ is an isomorphism. I am aware of many proofs of this fact but none use methods from ...
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106 views

Reedy model structure, simplicial sets and model categories

I have two questions, the first one is just wether the following statement is true or not? Is there a reference for this? The second one is much more open, it's basically what restricitions do we ...
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105 views

Kan extension along any functor implies adjoint exists

I trying to decide whether the existence of a, say, a left Kan extension along $i: C \to D$ for any functor $F: C \to E$ implies that $i$ has a right adjoint. I have proven the converse, and I know ...
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46 views

Showing that $\pi_n(X,v)$ satisfies inverse axiom.

Given a fibrant simplicial set $X$ (has lifting condition with all horns) and a vertex $v \in X_0$. I want to show that the simplicial homotopy group $\pi_n(X,v)$ satisfies the inverse axiom. ...
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0answers
31 views

Triangulation of fiber bundles

Let us suppose we are given a fiber bundle $(E,B,F,p)$ where all spaces involved are triangulable and compact. Assume we choose a triangulation for the base B. I believe it is possible to give E a ...
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110 views

Theorem 2.1.2.2 Higher Topos Theory

At the page 74 of HTT, there is the following theorem Let $S$ be a simplicial set, $\mathcal{C}$ a simplicial category, and $\phi: \mathfrak{C}[S] \rightarrow \mathcal{C}^{op}$ a simplicial functor....
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40 views

Simplicial intersection product

Let $X$ be both a simplicial set and a closed $n$-dimensional manifold. We have a duality isomorphism $H^k(X)\to H_{n-k}(X)$. Furthermore, let $Y,Y'\subseteq X$ be two subcomplexes and closed ...
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0answers
98 views

Simplicial set morphisms from an inner horn to a nerve are determined on the spine

This is a subject I'm completely new to, so I am a bit wary of my own proofs. I will follow the notations of Charles Rezk, Stuff about quasicategories, as of 5 April 2020. I want to prove the ...
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86 views

$\mathrm{\Gamma}$ free group functor of Barratt-Eccles

In the article A free group functor for stable homotopy theory, Barratt and Eccles define for each $X\in\mathsf{sSet}_{\ast}$, the free simplicial monoid $\Gamma^{+}X$. Proposition 6.2 states if ...
3
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1answer
158 views

Homology of the simplicial complex obtained from an octahedron by removing 4 faces.

Suppose you have the surface of an octahedron and you remove 4 of the eight faces as follows: If you remove one face then you don't remove all the adyacent faces and so on. You can look at this as ...
3
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0answers
34 views

The dimension of $H_i(X_n, \mathbb{Q})$ is linear in $n$ with a bounded nonnegative error, where the error is periodic?

Let $X$ be a finite simplicial complex with a simplicial map to $S^1$. Take covers of $X$ associated to the subgroups $n \mathbb{Z}$ of $\mathbb{Z} = \pi_1(S^1)$. This defines a sequence of spaces $...
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118 views

Can some Lie groups ($S^3$ in particular) be converted to simplicial groups?

I'm trying to understand the concept of Kan simplicial sets and simplicial groups in particular. My question is, in a nutshell: What is the relation between the group operation in a simplicial ...
3
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0answers
182 views

Cohomology of geometric realization of a simplicial topological space

Let $X$ be a simplicial topological space. We can consider to notions of cohomology of $X$. Denote by $|X|$ geometric realization of $X$. Then we can take just $H^*(|X| )$ (i.e. usual cohomology of $|...
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142 views

Fibrant (Kan complex) geometric meaning

A simplicial set $X$ is said to be fibrant (or Kan complex) if it satisfies the following homotopy extension condition for each $i$: Let $x_0,\dots,x_{i-1},x_{i+1},\dots,x_n\in X_{n-1}$ be any ...
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63 views

Classifying space of resolution of a n-regular hypergraph

Let $G =(V(G),E(G)))$ be a $n$-regular hypergraph. Let $Z_n$ be a cyclic group with generator $a.$ Definition : A resolution of $G$ is finite partially ordered set $C$ with $C_0$ is the set of ...
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0answers
229 views

How to compute (co)limits of enriched categories?

