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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45 views

Boundary Map of Bar Resolution vs. Face Map of the Nerve of a Group

For a discrete group $G$ I have the following two definitions, which I think are correct: The nerve of $G$ is $NG$, a simplicial set whose $n$-simplices are $G^n$ ($G^0$ being the trivial group $\{1\}...
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38 views

Example of right homotopy not transitive in Kan-Quillen model structure

I am learning about abstract homotopy theory and know that when $Y$ is fibrant then right homotopy is an equivalence relation on the set of maps from $X$ to $Y$. When $Y$ isn’t, then transitivity ...
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1answer
31 views

Connes cyclic set - construction explaination.

This is from a paper by Scholze and Nikolaus, THH, p145, second equation. The author defines two categories. $\Lambda_\infty$ a fullsubcaregory of Posets with $\Bbb Z$ action: Its objects are $\frac{...
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Combinatorics associated to the fillings of spines (and necklaces) in an (infinity,1)-category

I'm currently trying to solve the following problem : Consider $X$ an $(\infty , 1)$-category, defined here as a simplicial set which fills the inner horns $\Lambda^k_n \rightarrow \Delta^n$, for $k=...
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1answer
22 views

Question about orientations and loops of the $n-1$ faces of an n-simplex $\Delta^n$

$S$ is the $\Delta$-complex formed by considering $\Delta^n$ along with its faces. Let the vertices be $v_0, v_1, ... v_n$. Then $C_n(S) = \langle[v_0 v_1..v_n]\rangle$ and $C_{n-1}(S) = \langle\{[v_0....
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2answers
43 views

The Funtion Complex - Product Adjunction of simplicial sets and diagram chasing

I run into the following claim in the book Simplicial Homotopy Theory (in the proof of Proposition 5.2.). Given $i:K \rightarrow L$ inclusion of simplicial sets and $p:X\rightarrow Y$ fibration. ...
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33 views

Product of two anodyne maps is anodyne

Anodyne morphisms can be defined as the closure of horn inclusions $\Lambda^k_n\to \Delta^n$ along retracts, transfinite composition and pushouts. A nice result on these anodyne morphisms is that the ...
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1answer
43 views

Equivalent definition of Kan fibration

Let $\Lambda^n_k$ be the $k$-th horn of the standard $n$-simplex $\Delta^n$, and let $Y$ be a simplicial set. In Simplicial Homotopy Theory, pg. 10, we find that the coequalizer description of the ...
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32 views

Standard $n$-simplex is not a Kan complex [duplicate]

In Lurie's Kerodon it is mentioned as a remark that none of the standard $n$-simplices, i.e. the representable simplicial sets, are Kan complexes. I am curious to see a proof of this 'straight from ...
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3answers
143 views

Simplicial sets which are not Kan complexes

A Kan complex is a simplicial set satisfying the horn-filler condition. What examples are there of simplicial sets which do not satisfy the horn-filler condition? In particular, what simplicial sets ...
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17 views

Connected locally finite abstract simplicial complex is countable

I want to verify whether the arguments for the statement is true. I say that an abstract simplicial complex, $X$, is locally finite if $\deg(v)<\infty$ for all $v\in X(0)$, where $$ \deg(v)=\vert \{...
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1answer
79 views

A simplicial structure on symmetric groups

Do the symmetric groups admit a simplicial structure? By this, I mean a functor $X: \Delta^{op} \to \text{Sets} $ such that $X(n) = S_n$. More explicitly, one has to find functions (not necessarily ...
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18 views

Locally finite abstract simplicial complex has only finite simplices?

I usually think of a locally finite abstract simplicial complex as having that any simplex has only finitely many higher dimensional simplices containing it. Namely if $\sigma$ is a $d$-simplex, then ...
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1answer
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Doubt in the proof of Lemma 8.5 (chapter II) in Goerss & Jardine

In Lemma 8.5 (chapter II) of Goerss & Jardine's book, we are given the following pushout square in a category of cofibrant objects: $$\begin{array}{cc} A & \xrightarrow {u} & B \\ \...
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1answer
92 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
38 views

Does strict group completion preserve weak equivalences?

