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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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How to understand May's proof that counit map is a weak equivalence?

A similar question was asked about 4 years ago here, but received no answers, so I hope it is appropriate to post a new question. I am trying to read the singular homology section in May's Concise ...
Christian's user avatar
3 votes
1 answer
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Inclusion of a certain full subcategory of the category of elements of a simplicial set is a final functor

For a simplicial set $X$, let $\text{el X}$ be its category of elements, whose objects are pairs $([n], x\in X_n)$. Let $\text{(el X)}_{nd}$ denote the full subcategory comprising objects $([m], y\in ...
User1234's user avatar
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Simplicial $\pi_0$ as homotopy classes $\Delta^0 \to X$ using that $\pi_0$ is left-Quillen

In Higher Categories and Homotopical Algebra, Cisinksi defines the connected component functor as $$\pi_0 \colon \mathsf{sSet} \to \mathsf{Set}, \qquad \pi_0(X) = \mathsf{colim}_{n} X_n,$$ the left ...
qualcuno's user avatar
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Does geometric realization commute with finite limits?

I am trying to find out if geometric realizations i.e. the functor $|-|\colon\text{sSet}\to \text{Top}$ commutes with finite limits. In the following post the user claim that this is well known: https:...
Womm's user avatar
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Definition for map $T$ is unclear? - Hatcher pg. 122

This comes from Hatcher's Algebraic Topology textbook (pg. 122). See photo below. (Note: $b_\lambda([w_0,...,w_n])=[b_\lambda,w_0,...,w_n]$ where $b_\lambda$ defines the barycenter of singular $n$-...
JAG131's user avatar
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50 views

Geometric intuition behind induced homology homomorphisms

This is certainly overkill...but at the moment, I'm trying to strengthen my geometric understanding towards homology groups. The current trouble I've been having is in understanding the geometric ...
JAG131's user avatar
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Linearity assumption on $f_{\#}$? (Hatcher, pg. 110-111)

This is coming from pg. 110-111 of Hatcher's Algebraic Topology textbook. (See below photos.) At the moment, it is unclear why we are able to simply construct $f_{\#}$ such that linearity is ...
JAG131's user avatar
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Reference request for realizing a simplicial set as the homotopy colimit of its simplices

I know that $$X\simeq hocolim_{Simp(X)}\Delta^n,$$ where $Simp(X)$ is the category of simplices of $X$, I know this for example because of proposition 7.5 of the nLab's page for homotopy limits. ...
DevVorb's user avatar
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2 votes
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The geometric realization of a simplicial set does not determine it

I read in Gallauer's notes on infinity categories, in "Warning 1.9." without proof, that the geometric realization of a simplicial set does not determine it and wanted to make sure I ...
kindasorta's user avatar
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3 answers
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Showing that the inclusion of the 2-skeleton of a simplicial complex induces an isomorphism on fundamental groups

Suppose that $K$ is a simplicial complex and let $K_{(2)}$ denote its 2-skeleton. Show that that for all $x \in |K_{(2)}|$ the map $i_*: \pi_1(|K_{(2)}|,x) \to \pi_2(|K|,x)$ induces an isomorphism of ...
John Robertson's user avatar
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Cofibrations and cofibrant objects in a simplicial abelian category

For context, I am reading chapter III.2 of Goerss and Jardine's book on simplicial homotopy theory, but I am not an expert in model categories. In that chapter, they introduce the model structure on ...
SetR's user avatar
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1 answer
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Morphism inducing isomorphism in homotopy category is weak equivalence

In Goerss-Jardine's book "Simplicial Homotopy Theory" they prove in Lemma 4.1 of chapter II that for a simplicial model category $\mathcal{C}$ the statement $$\text{ If }f:X\to Y \text{ in } ...
Fabio Neugebauer's user avatar
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Describe all cones in $\mathbb R^n$ that are simplicial and cosimplicial.

