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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Characterization of boundary in simplicial sets
I am reading kerodon, and I am stuck on exercise 1.1.2.8., which characterizes the simplicial maps of $\partial \Delta^{n}$ on a simplicial set $S_{\cdot}$ as lists $(\sigma_{1},...,\sigma_{n})$ of $n-...
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what is the simplicial surface of the word $ab^{-1}c^{-1}a^{-1}cb$
In order to study simplicial surfaces I saw the classification theorem and I'm trying to find a homomorphic surfaces for some words one of them is 1)$ab^{-1}c^{-1}a^{-1}cb$.
$a^{-1}$ is the opposite ...
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What does a "discrete $\infty$-category" mean?
Often, category theorists refer to a quasi-category as "discrete". I found only the following formal definition for this term, which I find quite puzzling:
A simplicial set $X$ is discrete ...
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Clarify on Kerodon 1.1.3.11
For any integer $n\ge 0$, let $[n]$ be the totally ordered set of the integers $k$ such that $0\leq k\leq n$, with the usual order relation.
Here are the last lines from the proof of Proposition 1.1.3....
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Definition of a contractible category
A simplicial set can be interpreted both as a generalized space as well as a generalized category. While reading about bisimplicial sets I came across the statement that the spatial $\Delta^{1}$ is ...
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First exercise on Kerodon
I'm having troubles with the exercise 1.1.1.10 on Kerodon. In particular, I don't understand how to prove the uniqueness of the sentence in italics; the paragraphs before it are mainly for context, so ...
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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 \...
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Identifying the delaunay simplex the point resides in without computing all the possible simplices
My question is whether it is possible to identify the delaunay simplex the point $\mathbf{x}_{\textrm{query}}$ resides in without pre-calculating all the possible simplices by triangulation?
Can we ...
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Eilenberg-Zilber's lemma existence
Let $X$ a simplicial set, $x\in X_m$ an $m$-simplex. We say that $x$ is degenerate if $\exists s:[m]\to[n]$ an epimorphism such that $n<m$ and $y\in X_n$ such that $X(s)(y)=x$.
Now I want to show ...
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Epimorphism in simplex category is split
Consider $\Delta$ the simplex category, with objects $[n]=\{0,\dots,n\}$ and morphisms $f:[n]\to [m]$ such that $i<j\implies f(i)\leq f(j)$ (my definition is with $i<j$). I have shown that ...
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Let K be a simplicial complex and σ a simplex of K. Show that the link lk(σ) is a subcomplex of K, if lk(σ) is not empty.
I wanna know if I choose the right way to do my proof? it's correct?
To show that the link $\textrm{Lk}(\sigma)$ is a subcomplex of $\mathcal{K}$, if $\textrm{Lk}(\sigma)$ is not empty, we need to ...
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$\mathfrak{h}(\mathcal{G}^{\Delta^1})$ is the category of commutative squares in $\mathfrak{h}\mathcal{G}$
This is related to the question Homotopy category of a symmetric monoidal $(\infty,1)$-category is symmetric monoidal.
Let us fix an $(\infty,0)$-category $\mathcal{G}$ (a Kan complex). We want to ...
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Is every $\Delta$-complex realizable as Simplicial complex, ie triagulazable?
In the following I will use definitions and
constructions of simplicial complexes and $\Delta$-complexes/sets from Greg Friedman's An elementary illustrated Introduction to simplicial sets.
I'm ...
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Understanding the Beck-Chevalley condition (II)
The older question here on the site asks about the intuitive meaning of the Beck-Chevalley condition. Accidentally one of the answers has caught my eye.
It made me wonder if it is possible to describe ...
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"Compatible /admissible" maps of $\Delta$-complexes
In the following I'm going to use definitions and
constructions of simplicial complexes and $\Delta$-complexes/sets from Greg Friedman's An elementary illustrated Introduction to simplicial sets.
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Where does one learn how to apply categorical algebra and higher abstractions to algebraic topology?
Tl;Dr: I know higher category theory and algebra is used ubiquitously in advanced algebraic topology. However, every time I ask someone, or try to find out, how one actually learns to apply the higher ...
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An infinity-category is a functor, but is it a category in some way and is the simplex notions so universal?
For some time I had the impression that infinity-categories are just a generalization of higher order categories: categories whose arrows have arrows among them, and they have further arrows etc. ...
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Geometric realization of a simplicial set is Hausdorff
I am reading the book May's Simplicial objects in algebraic topology and I am trying to understand the proof for Theorem 14.1 - the geometric realization of a simplicial set is a CW-complex. He says ...