Let $i : \mathbf{Grp}\to \mathbf{Mon}$ denote the forgetful functor from groups to monoids. It has a left adjoint, $(-)^{gp}$, which one could call group completion. We have an induced functor $\...
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128 views

Understanding mapping spaces in higher category theory

I'm trying to understand the mapping space of two objects in an infinity category. Below is context and definitions, but please let me know if I have any misunderstandings. The source of everything ...
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1answer
36 views

The coproduct of any indexed collection of quasicategories is a quasicategory.

I'm reading the following document: https://faculty.math.illinois.edu/~rezk/quasicats.pdf The proposition I'm having trouble with is the following: The coproduct of any indexed collection of ...
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1answer
55 views

Proof that a simplicial group is a Kan complex

I'm trying to understand the following proof that a simplicial group is a Kan complex by Jardine, but I can't understand the bold statements: Suppose $S \subset [n]$ and $|S| \leq n$. Write $\Delta^n \...
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1answer
38 views

Connected Component of a simplicial set

From this note, I'm trying to solve 6.8: https://faculty.math.illinois.edu/~rezk/quasicats.pdf I will list the necessary definitions: And $e_0$ refers to $X(f)(e)$, where $f \colon [0] \rightarrow [...
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1answer
44 views

Tensor product of associative ring $A$ is flat over $A \otimes_k A^{op}$.

It is claimed in this notes line 4 pg 5 that If $A$ is flat over $k$, $A^{\otimes n}$ is flat over $A \otimes_k A^\text{op}$ for $n \ge 2$. I am stuck even at $n=2$. I tried some base changes but ...
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27 views

Subcategory of quasicategory- equivalent condition of subcomplex being a subcategory

I'm working on the following problem, from this link: https://faculty.math.illinois.edu/~rezk/quasicats.pdf I will list the relevant definition and notation first. And by $a_{ij}$, $a_{ij} = Cf(a)$, ...
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1answer
48 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
42 views

Are extensions of simplicial objects to functors $\mathsf{sSet} \to \mathsf{C}$ Kan extensions?

Suppose that we have a functor $F : \boldsymbol{\Delta}^\bullet \to \mathsf{C}$ with domain the full subcategory of simplicial sets given by representable functors. For example, for each $\Delta^n = \...
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39 views

Morphisms of a Simplex Category of a simplicial set.

In the Page 7 of the book Simplicial Homotopy theory by Jardine and Goerss they defined the simplex category of a simplicial set $X$ as the slice category $\Delta \downarrow X$. So objects of $\...
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1answer
82 views

Proof of homotopy invariance of homology : any way to make it better?

Suppose you want to prove that homotopic maps induce the same morphisms in singular homology. One way to do that is the following : you have your homotopy $X\times I \to Y$, apply $Sing$ to it to get $...
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82 views

Cosimplicial resolution and fibrations

A cosimplicial resolution of a functor $\gamma: \mathcal{C}\to \mathcal{M}$ is given by a functor $\Gamma: \mathcal{C}\to \mathcal{M}^{\Delta}$ such that for every $X\in \mathcal{C},$ $\Gamma(X)$ is ...
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1answer
51 views

Can the geometric realization increase the dimension?

Let $M$ be a smooth manifold of dimension $n$, and $\mathcal{U}$ any open cover of $M$, in particular $\mathcal{U}$ is not necessarily good or finite. Consider then the geometric realization of the ...
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1answer
74 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
68 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}^{...
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2answers
102 views

A motivating argument for studying determinants of $2\times 2$ matrices with prime entries in relation to Goldbach's conjecture.

Let $\Bbb{P}$ be the set of odd primes. Let $X_n$ for $n \geq 3$ be the Goldbach solution set $X_n = \{(p,q) \in \Bbb{P}\times \Bbb{P} : 2n = p + q \}$. Suppose that for combinatorial reasons we are ...
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1answer
44 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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57 views

Reference or counterexample: simplicial algebra isomorphism

I'm looking for either a reference, a quick proof, or a counter example to the following claim: Let $R_{\bullet}$ and $T_{\bullet}$ be simplicial $k$-algebras. Let $S_{\bullet}$ be the polynomial ...
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1answer
43 views

