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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Cocartesian edge gives contractible choice of filling in commutative diagram

This appears on p. 192 of Land's Introduction to Infinity-Categories which is the first page of his section on Straightening-Unstraightening. Let $p:\mathscr{E} \to \mathscr{C}$ be a cocartesian ...
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"Free" resolution of algebra

Given a ring $R$, we have an adjunction of the forgetful functor $CAlg_R\to Set$ and the free algebra functor $Set\to CAlg_R$, giving us an endofunctor $$T:CAlg_R\to CAlg_R, S\mapsto R[S].$$ In ...
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2 answers
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Iterated homotopy pullbacks

I would like a reference for the following fact, which I believe to be true. Consider the simplicial model category of Kan complexes with the Quillen model structure, and suppose given a commutative ...
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HA, 1.1.1.7, Lurie

In this remark, Lurie states that applying proposition HTT 4.3.2.15 twice, we deduce that $\theta$ is a kan fibration. How is this assertion deduced?
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The value of the left adjoint of the diagonal functor $\delta^*:\mathsf{bisSet}\to\mathsf{sSet}$ at $\Delta^m\boxtimes \Delta^n$

I'm currently reading the proof of Theorem 5.5.7 in Cisinski's book Higher Categories and Homotopical Algebra. There's one detail I don't understand, and I need someone's help. Let us introduce some ...
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1 answer
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Nerve of category is a functor

Let $\mathcal{C}$ be a category. We define a mapping $M:\Delta^{\text{op}}\rightarrow\textbf{Set}$ as follows. Write $M_0:=\text{Ob}(\mathcal{C})$ and for $n\geq 1$ let $M_n$ be the collection of $n$ ...
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Slice and Internal Hom; On the Definition of a Map $\underline{\operatorname{Hom}}(A,X/x)\to \underline{\operatorname{Hom}}(A,X)/x$

Recall that the join of two simplicial sets $X,Y$ are defined by $$(X\ast Y)_n=X_n\amalg X_{n-1}\times Y_0\amalg\cdots\amalg X_{0}\times Y_{n-1}\amalg Y_n,$$ with face and degeneracy maps defined ...
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Image, union and intersections in category of simplicial sets

Let $\Delta$ denote the simplicial category and $\textbf{sSet}$ the category $\text{Func}(\Delta^{\text{op}},\textbf{Set})$. If $X$ is a simplicial set, a simplicial subset of $X$ is a simplicial set $...
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Coequaliser of join inclusions of simplicial sets?

In simplicial sets, is there a simple description of the coequaliser of $$X \rightrightarrows X \star X$$ where the arrows are the left and right inclusions into the join?
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How is the internal hom in $\mathsf{sSet}/\mathcal C$ defined?

Fix a simplicial set $\mathcal C$. We have a category of simplicial sets over $\mathcal C$, denoted by $\mathsf{sSet}/\mathcal C$. This is supposed to have an internal hom, denoted $\underline{\mathrm{...
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Motivation for Quasicategories

Recall that a simplicial set $X$ is called a quasicategory if for every $0<k<n$, every map $\Lambda^n_k\to X$ admits an extension to a map $\Delta^n\to X$. It is good to have a concise ...
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Interchange map in simplicial sets is a monomorphism?

In the category of simplicial sets there is an 'interchange' map $$h : (A \times B) * (C \times D) \to (A * C) \times (B * D)$$ given by $$h = (\pi_A * \pi_C, \pi_B * \pi_D)$$ where $\times$ is the ...
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Face maps of simplicial replacement of diagrams

I am trying to wrap my head around the construction of the simplicial replacement of a diagram, e.g. following Rodríguez-González (Realizable Homotopy Colimits, 1.5, https://arxiv.org/pdf/1104.0646....
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Every simplicial presheaf is a colimit of objects $U\times \Delta^n$

Let $\mathcal C$ be a small category, the category of simplicial presheaf $sPre$ over $\mathcal C$ is the functor category $\mathcal C^{op} \rightarrow sSet$. Or equivalently simplicial objects in $\...
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How to show the exactness of the simplicial complex of free abelian groups generated by configurations of a finite set?

Given a finite set $X$ with $|X|=N>2$, we can construct a simplicial free abelian group $C_*(X)$ (which is a chain complex) defined as follows: for each $n\geq0$, $C_n$ is defined to be the free ...
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When does a map between abstract simplicial complexes induce a homotopy equivalence on their geometric realisations?

Let $\left(X_i,S_i\right)$ for $i=1,2$ be abstract simplicial complexes where $X_i$ are the vertex sets and $S_i\in\mathcal P\left(X_i\right)$ are the sets of simplices. Let $F\colon X_1\to X_2$ be a ...
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2 votes
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What is a flat Kan fibration?

What is the Kan fibration analog of a connection? of being `flat'? A Kan fibration is the simplicial analog of a topological fibration. What structure on a Kan fibration is the simplicial analog of ...
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2 votes
1 answer
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Order-preserving surjective maps between finite linear orders

Suppose I have two order-preserving, surjective maps $s \colon [m] \to [n] $ and $s' \colon [m'] \to [n]$. I need to show that there exists some $l \in \mathbb{N}$ and order-preserving, surjective ...
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Morphisms set of sub-$\infty$-category closed under equivalences

In Markus Land's Introduction to Infinity-Categories, he defines sub-$\infty$-categories the following way (p. 56): Definition. A sub-$\infty$-category $\mathscr{C}'$ of an $\infty$-category $\mathscr{...
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1 answer
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Composition in an $\infty$-category

In the following, I'm following Markus Land, Introduction to Infinity-Categories (p. 81). Let $\mathscr{C}$ be an $\infty$-category and $x,y,z \in \mathscr{C}$ be objects, then I want to understand ...
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Calculating the simplicial homology of a tetrahedron.

