# 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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### Are degeneracy maps always injective?

Are the degeneracy maps of a simplicial set always injective? I wonder, because I'd naively think that it wouldn't make sense to have two $n$-simplices which yield the same degenerate $n+1$-simplex. ...
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### The simplex category as a "free" category

Consider the simplex category $\Delta$. Its objects are the linearly ordered sets of the form $[n]=\{0<1<\dots<n\}$. Its morphisms are nondecreasing functions. There are two special classes ...
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### Intuition degeneracy maps

I'm trying to get an intuitive understanding for the notion of a simplicial set. Roughly, a simplicial set consists of a set $S_n$ of $n$-simplices for each non-negative $n$, and families of face maps ...
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### $n$-Simplices of Fiber product of Simplicial sets

Let $X,Y Z$ be simplicial sets with two maps $a: X \to Z, b: Y \to Z$ and I would like discuss how the $n$-simplices of the new simplicial set $X \times_Z Y$. I found only a construction for direct ...
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### k-point suspension of spheres

In "Complexes of Directed Trees" Kozlov defines the $k$-point suspension of a simplicial complex $X$ as $$susp_k(X) = \{k \text{ distinct points}\}*X,$$ were $*$ denotes the join of ...
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### Understand the internal hom simplicial set

I am currently reading the paper "a short course on $\infty$-categories" by M.Groth. The Theorem 1.18 in this paper states: A simplicial set $X$ is an $\infty$-category if and only if the ...
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### Every CW complex is homotopically equivalent to a simplicial complex

I saw a claim in an old version of the book of Fuchs and Fomenko saying that every CW complex is homotopically equivalent to a simplicial complex. I would like to see a proof of this claim. Hopefully ...
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### Is $\oplus$ the only monoidal structure on the simplex category?

Simplicial sets are presheaves on the simplex category $\Delta$, while augmented simplicial sets are presheaves on $\Delta_+$, the augmented simplex category. Because Day convolution allows us to ...
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### On the lifting property of canonical map between inner Hom

This question arises from the lecture notes. The notation $\perp$ means the thing on the left has the left lifting property w.r.t the thing on the right and, at the same time the right has right ...
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### How do you show that Lurie's straightening preserves colimits?

I am struggling to show that the straightening construction (2.2.1 in Higher Topos Theory) preserves colimits. More specifically let $M_X:=\mathfrak{C}X^\triangleright\sqcup_{\mathfrak{C}X}C^{op}$ for ...
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### Homotopy groups of simplicial sets commute with filtered colimits

Apparently it is well-known that homotopy groups of simplicial sets commute with filtered colimits. This is stated for instance in this MathOverflow question, and I've found a few papers that assume ...
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### Proof of Homotopy Addition Lemma

Let $X$ be a simplicial set. I'm looking for proof that $[d_0 y] - [d_1 y] + \ldots + (-1)^{n+1}[d_{n+1} y] \in \pi_n(X,x)$ is homotopic to 0, where $y \in X_{n+1}$ and $d_i$ are face maps. Thanks for ...
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### Barycentric subdivision of a simplex is a geometric simplicial complex

My question is part of a question which has been asked a few years ago here: Barycentric subdivisions of simplices yield a simplicial complex, but has not been answered to my satisfaction: The ...
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### Degenerate element in simplicial homotopy group

Let $A$ be a simplicial abelian group. Consider $\pi_n(UA, 0)$, where $U$ is the forgetful functor from Simplicial Abelian group to Simplicial Set. Let $x \in UA$ be a sum of degenerate element of $A$....
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### Question on cosimplicial objects

I am studying Descent theory and for this, I need some knowledge of cosimplicial objects. In the section 14.26.9 of the stacks projects, I could not find where the assumption of section is used. I ...
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### Homotopy on abstract simplicial complexes vs simplicial sets

Suppose $\Sigma$ is an abstract simplicial complex, suppose the vertices are ordered, and let $\tilde{\Sigma}$ be the corresponding simplicial set, where the set of $k$-simplices of $\tilde{\Sigma}$ ...
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### Homotopy Limit is the Limit in the Homotopy Category

