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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Is the skeleton-coskeleton adjunction $sSet$-enriched?

Let $n\geq 0$ be an integer. Is the adjunction $$ \mathbf{sk}_k\colon sSet \leftrightarrows sSet\colon\mathbf{cosk}_k $$ of the skeleton and coskeleton an $sSet$-enriched adjunction?
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703 views

cells of quotient CW complex

Let $X$ be a CW complex and $Y$ a CW subcomplex. If $X$ has no cell of dimension $n$, for some $n>0$, then $X/Y$ has no cell of dimension $n$. Is it true? Why?
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285 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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The fat realization of simplicial trivial $G$-bundle is principal G-bundle?

When I read Johan L.Dupont's book 'Curvature and characteristic classes', I couldn't understand his following claim. Claim Let $ \pi :E\rightarrow X$ be a principal G bundle and let $\mathfrak{U} =\{...
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29 views

Cohomology(ies) of simplicial sheaves

Let $X$ be a topological space and denote by $Ab(X)$ the category of abelian sheaves on $X$. My question is on the category of simplicial abelian sheaves $[\Delta^{op},Ab(X)]$. A natural way to define ...
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23 views

Long exact sequence of homotopy groups groups in simplicial sets - reference request

I believe it is well-known that for a based map $f:X\to Y$ of simplicial sets (possibly with some extra hypotheses on $X$ and $Y$), there is a long exact sequence $$ \ldots \to \pi_n(F)\to \pi_n(X) \...
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63 views

Concrete constructions in derived algebraic geometry

I am trying to understand basic constructions in derived algebraic geometry (such as derived Hilbert schemes or the derived stack of vector bundles). What is a good reference for learning about these? ...
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19 views

Simplicial nerve of a simplicial natural transformation

In Higher Topos Theory, Lurie describes the simplicial nerve functor taking as input a simplicially enriched category $\mathcal C$ and outputting the ∞-category $\mathrm{N}(\mathcal C)$. The ...
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Cofibrant model of a simplicial group is free

I am reading this by Goerss and Schemmerhorn. In Example 4.26 it says without proof that for a group $G$ regarded as a constant simplicial group and its cofibrant model $X$, $X_q$ is free for each $q$...
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Identifying the differentials of the cellular chain complex of a (fat) geometric realization

Let $X_\bullet$ be a simplicial set, $\|X_\bullet\|$ its fat geometric realization and $|X_\bullet|$ its geometric realization, see here for definitions. The fat geometric realization $\|X_\bullet\|$ ...
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Equivalent definition of Kan fibration

Let $\Lambda^n_k$ be the $k$-th horn of the standard $n$-simplex $\Delta^n$, and let $Y$ be a simplicial set. In Simplicial Homotopy Theory, pg. 10, we find that the coequalizer description of the ...
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Connected locally finite abstract simplicial complex is countable

I want to verify whether the arguments for the statement is true. I say that an abstract simplicial complex, $X$, is locally finite if $\deg(v)<\infty$ for all $v\in X(0)$, where $$ \deg(v)=\vert \{...
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Morphisms of a Simplex Category of a simplicial set.

In the Page 7 of the book Simplicial Homotopy theory by Jardine and Goerss they defined the simplex category of a simplicial set $X$ as the slice category $\Delta \downarrow X$. So objects of $\...
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85 views

Cosimplicial resolution and fibrations

A cosimplicial resolution of a functor $\gamma: \mathcal{C}\to \mathcal{M}$ is given by a functor $\Gamma: \mathcal{C}\to \mathcal{M}^{\Delta}$ such that for every $X\in \mathcal{C},$ $\Gamma(X)$ is ...
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57 views

Reference or counterexample: simplicial algebra isomorphism

I'm looking for either a reference, a quick proof, or a counter example to the following claim: Let $R_{\bullet}$ and $T_{\bullet}$ be simplicial $k$-algebras. Let $S_{\bullet}$ be the polynomial ...
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36 views

Delooping block diffeomorphism spaces

In Kuper's Lecture Notes on Diffeomorphism Groups pg. 195, Kuper defines the simplicial group of block diffeomorphisms $Diff^\flat_\partial (M)$ for a manifold with boundary $M$ as having n-simplices ...
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43 views

Kan extensions and restrictions

Let $\mathcal{C},\mathcal{D}$ be two small categories and $\iota:\mathcal{C}\rightarrow \mathcal{D}$ a faithful functor. Then take a representable functor $$\text{Hom}(-,D):\mathcal{D}\rightarrow \...
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Proving simplicial homology is preserved on mapping?

Assume we have a the $\mathbb Z\text{ modules}$ $S1 \equiv (F, E, V)$ with boundary maps $(\partial_{FE}: F \rightarrow E$, $\partial_{EV}: E \rightarrow V)$, with the condition that $\partial_{EV} \...
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25 views

How is a simplicial complex completely determined by a simplicial set?

