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Questions tagged [geometric-realization]

Questions about geometric realization of categories, simplicial objects and any other similar notion of geometric realization.

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Ressources for the swallowing lemma

I am doing a bachelor project in Algebraic K-theory, and have run into the swallowing lemma, in particular in the proof that the $Q$ and $S.$ construction yield the same and in the proof of the ...
DevVorb's user avatar
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2 votes
1 answer
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Fundamental Group of a Simplicial Space with trivial 0-skeleton

I am trying to find a reference or an explanation for the result that if $X_\bullet$ is a simplicial space with $X_0$ a point then $\pi_1(|X_\bullet|)$ is the free group on $\pi_0(X_1)$ $/ \sim$, ...
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Are the n-tuples of constructible numbers a model for Euclidean Geometry?

Constructible numers are an algebraic field which can be obtanined by a finite number of algebraic extensions of degree $2^k$ from $\mathbb{Q}$. I believe this is equivalent to define the set of ...
Davius's user avatar
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Geometric realization of a simplicial set depends functorially on the simplicial set

On Kerodon (subsection 1.1.8) the following definition is given for the geometric realization of a simplicial set: Let $S_{\bullet}$ be a simplicial set and let $Y$ be a topological space. We will ...
Steven's user avatar
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2 answers
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Colimit of simplicial set in bijection with path connected components of geometric realization

I want to show that if $X$ is a simplicial set we have a bijection $\text{colim}_{\Delta^{op}}X\cong \pi_0|X|$ with $|X|$ the geometric realization of $X$. So here $|X|=\coprod_{n\geq 0}X_n\times \...
raisinsec's user avatar
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How is the minimum embedding dimension of an abstract simplicial complex related to its simplices?

I was working on an example of an abstract simplicial complex and its geometric realization and I had a question. I think the question makes more sense when explained through the example. Some ...
Ethan's user avatar
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Can we ignore higher dimensional information when computing the geometric realisation of an $n$-dimensional simplicial set?

$\newcommand{\lan}{\operatorname{Lan}}\newcommand{\tr}{\operatorname{tr}}\newcommand{\H}{\mathsf{H}}\newcommand{\set}{\mathsf{Set}}\newcommand{\T}{\mathsf{Top}}\newcommand{\C}{\mathsf{C}}\newcommand{\...
FShrike's user avatar
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(Co)homology of partially ordered sets (posets)

I'm having trouble understanding some basic facts on algebraic topology. If I have a topological space $X$ and want to calculate the (co)homology of it, it seems to me to calculate singular (co)...
Júlio César M. Marques's user avatar
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166 views

Geometric Realization of Standard Combinatorial $n$-Simplex is Standard Topological $n$-Simplex

Let $\Delta$ be the simplex category with objects $[n]=\{0,...,n\}$, $n\geq 0$, and morphisms ordering preserving functions $[n]\rightarrow [m]$. Then let the standard categorical $n$-simplex be $Hom_{...
Jon Doe's user avatar
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Question on geometric realization of the torus.

Definition $:$ A semi-simplicial set is a sequence of discrete space $\{X_n\}_{n \geq 0},$ where $X_n$ is a discrete space consisting of $n$-simplices for $n \geq 0$ together with the face maps $d_i : ...
Anacardium's user avatar
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Finding the Canonical Polyhedron associated with a 3-connected simple graphs.

I am not a professional mathematician but I am a reasonably competent programmer and I am also no stranger mathematics, though I must say that my usual domain is closer to calculus and functions ...
urquiza's user avatar
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3 votes
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464 views

Understanding the geometric realization of a graph

I'm working through the beginnings of the graph theory necessary for pasting diagrams in $2$-categories using Johnson and Yau's 2-Dimensional Categories, and I'm not quite sure what the relation they'...
Alec Rhea's user avatar
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1 vote
1 answer
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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 $\|X_\bullet\|$ ...
Olivier Bégassat's user avatar
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Even solutions for a second order ODE

I need a check on the following exercise, especially a help in the second point, which I am not able to solve Consider the ode $$-u'' +u = u^3$$ Given that for all $t \in \mathbb{R}$: $$\frac{1}{2}|u'...
andereBen's user avatar
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7 votes
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Convex geometric realizability of abstract polyhedron with congruent isosceles obtuse triangular faces

Question below. Some background: Take an isosceles obtuse triangle of the form with $\alpha = \frac{n-1}{n}\pi$ for some $n \geq 3$ ($\beta=\frac{\pi}{2n}$) If you look at the class of convex ...
Simon Marynissen's user avatar
3 votes
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weak equivalence of the geometric realisation of a total singular complex and a topological space (from P.May concise course in Algebraic Topology)

In P.May's book "A concise course in Algebraic Topology", chapter 16, He establishes a weak equivalence between $\Gamma X = |S_*(X)|$ and $X$, where $X$ is a topological space, $S_*(X)$ is ...
진현희's user avatar
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1 answer
125 views

On the geometric realization of a finite abstract simplicial complex which is connected, orientable $3$-manifold without boundary

Let $\Delta$ be an abstract simplicial complex on finitely many vertices and $|\Delta|$ be it's geometric realization. (https://en.m.wikipedia.org/wiki/Abstract_simplicial_complex) If $|\Delta|$ is ...
uno's user avatar
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Simplicial set such that its geometric realization is a circle.

I got confused with the definition of a simplicial set and its geometric realization. Is it correct that if I take the following simplicial set then its geometric realization will be a circle? $X_0=\{...
whoami's user avatar
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Definition of a geometric realization of a simplicial space.

I am a little confused with the definition of a geometric realization. I follow "geometric realization of a simplicial topological spaces" link on ncatlab. It says that $|X|=\coprod_n X_n\times \Delta^...
whoami's user avatar
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2 votes
2 answers
262 views

On $n$-connected simplicial complex and vanishing of reduced homology

Let $X=|\Delta|$ be the geometric realization of an abstract simplicial complex $\Delta$. Let $k$ be a field. Assume that $X$ is path connected . Consider the following two conditions: (1) $\pi_i (X)...
user's user avatar
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1 vote
0 answers
274 views

(path)Connectedness of a simplicial complex Vs it's $1$-skeleton

Let $\Delta$ be an abstract simplicial complex https://en.m.wikipedia.org/wiki/Abstract_simplicial_complex and $|\Delta|$ be its geometric realization. Let $\Delta ^{(1)} :=\{F $ is a face of $\...
user's user avatar
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2 votes
1 answer
72 views

Embedding the $n$-crosspolytope as a $01$-polytope with edge length $\sqrt 2$

The regular $n$-crosspolytope is the $n$-dimensional generalization of the regular octahedron. I know that all of them can be realized as $01$-polytopes, that is, as a polytope with coordinates in $\{...
M. Winter's user avatar
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4 votes
1 answer
155 views

On a lemma regarding non-degenerate points of (geometric realizations of) simplicial sets

I'm currently trying to understand Lemma 14.2 of Peter May's Simplicial Objects in Algebraic Topology, which (paraphrasing) states that given a simplicial set $X$ and a point in the coproduct $(x,p) \...
qualcuno's user avatar
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4 votes
0 answers
54 views

How does a group action on a category pass to the geometric realization?

Let $Y_n$ be the category whose objects are pairs $(x,y)$, where $x$ belongs to the braid group $B_n$ and $y$ is a parenthesizing of the non-associative product of $n$ elements, for instance, $(x_1x_2)...
Javi's user avatar
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