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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Simplicial categories and simplicial objectcs. HTT Remark 1.1.4.2

In Lurie's HTT, Remark 1.1.4.2. says that: every simplicial category can be regarded as a simplicial object in the category $\textbf{Cat}$. Conversely, a simplicial object of $\textbf{Cat}$ arises ...
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1answer
70 views

Why 2 out of 6 property is stronger then 2 out of 3 property

Why (literally) is $2$ out of $6$ property stronger then $2$ out of $3$ property? I use this: if $hg$ and $gf$ are in $W$, so are $f , g, h,$ and $hgf$ if any two of $gf$, $g$ and $f$ are in $W$ ...
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2answers
426 views

horn of a simplex

I'm reading one book of simplicial homotopy. It's just amazing. But I am stuck at the very beginning of the book. He let a simplicial set be a contravariant functor between the category $ \Delta $ of ...
2
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1answer
77 views

Does every representable functor in $\text{Psh}(\mathcal{C}\times{\mathcal{\Delta}})$ have a weak equivalence to $h_{(c,0)}$?

Let $\text{sPsh}(\mathcal{C})$ be the category of simplicial presheaves, which I want to see as $$\text{sPsh}(\mathcal{C})=[\mathcal{C}^{\text{op}}\times\Delta^{\text{op}},\text{Set}]=\text{Psh}( \...
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1answer
43 views

Two-fold branched cover of Hexagon - Which map is meant

I am trying to understand an obvious example in a paper but do not get what is meant by: "X is a hexagon and $f:X \rightarrow \sigma^2$ is a two-fold branched cover (branched at the center of $\sigma^...
2
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1answer
55 views

$2$-skeleton of k-th horn of $\Delta^n$

I'm trying to show that $Sk_2(\Lambda_k^n)=Sk_2(\partial \Delta^n)=Sk_2(\Delta^n)$ for $n \geq 4$ is it true for $n=3$? any solution or reference is very much appreciated.
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1answer
88 views

For which families of subsets is the colimit of the Čech nerve realized by the union?

Let $(U_i\subset U)$ be a family of subspaces of a topological space $U$. Consider the Čech nerve of this family, given by the simplicial object furnished by taking iterated kernel pairs of the ...
2
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1answer
206 views

definition of a $\kappa$-small simplicial set

Let $\kappa$ be a regular cardinal. A set $X$ is $\kappa$-small, if $|X|<\kappa$. What does it mean for a simplicial set $X\colon \Delta^{op}\to Sets$ to be $\kappa$-small. I can imagine at ...
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1answer
160 views

$n$-skeleton and the category of finite simplicial complexes

Let $C$ be the category of finite simplicial complexes with simplicial maps. Then I want to define a functor $F : C \rightarrow C_n$, where $C_n$ is the subcategory of $C$ consisting of complexes up ...
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1answer
47 views

$\sum_{i=0}^n (-1)^i \delta_i(\sum_{j=0}^{n+1} (-1)^j \delta_j) = 0$, given that $\delta_i \delta_j = \delta_{j-1}\delta_i$ whenever $i < j$

$\sum_{i=0}^n (-1)^i \delta_i(\sum_{j=0}^{n+1} (-1)^j \delta_j) = 0$, given that $\delta_i \delta_j = \delta_{j-1}\delta_i$ whenever $i < j$ This problem shows up in the middle of dealing with ...
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1answer
714 views

Simplicial complex of a graph?

Starting with a graph $G$, form a simplicial complex $X$ which has $G$ as the 1-skeleton, and then has higher dimensional simplices whenever more than two vertices of $G$ are mutually adjacent. So any ...
2
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1answer
292 views

adjunction relation

Assume I have simplicial sets $X:\Delta^{op}\rightarrow Set$ and $Y:\Delta^{op}\rightarrow Set$, then I can form the simplicial set $\operatorname{Hom}(X,Y)_n := \operatorname{Hom}_{sSet}(\Delta^n\...
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1answer
52 views

the difference between chains and cochains

I have heard the following definitions, for concreteness I will refer a simplicial complex $K$ and a ring $R$, though the definitions can be extended: A chain is a formal linear combination of ...
2
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1answer
72 views

Proving the adjunction $\text{ev}_0 \dashv r:\mathcal{C}^{\Delta} \to \mathcal{C}$

I recall that $\Delta$ is the category whose objects are of the form $\textbf{n}=\{0,1,...,n\}$ and morphisms are (weakly) order preserving maps. Let $\mathcal{C}$ be a category, and let $\mathcal{C}^{...
2
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1answer
51 views

Proving that the set $\pi_n(X,v)$ is a group (Goerss and Jardine Theorem 7.2)

In theorem 7.2 of Goerss and Jardine's book Simplicial Homotopy Theory, the authors ask us to show that identity law and inverse law holds for the set $\pi_n(X,v)$. I am unable to prove these ...
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1answer
54 views

Adjunction on simplicial sets I

I am trying to understand the proof adjunction described in page 244-245, of Joyals Theory of Quasicategories. Background: $$i^*:S/I \rightarrow S/\partial I = S \times S$$ $S$ is category ...
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1answer
78 views

