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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1answer
56 views

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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1answer
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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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1answer
13 views

$n$-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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$\infty$-groupoids = Kan complexes

One direction is clear, a Kan complex has all its morphisms invertible hence it is an $\infty$-groupoid. Is it possible to show the opposite as a Corollary of the following theorem? Theorem. Let $p:X\...
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Bijection induced by Hom and inner Hom

For an $\infty$-category $X$ and a simplicial set $A\in sSet$. And for a category $\mathcal{C}$ we define $\mathcal{C}^\simeq$ whose objects are those of $\mathcal{C}$ and morphisms contain only ...
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1answer
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Common subdivision of two simplicial subcomplexes (on the way to topological invariance of simplicial homology)

In Munkres' Elements of Algebraic Topology Chapter 18, he aims to show that given a continuous map $h:|K|\to |L|$, there is a well-defined map $h_*:H_p(K)\to H_p(L)$, given by $f_*\circ(g_*)^{-1}$, ...
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stable homotopy category Ho(Spectra) of category of spectra

The category of spectra over CW-complexes has as objects sequences $E:= \{E_n \}_{n \in N}$ of CW complexes $E_i$ together with structure maps $S^1 \wedge E_n \to E_{n+1}$. The morphisms $f: E \to F$ ...
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26 views

Categories as simplicial object in Set VS simplicial object in Cat

It is well known that any category allow for the construction of a functor from the simplex 1-category $\Delta_0$. All of the axioms of a category translate simply in the functoriality requirements. ...
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1answer
32 views

internal hom between nerve of categories is nerve of functor category

The following statements are true:\ if $A,B$ are small categories, then: $\underline{Hom}(N(A),N(B))= N(Fun(A,B))$ if $A\in sSet$ and $B$ a small category, then $\underline{Hom}(A,N(B))\cong N(Fun(\...
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1answer
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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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1answer
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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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1answer
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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 ...
3
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1answer
89 views

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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1answer
41 views

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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0answers
31 views

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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51 views

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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0answers
97 views

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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30 views

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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1answer
69 views

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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0answers
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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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1answer
63 views

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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0answers
75 views

Question about Remark 2.4.1.9 in Higher Topos Theory

In Remark 2.4.1.9 in Lurie's Higher Topos Theory, he asserts that if $p:X\to S$ is an inner fibration, x a vertex of X, and $\overline{f}: \overline{x}'\to p(x)$ an edge of S ending at $p(x)$, then $...
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Can I have some examples regarding simplicial commutative rings in derived geometry?

Like how algebraic geometry work with SCR's, computed cotangent complexes, smooth and etale morphisms, etc. Also how does the intuition that higher homotopy groups are higher infinitesimals work?
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How does the graded ring structure of simplicial commutative ring work?

As I understand, for $A$ a simplicial commutative ring, by Dold-Kan, an element of $\pi_i(A)$ should consist of a "center piece" that is $\in A_i$ and $0$ for all things on the boundary, ...
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Pullback reflects inner fibration

An inner fibration is a map have right lifting property against inner horn inclusions $\wedge^n_i \to \triangle$, $0 < i < n$. The problem is the following. Suppose we have the following ...
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0answers
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Prove post-composition map is isomorphism.

Let $f \colon X \to NC$ be a natural transformation. ($C$ a category and $N$ is nerve functor from Cat to sSet) Consider post-composition map $Hom(\wedge^n_1, X) \to Hom(\wedge^n_1, NC)$. Assume that ...
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0answers
37 views

Nerve of functor between small categories is an inner Kan fibration?

Let's say if $u:C\to D$ is a functor between small categories. I am wondering why the nerve $$N(c):N(C)\to N(D)$$ becomes an inner Kan fibration? An inner Kan fibration is a morphism with the right ...
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0answers
26 views

Simpliciality of projective model structure on simplicial cofibrantly generated model categories

In Model Categories and Their Localizations, Definition 11.7.2, for $M$ a simplicial cofibrantly generated model category, $C$ a small category, Hirschhorn gives the following simplicial structure on $...
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0answers
36 views

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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26 views

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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0answers
34 views

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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1answer
38 views

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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Simplicial structure on chain complexes

It is known that the homotopy category of chain complexes has the right action by the homotopy category of simplicial sets. For any simplicial sets, $X$, one can define the associated chain complex $...
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0answers
21 views

Meaning of simplicial homotopies outside of SSet

If $X, Y$ are simplicial sets, and $f, g: X\to Y$ are two simplicial maps between them, we can define a homotopy of simplicial sets $h: X\times I\to Y$ or, equivalently, $h': X\to Y^I$. The latter ...
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41 views

Boundary of a simplicial set

The standard $n$-simplex is defined as the simplicial set $\Delta^n$ with $\Delta^n_m = \text{Hom}_\Delta([m],[n])$. The boundary of $\Delta^n$ can be defined as the simplicial set $\partial \Delta^n$ ...
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Homotopy equivalence of simplicial sets

From Lurie's Kerodon Remark 3.1.6.6 see : https://kerodon.net/tag/00U5. It is said that a morphism of simplicial set $f : X \rightarrow Y$ is a homotopy equivalence if and only if for every ...
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0answers
39 views

When does a hypercover imply hyperdescent?

Suppose we have a site $\mathcal{C}$ and $S$ is an object of $\mathcal{C}$. Let $F$ be a sheaf of abelian groups on $\mathcal{C}$ and let $S_\bullet$ be a simplicial hypercover of $S$. My question is: ...
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43 views

Homology of a Delta set and singular homology of its geometric realization

In Greg Friedman's article "An elementary illustrated introduction to simplicial sets" (2016), there is the following definition: Given such a Delta set $X$, we can form a chain complex $C_\...
3
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1answer
52 views

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 ...
2
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1answer
43 views

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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1answer
39 views

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 ...
5
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0answers
62 views

Cech Nerve Good Cover

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 ...
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0answers
91 views

Simplicial schemes vs simplicial presheaves

My question is not very precise, I apologize in advance. Essentially I am wondering in the context of "homotopy theory for schemes, in the broad sense, be it in motivic homotopy theory or $\...

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