I am trying to understand the homotopy limit. This question naturally appears to my mind. Let $I$ be a small category and $\mathcal{X}$ is an $I$-diagram of simplicial sets. There is a functor from ...
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### Weak Homotopy Equivalence in Simplicial Sets

Call a map $X \to Y$ of simplicial sets a weak Homotopy equivalence if it induces a bijection $\pi_0(\text{Map}(Y, Z)) \to \pi_0(\text{Map}(X, Z))$ for every Kan complex $Z$. Can one drop the $\pi_0$ ...
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### Moore Complex is quotient of alternating face map complex by the degenerate complex

I'm trying to understand the following lemma from Weiebl. https://imgur.com/a/M87KCbv (found from here: https://people.math.rochester.edu/faculty/doug//otherpapers/weibel-hom.pdf) I will list the ...
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### What is the motivation for defining and working with the simplex category?

The simplex category $\mathbf{\Delta}$ is, for our purposes, the category whose objects are $[n]=\{0,1, \dots , n-1, n\}$ for each $n = 0,1,2 \dots$, and whose morphisms are (all) order-preserving ...
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### Decomposing an order-preserving injection between finite sets as a product of coface maps

Write $\mathbf{n}:=\{0<1<\ldots<n\}$. Let $1_{\mathbf{n}}\ne f:\mathbf{n}\rightarrow\mathbf{m}$ be an order-preserving injection. Set $\mathbf{m}-f(\mathbf{n}):=\{i_k<\ldots<i_1\}$. ...
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### Join/slice adjunction and lifting problems

Let $C \in sSet$ be a quasicategory. $*$ denote the join of simplicial sets. Then consider $- * X \colon sSet \to sSet_{X/}$, where $sSet_{X/}$ is the slice category of simplicial set under $X$, with ...
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### Homotopy group of geometric realization is isomorphic to the original homotopy group.

It is a fact that if you take a Kan complex $X$, its homotopy group is isomorphic to the homotopy group of $\left|X \right|$, the geometric realization of $X$ as topological space. I'm looking for ...
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### Hypercovers via Resolution of Singularity

I'm trying to understand the proof of Thm 4.16 in B. Conrad's notes on Cohomological Descent. A special case of this theorem is the following form: If $k$ is a field of characteristic zero and $S$ is ...
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### Is the composition of hom-spaces entirely determined by the tensor in a simplicial model category?

Let $\mathcal{C}$ be a simplicial model category, as defined in https://ncatlab.org/nlab/show/simplicial+model+category. We can prove that the $\mathrm{Hom}$ functor is determined by the tensor, ...
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### Connected Components of the Homotopy Pullback

Let $K$, $K'$ and $L$ be simplicial sets and be a homotopy pullback square. Question: I think that in this situation, the natural map $\pi_0 (L') \rightarrow \pi_0(K') \times_{\pi_0(K)} \pi_0(L)$ is ...
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### Kerodon 2.4.4.10.: pushouts of graphs associated to a pushout of simplicial sets

I don't understand a nuance in Theorem 2.4.4.10. of Kerodon. To not burden the question with exposition, I will only provide information for the relevant part. Fix a natural number $m$. For a ...
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### Simplicial orbit realization functor.

In their paper, Singular functors and realization functors, Dwyer and Kan define a "realization functor" $\mathbf{O}\otimes - \colon \mathbf{S}^{\mathbf{O}^{op}}\to \mathbf M$ for a ...
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### Simplicial set map from $\Delta[n]$ to $K$ induced by an $n$-simplex $\sigma \in K_n.$

I am reading "rational homotopy theory" by Yves Felix, Stephen Halperin and Jean-Claude Thomas on simplicial sets and simplicial cochain algebras. There is a lemma 10.3 page 118 which states ...
I'm trying to understand the notion of a Čech Nerve in simplicial presheaves. Suppose we have a site $C$, and we consider its category of simplicial presheaves $\mathsf{sPshv}(C)$. This is a ...