I want to calculate the simplicial homology of the $\Delta$-complex figure on the right: I believe that it consists of $2$ 0-simplices, four $1$-simplices and three $2$-simplices. So I am now in the ...
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1 answer
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Dold-Kan correspondence, Model structures and Homology

I am fairly new to the concept of model categories, simplicial sets, etc. And so there is some questions, which may be obivuous, that I need to clarify. Consider the cateogry of simplicial abelian ...
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Computation of a specific $k-$horn

I have been learning about simplicial sets and at some point we define $\Lambda^k_n$ which is the $k-$th horn of $\Delta[n]$. Now we define it has being the boundary minus the image of $d^n_k$. I ...
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A result on a fibration that is an epimorphism of simplicial sets

Consider $p:E\rightarrow B$ to be a fibration of simplicial sets. I want to see that if $p$ is an epimorphism and $E$ is fibrant then $B$ is also fibrant. I have come to the conclusion that to solve ...
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1 vote
1 answer
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Multiplicative structure on $\mathbb{S}^1$ in $\mathsf{sSet}$

I have recently been tasked with reading up on simplicial sets and I'm trying to define some concrete maps in the category to try to get my head around how things relate to stuff I already know. In ...
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2 votes
1 answer
68 views

Composition in quasi-categories

Let $C$ be a quasi-category. Then $C^{\Delta^2}\to C^{\Lambda_1^2}$ is a trivial fibration and we may choose a section $s$. The map $C^{\Lambda_1^2}\xrightarrow s C^{\Delta^2} \to C^{\{0,2\}}$ defines ...
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Product of standard simplices based on Eilenberg-Zilber theorem

I am trying to find an explanation for the claim that the Eilenberg-Zilber theorem implies the isomorphism $$\mathsf{Hom}_{\mathbf{sSet}}(\Delta\left[n\right] \times \Delta\left[1\right], L) \cong \...
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Singular simplicial sets that are (nerves of) categories

The simplicial sets that are both Kan complexes and nerves of a category are exactly the groupoids. Simplicial sets of the form $\mathrm{Sing}_\bullet(X)$ are a proper subclass of Kan complexes. ...
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1 answer
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Does this specific identity follow from the simplicial identities?

In a simplicial set, if $r<s$, does it follow that the face maps satisfy $$d_s\circ d_r = d_{r+1}\circ d_s?$$ Motivation: I want to show that for all $i<j$, $$d_{n-i}\circ d_{n-j}=d_{n-j+1}\circ ...
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2 votes
3 answers
452 views

Do we distinguish two singular simplices if they have different vertex orders?

We define a $\textbf{singular $n$-simplex}$ in $X$ to be a continuous map $\sigma:\Delta^n\to X$ where $\Delta^n$ is the standard $n$-simplex. Now, as an example, Let $X$ be a singleton $\{p\}$. Then ...
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1 vote
1 answer
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Do the levels $0$, $1$, and $2$ of a simplicial set uniquely determine a $\leq 2$-dimensional simplicial set?

Suppose $S$ and $T$ are simplicial sets such that $S_0 = T_0$, $S_1 = T_1$, and $S_2 = T_2$, the maps $d_0, d_1 : S_1 \to S_0$ agree with the maps $d_0, d_1 : T_1 \to T_0$ respectively, the map $s_0 :...
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Are there nondegenerate simplices that "look like" degenerate simplices?

Is there a simplicial set $X$ in which there is an edge $(f\colon C\to D)\in X_1$ (i.e., $d_0(f)=D$ and $d_1(f)=C$) and a 2-simplex $\tau\in X_2$ such that $$d_0(\tau)=s_0(D),\quad d_1(\tau)=f, \quad ...
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0 votes
1 answer
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Complex of $n$-balls in analogy of simplicial complexes

I would like to know if there is a version of complexes (as in simplicial complexes/sets) with the role of simplices played by $n$-balls. I thought of the following: the analogue of a (-1)-simplex(?) ...
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1 vote
1 answer
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Is there a simplicial set which has a nondegenerate $k+1$-simplex but all whose $k$-simplices are degenerate?

Is there a simplicial set $S_\bullet$ which has a nondegenerate $k+1$-simplex but all whose $k$-simplices are degenerate? I tried proving that whenever all faces $d_i\sigma\in S_k$ of a $k+1$-simplex $...
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0 votes
1 answer
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Proving that the singular simplicial set is a Kan complex

I am stuck with proving the following: For each topological space $X$, the singular simplicial set $\mathrm{Sing}_\bullet(X)$ is a Kan complex. By the definition of Kan complex, we need to do the ...
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1 vote
1 answer
38 views

Must the degenerate simplices of a nondegenerate simplex be different?

Let $\sigma$ be a nondegenerate $n$-simplex in a simplicial set. Does it follow that the degenerate simplices $s_0(\sigma)$ and $s_1(\sigma)$ are different? For instance, consider $\Delta^2$, the ...
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1 vote
2 answers
101 views

Constructing the geometric realization of a simplicial set

I'm trying to understand the construction of the geometric realization of a simplicial set. The goal is to solve the following problem: For each simplicial set $S_\bullet$, find a space $|S_\bullet|$,...
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3 votes
1 answer
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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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3 votes
1 answer
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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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3 votes
1 answer
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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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1 vote
1 answer
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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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0 votes
2 answers
55 views

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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2 votes
1 answer
112 views

$\infty$-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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2 votes
0 answers
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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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5 votes
1 answer
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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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31 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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1 vote
1 answer
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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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  • 152
3 votes
1 answer
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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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  • 152
2 votes
0 answers
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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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