Equivalence between the Dwyer-Kan loop groupoid and the fundamental groupoid

Let $X$ be a homotopy 1-type (a space with vanishing homotopy groups above degree one). It is a classical fact that $X$ can be recovered completely from its fundamental groupoid. On the the other ...
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1answer
107 views

Pushout-product of anodyne extensions is again anodyne

Currently, I'm reading Simplicial Homotopy Theory by Jardine and Goerss and I got stuck in the proof of the theorem about pushout-products of anodyne extensions (corollary 4.6 in the book). Namely, I ...
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1answer
193 views

Horn of a Simplicial Set

My question refers to description of horns $\Lambda^n_k$ for Kan Fibrations in Laures' and Szymik's "Grundkurs Topologie" (page 227). Sorry, there exist only a German version. Here the relevant ...
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1answer
185 views

What is the fundamental category?

Given a category $\mathcal{C}$, we have a nerve functor $$\mathrm{N} \colon \mathbf{Cat} \to \mathbf{Set}_{\Delta}$$ that assigns to $\mathcal{C}$ its nerve $\mathrm{N}(\mathcal{C})$. This functor ...
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1answer
111 views

Which simplicial sets are filtered colimits of standard simplices?

The question is all in the title : every simplicial set is a colimit of the standard simplices $\Delta^n$, but I'm wondering which ones are filtered or directed colimits of these, if there's a nice ...
2
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1answer
98 views

Notation for Geometric realization of simplicial sets

I am confused about some notation in the quick way of constructing the geometric realization of a simplicial set. Consider the simplex category $\Delta \downarrow X$ of a simplicial set $X$. The ...
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1answer
103 views

Proof of Lemma 5.7 in Goerss & Jardine

I am trying to understand the proof of Lemma $5.7$, chapter $4$ page $237$, in 'Simplicial Homotopy Theory', Goerss & Jardine. The lemma: Suppose that $X:I\rightarrow \textbf{sSet}$ is a ...
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1answer
228 views

Understanding pointed simplicial sets

For pointed simplicial sets there are two equivalent definitions of the basepoint. Let $\Delta^0$ be the simplicial set with only one vertex in each degree. Let $X$ be a simplicial set. Then a ...
2
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1answer
471 views

What does it mean that interior of a simplex is a vector space?

Let the $n-1$ simplex be denoted as: $$S_{n-1} = \{x = (x_1, \ldots, x_n) | \sum\limits_{i = 1}^nx_i = 1, 0\leq x_i \leq 1\}$$ Then the interior of a simplex is simply: $$S_{n-1}^\circ = \{x = (x_1,...
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1answer
71 views

Free functor $\mathsf{sSet} \to \mathsf{sGrp}$ on homotopy groups

The free functor $\mathsf{sSet} \to \mathsf{sAb}$ corresponds to the Hurewicz map on homotopy groups. What about the free functor $\mathsf{sSet} \to \mathsf{sGrp}$? What effect does it have on ...
2
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1answer
145 views

Enriching categories of simplicial objects

Let $C$ be a cocomplete category and $Simp(C)=C^{\triangle^{op}}$ the category of simplicial objects in $C$. I want to show that $Simp(C)$ is simplicially enriched but I don't understand how the ...
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1answer
155 views

Simplicial homotopy's exponential law

The following was cited from Simplicial Homotopy Theory by John F. Jardine and Paul Gregory Goerss. We have an adjunction $$\hom_{\mathbf{S}}(X\times Y,Z)\simeq \hom_{\mathbf{S}}(Y,\mathbf{Hom}(X,Z))$$...
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1answer
184 views

Geometric realisation of simplicial map

Typically texts give a good definition of the geometric realisation $|\Delta|$ of a simplicial complex $\Delta$. I'm supposing that the geometric realisation forms a functor $|-|:\text{sComp} \to \...
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1answer
292 views

Examples of finite simplicial sets

Let $K$ be a simplicial set. A simplex $x\in K_{n}$ is said to be non degenerate if it is not the degenerancy of a $n-1$ simplex, i.e if there is no $y\in K_{n-1}$ such that $s_{i}y=x$. A simplicial ...
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1answer
96 views

Are these pointwise cofibrant cosimplicial objects cofibrant in the Reedy model structure?

Suppose I have a Quillen pair $F \dashv G$ with $F:\text{Psh}(\mathcal{C}\times{\Delta}) \to \mathcal{M},$ and consider also the category of cosimplicial objects in $\mathcal{M}$ denoted $\mathcal{M}^...
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1answer
57 views

Confusion in the definition of geometric realization of a simplicial set as a colimit.

In the answer given by @Kevin Arlin in the MSE question https://math.stackexchange.com/a/2994934/820022 if I am not mistaken the geometric realization of a simplicial set $X$ is defined as a colimit ...
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1answer
61 views

Existence of certain factorization of simplicial map

The following is an image of a proof from Hovey's Model Categories: How exactly do we know that $s\restriction_{\partial{\Delta[n]}}$ factors through $X_n$? Since $\partial{\Delta[n]}$ has only ...
2
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1answer
226 views

How to compute a derived tensor product?