I was wondering how a simplicial complex is completely detremined by a simplicial set. I say let $K$ be a simplicial complex. Then $K=(V,S)$ where $V$ is a set and $S$ is a collection of finite non-...
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14 views

Dyadic division of a simplex - is it possible in dimension > 2?

Assume T is a triangle. We can join the middle points of the sides of T, and this form a division of T into 4 triangles, homothetic to the original triangle, and with sides equal to sides(T)/2. Let me ...
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35 views

Introduction to $E_\infty$-algebras

To give an idea on how limited my background is: a few days ago, I have heard the term operad for the first time. I am trying to understand a text about dg-, $A_\infty$- and $E_\infty$-algebras in the ...
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Proving that for every element $x\in X_n$ there is a unique $m$, surjection $f\in\Delta(n, m)$, and nondegenerate $y\in X_m$ such that $x=f^*(y)$.

Let $X$ be a simplicial set, and $|X|=X\otimes_\Delta\Delta^{\bullet}$ its geometric realization. We say $x\in X_n$ is degenerate if there is a surjection $f\in\Delta(n,m)$ with $m<n$ and $y\in X_m$...
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Can an $n$-orthoplex be shrunk to a point like an $n$-cube?

An inverse process of expanding a point to a 4-cube is shown in [https://en.wikipedia.org/wiki/Hypercube]. I assume that it can't be done for $n$-orthoplex, as having $$2^{k+1}\binom{n}{k+1}$$ $...
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35 views

hom between simplical sets $\Delta^1 \times \Delta^1$ and $\Delta^1$?

I'm stuck on a very simple question: Let $\Delta^1$ be the simplicial set, so $(\Delta^1)_n = Hom([n], [1])$ is the set of nondecreasing functions $f:\{0, 1, \ldots, n\} \to \{0,1 \}$. Let $sSets$ be ...
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46 views

Conditions for creating a void homeomorphic to $n$-cube or to $n$-orthoplex within a discrete manifold

Assume an $n$-dimensional discrete manifold $M$ ($n$-dimensional simplicial complex in which each ($n-1$)-simplex has exactly two adjacent $n$-simplices or only one if it is on a boundary of $M$) ...
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25 views

Simplicial space of a total space of a classifying bundle for $G$

I am reading lecture notes on topology and the total space $E(U(N))$ is given as a geometric realization of a simplicial space $$E(U(N))=|[n]\rightarrow U(N)^{n+1}|$$ Here I am confused because 1) ...
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26 views

Computations using “Stover's spectral sequence”

In this article from 1990, Stover describes a specral sequence which converges to the higher homotopy groups of the homotopy colimit of a diagram $\underline{X}$ of topological spaces. The second ...
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29 views

An element in the simplicial homotopy group

I am confused with this construction in Goerss Jardine's Simplicial Homotopy Theory, pg. 26+27 Is not $v$ a $0$-simplex? So I suppose we define $v_i$ as the composition $\Delta^n \rightarrow \Delta^0 ...
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35 views

Why if $n\geq1$ and $0\leq i\leq j\leq n$ then $F_n^j\circ F_{n-1}^i=F_n^i\circ F_{n-1}^{j-1}$?

Why if $n\geq1$ and $0\leq i\leq j\leq n$ then $F_n^j\circ F_{n-1}^i=F_n^i\circ F_{n-1}^{j-1}$? We know that an n-simplex $\mathbb{\Delta}_n$ is a subspace of $\mathbb{R}^{n+1}$ given by $\mathbb{\...
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How to see that $S(\sigma) = \text{Hom}_{\Delta}(\sigma, [0,1])$ maps to the category $\tilde{\Delta}^{\text{op}}$.

Let $\Delta$ be the simplicial category. Let $\tilde{\Delta}$ be the subcategory of non-empty totally ordered sets as objects and order-preserving maps that also preserve the smallest and largest ...
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47 views

Mapping spaces for pro-objects in a simplicial model category

If $C$ is a simplicial model category (I have simplicial commutative rings in mind), then one can simplicially enrich pro-$C$ over sSet by taking $$\lim_j \mathrm{colim}_i \underline{\mathrm{Hom}}_C(...
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35 views

$\mathbf{sSet}$-enriched Algebraic Theories

If $\mathcal{L}$ is the $\mathbf{sSet}$-enriched subcategory of $\mathbf{sSet}$ whose objects are finite coproducts of the terminal simplicial set $\Delta^0 = \Delta(-,[0]) = *$, identify the object $\...
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81 views

Is the geometric realization of a simplicial abelian groups is a topological group?