For any finite set of vectors $(v_1, \ldots, v_k \in \mathbb{R}^n)$, let's define: $Cone(v_1,…,v_k)= \{λ_1v_1+…+λ_kv_k,∣,λ_1,…,λ_k \ge 0\}$ and $Poly(v_1,…,v_k)= \{ x \in \mathbb{R}^n ∣(v_i, x) \ge 0, ...
PikaPika's user avatar
1 vote
1 answer
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Weak equivalence of filtered Colimit

Given a model category $C$, I have two functors $F,G:\mathbb{N}\rightarrow C$, where see $\mathbb{N}$ as sequence category. Question: Given a natural transformation $J:F\rightarrow G$ and suppose the $...
Mukilraj K's user avatar
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Realization of singular sets preserving homeomorphism type

Let $X$ be a topological space. The simplicial set $\operatorname{sing}(X)$ has as its $n$ simplicies all singular $n$-simplices $\Delta^n \to X$. The realization of the singular simplicial set is ...
Dennis's user avatar
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2 votes
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Segal subdivision

I call Segal subdivision the endofunctor of simplicial objects in a category $\mathcal{C}$ induced by the doubling endofunctor of $\Delta^{op}$ sending $x_0<\cdots<x_n$ to $x_0<\cdots <x_n&...
DevVorb's user avatar
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1 vote
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Triangulating Product of Simplicial Complexes

I am currently working on a problem for which I believe the following result is crucial. The result of this problem was discussed in this post. Product of simplicial complexes? However it is not ...
slowlight's user avatar
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Showing that a simplicial map in McCarthy's paper on additivity is a homotopy equivalence

I am reading Prof. McCarthy's paper proving additivity (1992). In this paper he proves a version of Quillen's theorem A for simplicial sets (this is the unique result labeled as proposition of the ...
DevVorb's user avatar
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6 votes
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129 views

Plus construction on Simplicial Sets?

Write $\mathsf{sSet}$ for the category of simplicial sets and $\mathsf{Top}$ for the category of topological spaces. I would like to know if there a functor $\mathsf{sSet}\to\mathsf{sSet}$ that ...
wind's user avatar
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1 answer
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Homotopy orbits

Let $\mathcal{C}$ be an $\infty$-category, and let $G$ be a group. Denote by $\mathcal{C}^{\text{BG}}$ the functor $\infty$-category $\text{Fun}(\text{BG}\longrightarrow \mathcal{C})$. The homotopy ...
kindasorta's user avatar
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3 answers
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Fundamental group of simplicial space

This question asks the same as mine, but it was unsuccessful in getting an answer, so I try again. For context I am reading Weibel's K-book and am struggling with proposition 8.4 which computes $K_0$...
DevVorb's user avatar
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1 vote
1 answer
70 views

Morphism $\infty$-categories

Let $\mathcal{C}$ be an $\infty$-category, and let $X, Y$ be a pair of its objects. It is said that $\text{Mor}_{\mathcal{C}}(X,Y)$ has again the structure of an $\infty$-category. What is this ...
kindasorta's user avatar
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1 answer
37 views

Objects in an $\infty$-category

I tried reading about the definition of a pre-additive $\infty$-category, there it is said that a pointed $\infty$-category is pre-additive if all finite products and co-products exist, and the ...
kindasorta's user avatar
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0 votes
1 answer
78 views

How to show that the computed kernel is isomorphic to $\mathbb{Z}^2$ in a simplicial complex?

I am working on calculating the first homology group $H_1(X)$ for a simplicial complex X, and I have found the kernel of the boundary operator $\partial_1: C_1 \rightarrow C_0$ (chain complex) to be ...
TheoMarsy's user avatar
2 votes
0 answers
61 views

Discrete cocycle datum of a principal $G$-bundle

Let $X$ be the topological realization of a finite simplicial complex, $G$ a finite group and $p: P \to X$ a principal $G$-bundle. Let's recall the standard fact that more generally for any numerable ...
JackYo's user avatar
  • 179
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2 answers
57 views

Does geometric realization of a Δ-set collapse connected simplices to a single point?