Let $A_*$ and $B_*$ be simplicial algebras over a simplicial commutative ring $R_*$. I would like to understand how one explicitly computes the derived tensor product $A_* \otimes^L_{R_*} B_*$. More ...
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1answer
55 views

HTT, 5.4.1.1, Lurie.

Lemma. 5.4.1.1 Lurie states that if $C$ is a simplicial category and $f_0:\partial \Delta^n \rightarrow N(C)$ is a map. $X=f_0(\{0\}), Y= f_0(\{n\})$, there is an induced map $$g_0:\partial (\Delta^1)^...
2
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1answer
34 views

What is the hom space in over category of simplicial sets?

Let $Set_\Delta$ denote category of simplicial sets, which is enriched in $Set_\Delta$. Let $B$ be a simplicial set. We can form the over category. $(Set_\Delta)_{/B}$. Then is this also enriched ...
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1answer
57 views

What are small $(\infty,1)$ categories?

The term first appeared in Chapter 3 of Lurie's HTT, p144. Where he says The simplicial category $Cat^\Delta_\infty$ has as objects (small) $\infty$-categories. What are small $\infty$-...
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1answer
65 views

What is the relation of a monoid and a topological monoid

Let $X$ be a simplicial set. We say that $X$ is a $H$-space if it has a map $m:X\times X\to X$ and a point $e\in X$ which is a homotopy identity, that is, the map $m(e,-),m(-,e):X\to X$ are homotopic ...
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1answer
91 views

Do colimits/limits exist in category of enriched categories?

This question may be too general. I am interested in references or proofs for special cases. I follow the definition in Chapter I of Kelly's Enriched Category. Let $V$ be a monoidal category theory. ...
2
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1answer
81 views

How to identify commutative diagrams by exponential law

In the category of simplicial set $S$, we have a bijection $ev_*:\hom_S(K,\operatorname{Hom}(X,Y))\to \hom_S(X\times K , Y)$ I wonder how to identify commutative diagrams using this bijection. ...
2
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1answer
53 views

Weak $n$-Category

I'm trying to obtain an intuition for weak n-categories based on explanations from: https://en.wikipedia.org/wiki/Higher_category_theory#Weak_higher_categories https://ncatlab.org/nlab/show/n-...
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1answer
229 views

Is the map $\Lambda^n_n \sqcup_{\Delta^{\{n-1,n\}}} J \to \Delta^n \sqcup_{\Delta^{\{n-1,n\}}} J$ left anodyne?

This question is about objects studied in chapter 2 of Lurie's Higher Topos Theory. Let $p : X \to S$ be a left fibration, and let $e \in X_1$ be an edge of $X$ such that $p(e)$ is an equivalence in $...
2
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1answer
113 views

Quillen equivalence between sSet (Joyal's model structure) and sSetCat (Bergner's one)

Let me consider two model categories: $ \mathsf{sSet} $: the category of simplicial sets with Joyal model structure, $ \mathsf{sSetCat} $: the category of simplicially enriched categories with ...
2
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1answer
108 views

Is the overcategory $C_{/p}$ a subcategory of the $\infty$-category $\operatorname{Fun}(K^{\vartriangleleft}, C)$?

Let $p: K \to C$ be a simplicial morphism from a simplicial set $K$ to an $\infty$-category $C$. $\newcommand{\catSSet}{\mathtt{SSet}}\DeclareMathOperator{\Fun}{Fun}$ The over $\infty$-category $C_{/p}...
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1answer
207 views

Completing a delta set to a simplicial set

I am reading "An elementary illustrated introduction to simplicial sets" by Greg Friedman, available online here. He defines Delta sets (or semi-simplicial sets) as a generalisation of a simplicial ...
2
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1answer
149 views

Fundamental groupoid of a Kan complex

Let $X\in \operatorname{sSet}$ be a Kan complex. This gives rise to the fundamental groupoid $\Pi(X)$ of $X$. I am having trouble seeing why the composition in $\Pi(X)$ is well defined. If $y\in X[2]$...
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2answers
130 views

Milnor's proof of $|X\times Y| \cong |X|\times |Y|$

I'm on my way reading this article of Milnor about the geometric realisation of a "(complete) semi-simplicial complex" ( = simplicial set, in modern terms). I encountered a problem in a passage of ...
2
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1answer
123 views

Is there a sensible way to form the geometric realization of a “simplicial space up to homotopy”?

I am new to simplicial methods, and have some naive questions. As such, the questions may be malformed and the terminology might not even be right. I assume these are answered in some reference and ...
2
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1answer
110 views

Why $|N(P_{i, j})| \cong [0, 1]^n$ as stated in page 21 of HTT?

maybe this is an idiot question, however I could not solve this after thinking for a while. I added the tag about higher categories simply because of the nature of the question, however this is just a ...
2
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1answer
55 views

explicit equivalent relation in the expression of the classifying space of a monoid

Let $M$ be a topological monoid. $M$ can be considered as a category internal to topological spaces and has a simplicial space $N_\bullet(M)$ as its nerve. (It's also called the internal nerve.) The ...

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