Let $\mathscr{V}$ be a monoidal category. Let $\mathbf{Cat}_{\mathscr{V}}$ be the category of (small) categories. I would like to know how to compute (co)limits in $\mathbf{Cat}_{\mathscr{V}}$. This ...
3
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1answer
89 views

Universality of the Simplex Category. Proving Functoriality of the Map.

Let $B$ be a strict monoidal category, and $\left \langle c,\mu ',\eta ' \right \rangle$ a monoid in $B$. Now suppose we consider the simplex category $\left \langle \triangle ,+,0 \right \rangle$, ...
3
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0answers
174 views

Is the projection from the cyclic nerve to the simplicial nerve a cyclic map?

I’m trying to understand Loday’s proof of Theorem 7.3.11 in “Cyclic Homology” (2nd ed 1998). I have a problem right at the beginning where it says: The projection map $\text{proj}:\Gamma_\bullet G \...
3
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0answers
231 views

Associativity of the tensor product of dendroidal sets

For any smal category $A$, I shall write $\widehat A$ for the category $[A^{\text op}, \mathbf{Set}]$ of presheaves on $A$, and $y_A\colon A \to \widehat A$ for the Yoneda embedding relative to $A$. ...
3
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0answers
88 views

When may the bar resolution be truncated for calculating $\operatorname{hocolim}$?

Suppose $D$ is a small category and $F\colon D\to sSets$ a functor. I want to calculate the simplicial set $$\operatorname{hocolim}F$$ via the bar construction: $\operatorname{hocolim}F$ is the ...
3
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0answers
159 views

Right Kan extension along a diagonal functor.

Denote $\newcommand{\simpcat}{\boldsymbol\Delta} \simpcat$ the simplex category (category of finite ordinals with non decreasing maps). The diagonal functor $\delta \colon \simpcat \to \simpcat\times\...
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0answers
95 views

An equality of some equalizers of simplicial sets.

$\newcommand{\cosimp}[1]{{#1}^\bullet} \newcommand{\sSet}{\mathsf{sSet}} \newcommand{\Set}{\mathsf{Set}}$ [I realize that the post is imposing.However, all the content of the problem is in the gray ...
3
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1answer
308 views

augmented algebras and their morphisms

Let $R$ be a commutative unital ring and $A$ an associative (unital) $R$-algebra. What is an augmented $R$-algebra? A (unital) $R$-algebra $A$, together with a (unital) ring morphism $\varepsilon: A\...
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0answers
86 views

Does the $\mathcal{E}G$-construction arise form a comonad?

Let $G$ be a simplicial group and consider the adjunction $$ Fr:sSets\rightleftarrows G-sSets:U $$ where the left adjoint $Fr$ maps a simplicial set $X$ to the free $G$-simplicial set $G\times X$ and ...
3
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0answers
145 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
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0answers
61 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
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0answers
346 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
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3answers
156 views

Simplicial sets which are not Kan complexes

A Kan complex is a simplicial set satisfying the horn-filler condition. What examples are there of simplicial sets which do not satisfy the horn-filler condition? In particular, what simplicial sets ...
2
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3answers
2k views

Show simplicial complex is Hausdorff

I have a simplicial complex $K$ and I need to show that its topological realisation $|K|$ is Hausdorff. And $K$ need not be finite. I have very little idea on how to get started on this. Only that if ...
2
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3answers
694 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
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1answer
137 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. ...
2
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1answer
239 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 Y_{\bullet}...
2
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1answer
78 views

Is the homology of the Moore complex associated to a classifying space $BG$ the group homology of $G$?

Let $G$ be a group. We define its standard resolution by setting $$ C_n = \mathbb{Z}G^{n+1} $$ to be the free $\mathbb{Z}$-module on $G^{n+1}$ for each $n \geq 0$, with a $\mathbb{Z}[G]$-module ...
2
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1answer
246 views

Understanding the exponential object in slice categories

Suppose $\mathcal C$ is a locally cartesian closed category (take simplicial sets for example), then for $B$ an object of $\mathcal C$, the slice category $\mathcal C/B$ is cartesian closed and thus ...

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