Correspondence of left and right homotopies produces a homotopy of maps $A\times I \to X^I$ (Hovey, Lemma 6.1.5)

My question pertains to a specific piece of the proof of Lemma 6.1.5 in Hovey's Model Categories (page 160 in pdf, 150 by numbering). I'll briefly recall the definitions and set up my question first, ...
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34 views

Delooping block diffeomorphism spaces

In Kuper's Lecture Notes on Diffeomorphism Groups pg. 195, Kuper defines the simplicial group of block diffeomorphisms $Diff^\flat_\partial (M)$ for a manifold with boundary $M$ as having n-simplices ...
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39 views

Kan extensions and restrictions

Let $\mathcal{C},\mathcal{D}$ be two small categories and $\iota:\mathcal{C}\rightarrow \mathcal{D}$ a faithful functor. Then take a representable functor $$\text{Hom}(-,D):\mathcal{D}\rightarrow \...
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33 views

The infinite loop space associated to the spectrum associated to a special $\Gamma$-space

I am currently reading M. Mandel's paper `An Inverse $K$-theory Functor' to extract some results I may wish to use in my masters research project, and I am stuck on one particular claim. On the top of ...
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0answers
61 views

Geomtric Realization as Colimit

So we know that geometric realization of simplicial set $X$ is just colimit of this functor F as seen here: Notation for Geometric realization of simplicial sets. And we can also state the geometric ...
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0answers
55 views

motivation and applications of barycentric subdivision of a simplicial complex

Question is as in the title: What is the motivation and applications of barycentric subdivision of a simplicial complex? Given a simplicial complex $K$, the barycentric subdivision of $K$ is another ...
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1answer
36 views

Degenerate simplex is the degeneracy of a unique non-degenerate simplex

Preface: Note that although this is a proof verification question, I'd also be happy with a link to a reference outlining a proof of the following statement. As far as I can tell this fact has not ...
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0answers
32 views

Is underlying hypergraph isomorphism of simplicial 'stuff'' equivalent to topological equivalence?

Assume that a piece-wise linear entity can be (heterogeneously) triangulated into a simplicial structure. Does not the underlying hypergraph (without the positional information of simplices) ...
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1answer
57 views

Kan extension and representable functors

Consider some category $\mathcal{C}$ with some full subcategory $\mathcal{G}$ such that $\mathcal{G}$ generates $\mathcal{C}$ (the example I have in mind the $Alg_A^{free}\subset Alg_A$). A functor $F:...
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1answer
90 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 ...
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1answer
62 views

There is a sometimes fully faithful functor from simplicial commutative algebras to differential graded algebras. What is this functor explicitly?

In Jacob Lurie's paper "Derived Algebraic Geometry," in 2.6 $E_{\infty}$-Ring Spectra and Simplicial Commutative Rings, page 25, there is the following claim: In general, we have functors $\...
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1answer
188 views

A Mayer-Vietoris argument I don't get — From a paper by Bousfield-Gugenheim

I working through the article On PL De Rham theory and rational homotopy type by Bousfield-Gugenheim. I am having troubles completing the proof of lemma 5.3, where they claim that the fact that a ...
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1answer
74 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. ...
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1answer
31 views

Is normalized chain complex functor the unique Quillen equivalence?

I don't have a grasp of model categories. I asked the question through Quillen equivalences in order to make it as general as possible. This might be too general to answer and/or might be above my ...
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1answer
35 views

Strong Homotopy Equivalence of $Sd^2 \Delta^n$ and $\Delta^n$

There is a double iteration of the last vertex map yielding $Sd^2\Delta^n \to \Delta^n$. My question is whether this is a strong homotopy equivalence. One has to find a homotopy inverse and the only ...
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1answer
65 views

Can nonisomorphic groupoids have homotopy equivalent classifying spaces?

We know that two discrete groups having the same classifying space up to homotopy are isomorphic. One can just take fundamental groups and conclude. The situation with topological groups is subtler. ...
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2answers
43 views

Why are objects in the simplicial category cosimplicial?

From nLab, A cosimplicial object in C is similarly a functor out of the opposite category, Δ→C. Δ here means the simplicial category. That implies that the the identity functor on the simplicial ...

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