Reading http://math.uchicago.edu/~amathew/SCR.pdf I found the result that ask for but for commutative rings. I'm looking for references of any of this facts.
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74 views

weak homotopy equivalence of simplicial sets are quasi-isomorphisms

I'm trying to prove that a weak equivalence $f: X \rightarrow Y$ of simplicial sets induces a quasi-isomoprhism $Z[X] \rightarrow Z[Y]$ where $Z[X]$ is the chain complex of free abelian groups of $X_n$...
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Do I correctly understand about the facts on the bar resolution?

$\DeclareMathOperator{\Hom}{Hom}$ I'm struggling on understading Weibel, p.283 He said in The Bar Resolution 8.6.12, ... Since the homotopy groups of the simplicial $R$-module $\bot_* M$ may be ...
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24 views

Trouble with changing the fractions into matrix-vector from

I am curious about is there any way to turn the following expression into more elegant matrix-vector forms? This desired transform could contain a matrix-vector product. $$f_1=\frac{DGX+KML}{DB-ML\...
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114 views

Homotopy type of simplicial complexes

I am confused about the following question: Let $K_{\bullet}$ be a simplicial complex. Let $( \sigma, \tau )$ be a pair of its simplices with $\sigma$ the only codimension $1$ coface of $\tau$. ...
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75 views

$\Omega X$ when X is a not fibrant simplicial set?

How can I define $\Omega X$ when $X\in\mathsf{sSet}_{\ast}$ but it is not fibrant?. Actually, I would like to formulate the 'Group completion theorem' in simplicial setting, I mean $H_{\ast}(M)[\...
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58 views

Equivalence between $SP_{h}(X)$ and $\Gamma^{+}(X)$ in simplicial context

According to the article The homotopy infinite symmetric product represents stable homotopy theory, Proposition 4.5 states: There is an equivalence of topological monoids $$\pi:SP_{h}(X)\...
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61 views

$\pi_{n}(X_{G})$ when $G$ is finite and acts freely on $X$

When $X\in\mathbf{sSet}_{\ast}$ and $G$ a finite group (it may be considered a constant simplicial group) that acts freely on $X$. My question is Can I express $\pi_{n}(X_{G})$ in terms of $\pi_{n}(X)...
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183 views

Converting a simplicial set into a simplicial complex

Let $\Sigma$ be a~simplicial set with finitely many nondegenerate simplices, homeomorphic to a $d$-sphere. I would like to construct some ''derived'' simplicial set $\Sigma'$ that is already a ...
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1answer
40 views

Intuition about axiomatic description of gluing

I am currently reading "Methods of homological algebra" by S. I. Gelfand and Yu. I. Manin. I was wondering if someone could explain the intuition about why $X(id) = id$ and $X(f \circ g) = X(f) \...
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47 views

Five lemma for fibre sequences in simplicial sets

Suppose I have two fibre sequences $F\to L\to K$ and $F'\to L'\to K'$ fitting into the commutative diagram: $\require{AMScd}$ \begin{CD} F' @>>> L' @>>> K'\\ @VV{\cong}V @VVV @VV{\...
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140 views

Why does a 2 simplex define a chain homotopy?

Given a topological space and two maps from $[0,1]=|\Delta^1|$, say $f$ and $f'$ to $X$, a homotopy using an element of the 2-simplices of the singular set of $X$, such that its edges are given by the ...
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58 views

1/nb, (given b=1/150) answer is: 150/n. How??

P = 1/nb + c Given: b=1/150 and c = 17,000 The answer is: P= 150/n + 17,000, but how? How do 1/(n*1/150) turn into 150/n? Can someone please explain? Thank you!
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76 views

Is every simplicial cancellative commutative monoid a Kan complex?

Simplicial commutative monoids can fail to be Kan complexes. I don't know if this also fails assuming they are cancellative (the usual proof for simplicial groups does not seem to work). We say that a ...
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157 views

Why is the second example not a Simplicial Complex?

This is my first encounter of simplex and simplicial complex, hence I am not very sure about the concepts. In the definition of Simplicial Complex , I am not sure why is the second picture not a ...
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18 views

Pointwise fofibrations in $\mathcal{C}^I$ as saturation of a set

Let $\mathcal{C}$ be a cofibrantly generated simplicial model category, say with its set of cofibrations $\beta$-saturated and generated by a set of $\beta$-small morphisms $\mathcal{M}_0$, and $I$ a ...
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1answer
140 views

What are simplicial $\infty$-groupoids

I'm trying to understand the object $\text{Fun}(\Delta^{\text{op}},\textbf{Grpd})$ as mentioned in 2.9 of http://arxiv.org/pdf/1512.07573v2.pdf. So far I have tried to use Higher Topos Theory by ...
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67 views

understanding a statement in Weibel's “The K-book” about bisimplicial sets

Here is a theorem from Weibel's The K-book, Chapter IV Theorem 3.6.1. Let f : X → Y be a map of bisimplicial sets. (i) If each simplicial map $X_{p,∗} → Y_{p,∗}$ is a homotopy equivalence, so is BX → ...

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