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Proving that the geometric realisation of a minimal fibration is a Serre fibration - have I got the details right?
$\newcommand{\O}{\mathcal{O}}$In the book: "Simplicial Homotopy Theory", by Goerss-Jardine, they 'prove' that every minimal fibration $q:X\to Y$ of simplicial sets $X,Y$ has $|q|:|X|\to|Y|$ ...
- 26k
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Suspension functor of pro-spaces to pro-spectra preserves weak equivalences
Suppose we have the category $\mathrm{sSet}$ of simplicial sets and $\mathrm{sSpectra}$ of simplicial spectra. Then we have the functor $\Sigma^{\infty} \colon \mathrm{sSet} \to \mathrm{sSpectra}$, $X ...
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How can I show my multiplication for $\pi_n(\Omega X,\ast)$ is unital? Lemma $7.6$, Goerss-Jardine
I've come up with a candidate construction, however I'm stuck on one last little detail. I'm also bothered by an anxiety that my construction is not the 'right one' and that I've seriously ...
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Difficulty with lemma $7.4$ of Goerss-Jardine: a simplex $\alpha$ is nullhomotopic to $v$ iff. there is a simplex with boundary $(v,\cdots,v,\alpha)$
We are given a nonempty fibrant simplicial set $X$ (a Kan complex) and $v\in X_0$ is any vertex. We are $\alpha\in X_n$ and the map $\alpha:\Delta^n\to X$ is homotopic to $v:\Delta^n\overset{!}{\...
- 26k
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"Small" simplicial group equivalent to the circle
Let $G$ be a topological group, call $\mathcal{H}\in sGrp$ simplicial model of $G$ if there is a homomorphism of topological groups $|\mathcal{H}|\to G$ which is also a homotopy equivalence. In a ...
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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{\...
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Is there a 'coherence' theorem for simplicial sets? On uniquely lifting natural transformations of $n$-truncations to full natural transformations
$\newcommand{\set}{\mathsf{Set}}\newcommand{\op}{^{\mathsf{op}}}\newcommand{\sk}{\operatorname{sk}}\newcommand{\tr}{\operatorname{tr}}\newcommand{\cosk}{\operatorname{cosk}}\newcommand{\nat}{\mathsf{...
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Difficulty using the (co)limit formulae to construct the $n$-(co)skeleton left and right Kan extensions for truncated simplicial objects
Tl;Dr - I’m struggling to show that the $n$-skeleton is a Kan extension, from the basic limit formula (this should be possible, as it was “left to the reader” in my book). I’m also struggling to even ...
- 26k
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What is the takeaway of cofibrant generation?
I have recently begun reading about model categories. In particular, I have been using Balchin's A Handbook of Model Categories as a reference, and the following quote has been quite perplexing.
A ...
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Equivalent definitions of suspension of simplicial set.
For a pointed simplicial set $X$, the suspension $\Sigma X$ is calculated as the smash product $S^1\wedge X$, where $S^1=\Delta^1/\partial\Delta^1$. In the literature, there is another definition: $\...
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Confusion about Hypercoverings
I have several questions about concept of hypercovering
that should be seen as natural generalisation of the Čech nerve of a cover.
Let fix an open cover $\mathcal{U} \to X$ of a space $X$. What a
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Is Milnor's join the realization of a simplicial set?
I am reading the famous papaer Classifying spaces and spectral sequences by Segal and I am a little confused by something.
I am familiar with Milnor's join construction of classifying spaces. Let us ...
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Iterated colimits in $\infty$-categories
The Problem
Let $\mathcal{C}$ be an $\infty$-category and $K$ a simplicial set. I believe that the following statement is true:
The full subcategory $\operatorname{Fun}^c(K^\triangleright,\mathcal{C})...
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Is the inclusion of the "smooth singular Kan complex" a homotopy equivalence?
Let $X$ be a smooth manifold.
Let $L$ be the singular simplicial set of $X$. We know $L$ is a Kan complex.
Let $K \subset L$ be the "simplicial subset" consisting of only those singular ...
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Elementary properties of the "forgetful" functor from simplicial sets to semisimplicial sets?
Would anyone know a reference which explains whether the forgetful/restriction functor from simplicial to semisimplicial sets is, or fails to be: full, faithful, essentially surjective, etc?
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Is there a "cubical" version of quasicategories?