According to Wikipedia, the geometric realization of a Δ-set is defined as the following quotient space: Each Δ-set has a corresponding geometric realization, defined as $|S|=\left(\coprod _{{n=0}}^{{...
GolDDranks's user avatar
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The formal definition of a Δ-set doesn't guarantee orientation, and its implications for gluing?

It seems to be that the formal definition of a Δ-set doesn't forbid identifying different faces of a simplex, that is, face maps $d_i$ and $d_j$, $i \ne j$ may map element $a : S_n$ to a same element $...
GolDDranks's user avatar
3 votes
3 answers
345 views

Definition of a Δ-set?

According to Wikipedia, Formally, a Δ-set is a sequence of sets $\{S_{n}\}_{{n=0}}^{{\infty}}$ together with maps ${\displaystyle d_{i}\colon S_{n+1}\rightarrow S_{n}}$ with ${\displaystyle i=0,1,\...
GolDDranks's user avatar
1 vote
1 answer
69 views

Proof criticize: $Sing(X)$ is fibrant

I am trying to prove the question in the title where definitions are given below. Definition (fibration). A map $f:X\to Y$ in the category of simplicial sets is a fibration if it satisfies right ...
Mizutsuki's user avatar
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3 votes
1 answer
72 views

Question about automorphisms of simplicial category $\mathbf{\Delta}$

Let $\rho: \mathbf{\Delta} \to \mathbf{\Delta}$ be the functor defined as the identity on objects and by the formula $$ \rho(f)(i)=n-f(m-i) $$ for any map $f:[m] \to [n]$, with $0 \leq i \leq m$. Then ...
Siyuan Yin's user avatar
1 vote
1 answer
41 views

Difference between the characterizations of an $\infty$-category and nerves

In Kerodon, Propposition 1.2.4.1 Proves that a simplicial set $S_\bullet$ is isomorphic to a nerve of an ordinary category if and only if every horn $\Lambda^n_i$ in $S_\bullet$, with $0<i<n$, ...
14159's user avatar
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0 votes
1 answer
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Connected components of a simplicial set and homology

Given a simplicial set $X$, we define $\pi_0(X)$ to be the coequalizer of $d_0, d_1: X_1 \to X_0$. I have to show that $\pi_0(X) \cong \pi_0( | X| )$. Does this mean that we are taking $X_0/ \sim$, ...
idontknow's user avatar
0 votes
1 answer
40 views

Homotopy equivalent Kan complexes have homotopy equivalent "path spaces"

$\DeclareMathOperator{\Fun}{Fun}$ Let $f\colon X\to Y$ be a homotopy equivalence of Kan complexes (in the sense of Definition 3.1.6.1. in Kerodon), with a homotopy inverse $g\colon Y\to X$. I want to ...
14159's user avatar
  • 945
1 vote
1 answer
53 views

Nondegenerate simplices of simplicial torus and homology

I would like to use the normalized chain complex approach to compute the homology of $\Delta[1] / \partial \Delta[1] \times \Delta[1] / \partial \Delta[1]$. In order to do that, I first have to ...
idontknow's user avatar
4 votes
2 answers
130 views

Homology computations of $\partial \Delta[n]$ using normalized chain complexes

I have to compute the homology group of the abstract simplicial set $\partial \Delta[n]$ using normalized chain complexes. I know that the nondegenerate $k$-simplices of $\partial \Delta[n]$ are $\...
idontknow's user avatar
1 vote
2 answers
126 views

Category whose classifying space is $S^n$

Suppose one wants a finite category $C_n$ (finite sets of objects and morphisms), for which $\left\vert NC_n \right\vert\simeq S^n$. I think one such $C_n$ can be built as follows : Take $\partial \...
t_kln's user avatar
  • 1,048
6 votes
1 answer
206 views

Geometric realization of the nerve of a category and group completion

I am trying to solve an exercise that has as a goal the description of the group completion of a monoid. In order to do that, we start with the definition of the nerve of a small category $\mathcal{C}$...
Alice in Wonderland's user avatar
5 votes
1 answer
324 views

Why does the bar construction model the classifying space in both topology and AG?

