Mathematics

Questions tagged [geometric-realization]

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

Geometric realization is the operation that builds from a simplicial set $X$ a topological space $|X|$ obtained by interpreting each element in $X_n$ – each abstract $n$-simplex in $X$ – as one copy of the standard topological $n$-simplex $Δ^n_{Top}$ and then gluing together all these along their boundaries to a big topological space, using the information encoded in the face and degeneracy maps of X on how these simplices are supposed to be stuck together. It generalises the geometric realization of simplicial complexes.

27 questions
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Homotopy equivalence of realization of nerve of a cover.

I was reading the paper "Homology fibration and group completion theorem" by Segal and McDuff. During the proof of Proposition 6 (of local-to-global flavor) he states the existence of spaces $E_\mathfrak{B}$ , $|\mathfrak{B}|$ homotopy equivalent to…
Alessandro Fenu
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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 polyhedra such that each face is congruent to this…
Simon Marynissen
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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, $(x_1x_2)x_3$ ; and there is exactly one arrow between…
Javi
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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 $(x,p) \in \coprod_{n \geq 0} X_n \times |\Delta^n|$ ,…
qualcuno
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Realization of $(\mathbb{N}, | )$

Consider the poset $P$ of strictly positive integers, where $d$ is smaller than $m$ if $d$ divides $m$ . We can view this as a category, and then take the geometric realization, and obtain a space which might have some interesting homotopy. One…
DevVorb
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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 definitions before the example: An abstract n-simplex ,…
Ethan
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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{\psh}{\mathsf{Psh}}\newcommand{\op}{{^\mathsf{op}}}…
FShrike
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Basic question of Cartan decomposition of affine Grassmannian

Let $F=\mathbb{C}((t))$ and $O=\mathbb{C}[[t]]$ . Take $G=SL_n$ , the affine Grassmannian of $G$ , $Gr_G=G(K)/G(O)$ . Then we have Cartan decomposition: $$ Gr_G=\bigsqcup_{\lambda}G(O)t^{\lambda}G(O)/G(O), $$ where $\lambda$ runs over the dominant…
zhichengzhang
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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're using to quotient by for their geometric…
Alec Rhea
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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 the total singular complex, and $||$ denoting the geometric…
진현희
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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 $\pi_0(X_1)$ $/ \sim$ , where $\sim$ is the relation that $\partial_1(x) =…
treutm14
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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) $\pi_i (X)=0, \forall 1\le i\le n$ (2) $\tilde H_i(X,k)=0,…
user
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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 $\{0,1\}^d$ for some $d\ge n$ . Question: Which…
M. Winter
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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 Waldhausen localization or fibration theorem. I was…
DevVorb
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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 constructible numbers $\mathbb{K}$ as: $$\mathbb{K} =…
Davius
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