Definitions:
A Kan complex is a simplicial set having the horn-filling condition;
A quasicategory is a simplicial set having the inner-horn-filling condition.
My question is:
I was wondering if ...
- 2,307
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Would anyone know an example of a simplicial set having the "spine-lifting" property while not being a quasicategory?
Definition:
By a simplicial set we mean a Set-valued presheaf on the category $\mathbf{\Delta}$ of nonempty finite totally-ordered-sets with non-strict order-preserving functions between them. The ...
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Could anyone give an example of a quasicategory which does not have the "spine lifting" property?
Definition:
By a simplicial set we mean a Set-valued presheaf on the category $\mathbf{\Delta}$ of nonempty finite totally-ordered-sets with non-strict order-preserving functions between them. The ...
- 2,307
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Does the nerve of a category have the following stronger lifting property?
Definition:
For integer $k\geq0$ we let $\Delta^k := \mathrm{Hom}_{\text{FinitePosets(Nonstrict)}}(-\;,\;[k]=\{0,1,\ldots,k\})$ denote the usual simplicial-set model of the $k$-simplex.
For integer $...
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Is the homotopy category of the quasicategory of quasifunctors between 1-categories $C,D$ equal to the 1-category of 1-functors $C\to D$?
We use the ("geometric") model of $(\infty,1)$-categories as quasicategories.
Notation: For a 1-category $C$, let $\tilde{C}$ denote the $(\infty,1)$-category / quasicategory given by the ...
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Does the simplicial-set internal hom of quasicategories give a quasicategory?
The internal hom structure on simplicial sets is described here. This provides us with a functor $h : C^{\text{op}} \times C \rightarrow C$ where $C := \mathrm{Sets}^{\mathbf{\Delta}^{\text{op}}}$.
My ...
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Simplicial Manifold: Are the face maps submersions?
To complete the question in the title:
I'm currently working on simplicial manifolds, using the definition stated here: https://arxiv.org/pdf/2112.01417.pdf , page 4.
That is, the simplicial manifolds ...
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How to prove that a map is a Kan fibration
For simplicial sets $X$ and $Y$, let us denote by $\underline{\mathrm{Hom}}(X, Y)$ the simplicial set of morphisms $X \to Y$. If $p: X \to Z$ and $q: Y \to Z$ are morphisms of simplicial sets, let us ...
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How to construct a Segal category from a quasicategory?
The following are both models for $(\infty,1)$-categories:
quasicategories;
Segal categories.
Given the above, I was wondering how to set up an equivalence (isomorphism?) between quasicategories and ...
- 2,307
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How to prove that homology of simplicial sets is homotopy invariant?
It seems to be well known that taking the homology of a simplicial set is homotopy invariant, meaning that a homotopy of simplicial sets induces a chain homotopy of the corresponding chain complexes, ...
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clarification on why BG has no action
I got confused by the page (Are these two definitions of $EG$ equivalent?) about the following.
Doesn't this also define an action on BG?
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Are quasicategories (resp. kan complexes) (co)reflective in simplicial sets?
This is definitely well known to experts, but I'm struggling to find a reference.
It seems intuitive that we should be able to "complete" any simplicial set into a quasicategory or a kan ...
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Understanding the definition of a weak homotopy equivalence between simplicial sets, and Joyal equivalences between Kan complexes.
Let $f: X \longrightarrow Y$ be a map of simplicial sets.
Kerodon, definition 3.1.6.12, defines $f$ to be a weak homotopy equivalence if for every Kan complex $Z$, the map induced by precomposition $\...
- 466
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0
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47
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Calculating Colimits in The category of Simplicial Sets
I am struggling to understand how to take colimits in the category of simplicial sets. To make my notation clear, I will use $\delta_i : [n-1]\to[n]$ to denote the $i$-th face map (it picks out the ...
- 1,766
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Universal Property of The Simplex Category Δop
I tried formulating the universal property of the simplex category. There seemed to be two different options to choose for the universal property of $\Delta^{op}$. I'm having trouble making ...
- 1,766
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Calculating the limit over $\partial \Delta ^2$
Suppose we have the diagram of spectra $$A \to B \to C$$
Then taking the limit of this diagram is essentially taking the limit over the Nerve of the category $$\cdot \to \cdot \to \cdot$$
which I ...
- 6,068
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Simplicial maps defined on non degenerate simplices
I'm studying "Introduction to $\infty$-categories", by Markus Land, and I found myself in need of constructing simplicial maps from their values on non singular simplices.
This is what I'd ...
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