For a topological group $G$, we can construct the classifying space $BG$ as the geometric realization of the nerve of $G$. I have seen a very similar assertion in the context of algebraic geometry: ...
Hyunbok Wi's user avatar
2 votes
1 answer
60 views

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 $\pi_0(X_1)$ $/ \sim$, ...
treutm14's user avatar
3 votes
2 answers
197 views

Geometric realization of simplicial sets via nondegenerate simplices

I have just started studying simplicial sets, and I was given the definition of geometric realization of a simplicial set X as $$|X| = (\coprod_{n \in \mathbb{N}} X_n \times \Delta_n) / \sim$$ with $(...
Alice in Wonderland's user avatar
2 votes
0 answers
29 views

Is the Moore filler of a degenerate simplex itself?

In the proof that a simplicial group G is always Kan (see e.g. https://ncatlab.org/nlab/show/simplicial+group#AsKanComplexes ), you can always fill a $(m, j)$-horn $\lambda$ to a $m$-simplex $x$ with ...
Chenchang Zhu's user avatar
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0 answers
7 views

What is (the current prefered definition) of a cleavage of a TCP = twisted cartesan Product?

Cleavage of a TCP refers to unique lifting of fillers of simplicial $k$-horns in an $n$-simplex in the base of a TCP. Two definitions appear to disagree: $k < n$ versus $k \leq n$
jim stasheff's user avatar
0 votes
1 answer
25 views

Simplicial manifolds which do not satisfy Kan condition locally?

Similar to Kan condition for simplicial sets, there are also Kan condition for simplicial manifolds, that is, we ask the horn projection $p^k_j: X_k \to Hom(\Lambda[k,j], X)$ to be a surjective ...
Chenchang Zhu's user avatar
2 votes
1 answer
77 views

Is pullback along a simplex really a fibre?

I know that the question could be posed for any presheaf category, but simplicial sets are the only one having a name for natural transformations from representables, so I'll stick with that. So, go ...
whatisandwhatshouldneverbe's user avatar
0 votes
0 answers
81 views

Question about construction of cup-$i$ products (Mosher & Tangora)

I'm currently reading about the construction of Steenrod squares in Mosher & Tangora, which starts with defining the cup-$i$ product. In the excerpt below, $W$ is a chain complex with $$W_i = \...
Hrhm's user avatar
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2 votes
1 answer
59 views

Morphisms from simplicial sets into groupoids are determined by what?

For this question, one can assume that the simplicial sets come from oriented simplicial complexes (you can assume the orientation induces an orientation on the geometric realization). A morphism from ...
JLA's user avatar
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0 votes
0 answers
15 views

How the figures and vertices of simplicial complex are related to Hom([m],[n]) elements of standard n-simplex (trying to grasp Kan complex in it)?

Kan complex has nice pictorial interpretation in the simplicial complex, from the other side there is standard $n$-simplex. I am trying to understand how simplicial complex (and Kan horn in it and Kan ...
TomR's user avatar
  • 1,323
0 votes
1 answer
61 views

Definition of intersection of simplices

A singular $n$-simplex $\alpha: \Delta^n\rightarrow \Delta^p$ in $\Delta^p$ is called affine if $\alpha(\sum_{i=0}^n t_i e_i)= \sum_{i=0}^n t_i\alpha(e_i)$ holds for all $t_i$ with $\sum_{i=0}^n t_i=1$...
Margaret's user avatar
  • 1,777
2 votes
0 answers
73 views

Terminology for "transposition" of monomorphism to epimorphism in simplex category?

Recall that the simplex category $\Delta$ is dual to the category of intervals $\mathbb{I}$. By $\Delta$ I mean the category of finite ordinals $\mathbf{n} \in \omega$ with monotone functions between ...
Brendan Murphy's user avatar
0 votes
0 answers
17 views

Spectral sequence for a truncated semi cosimplicial space

Consider the simplicial indexing category $\Delta$. Now, let's denote the subcategory consisting of injections as $\Delta_{inj}$. When we're dealing with a cosimplicial space, which is essentially a ...
happymath's user avatar
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