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.

Filter by
Sorted by
Tagged with
0
votes
0answers
30 views

Non simplicial model categories

What are some interesting/notable examples of non simplicial model categories? What are some notable examples of model categories which cannot be seen to be Quillen equivalent to a simplicial one? ...
1
vote
0answers
16 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 ...
4
votes
1answer
72 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 ...
0
votes
1answer
32 views

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 \\ \...
1
vote
0answers
67 views

Contractible bisimplicial sets

I'm reading a paper where I find that in the category of simplicial presheaves on $\Delta,$ i.e. $\text{sPsh}\left(\Delta\right)=[\Delta^{\text{op}},\text{sSets}],$ the object $\Delta^n$ is not ...
2
votes
1answer
34 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 $\...
2
votes
0answers
115 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 ...
1
vote
1answer
34 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 ...
1
vote
1answer
31 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 [...
1
vote
1answer
46 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 \...
0
votes
1answer
40 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 ...
2
votes
1answer
87 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}^...
2
votes
0answers
78 views

Representable functors are cofibrant

Let $\text{Psh}(\Delta\times{C})$ be the category of simplicial presheaves, and $y$ the Yoneda embedding $y:\Delta\times{C} \hookrightarrow \text{Psh}(\Delta\times{C}).$ How do you show that a ...
1
vote
0answers
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 $a_{ij}$, $a_{ij} = Cf(a)$, ...
2
votes
1answer
44 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 ...
1
vote
1answer
36 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 = \...
0
votes
0answers
34 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 $\...
0
votes
0answers
80 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 ...
1
vote
1answer
50 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 ...
2
votes
1answer
71 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}( \...
2
votes
1answer
66 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}^{...
3
votes
2answers
94 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 ...
0
votes
0answers
56 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 ...
1
vote
1answer
40 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, ...
0
votes
0answers
31 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 ...
0
votes
0answers
36 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 \...
2
votes
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 ...
1
vote
0answers
29 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 ...
1
vote
0answers
56 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 ...
1
vote
0answers
51 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 ...
2
votes
1answer
34 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 ...
1
vote
0answers
27 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) ...
1
vote
1answer
52 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:...
1
vote
1answer
56 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 $\...
1
vote
1answer
30 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 ...
1
vote
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 ...
7
votes
0answers
54 views

Is the nerve of a symmetric monoidal category a K-theory space?

It's an amazing fact that if the $C$ is a symmetric monoidal category so that its components form a group, then $NC$ is an infinite loop space. Now if we have a Waldhausen category $D$, a category ...
0
votes
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 ...
2
votes
0answers
39 views

Snake lemma, pictorially on simplicial complexes

I'm trying to understand what the snake lemma 'computes' on small examples. Consider this: It seems to me that given two 'defective' / 'incomplete' simplicial complexes that are related through a ...
5
votes
1answer
77 views

Definition of the filtered normalized chain complex $N_0A$

At the page 163 of the book Simplicial Homotopy Theory by Goerss and Jardine, the first sentence starts with "Observe that $N_0A= A$, that ...". According to the further calculation in that ...
0
votes
0answers
25 views

Proving simplicial homology is preserved on mapping?

Assume we have a the $\mathbb Z\text{ modules}$ $S1 \equiv (F, E, V)$ with boundary maps $(\partial_{FE}: F \rightarrow E$, $\partial_{EV}: E \rightarrow V)$, with the condition that $\partial_{EV} \...
3
votes
0answers
56 views

How to divide a unit space into many simplices?

I'm sorry, it may be simple and stupid but I didn't find any relative solutions on the Internet. Given the unit hypercube $C$ in the Euclidean space $R^n$, how to divide (or we can say "partition", I ...
0
votes
0answers
24 views

How is a simplicial complex completely determined by a simplicial set?

I was wondering how a simplicial complex is completely detremined by a simplicial set. I say let $K$ be a simplicial complex. Then $K=(V,S)$ where $V$ is a set and $S$ is a collection of finite non-...
0
votes
0answers
14 views

Dyadic division of a simplex - is it possible in dimension > 2?

Assume T is a triangle. We can join the middle points of the sides of T, and this form a division of T into 4 triangles, homothetic to the original triangle, and with sides equal to sides(T)/2. Let me ...
0
votes
0answers
30 views

Introduction to $E_\infty$-algebras

To give an idea on how limited my background is: a few days ago, I have heard the term operad for the first time. I am trying to understand a text about dg-, $A_\infty$- and $E_\infty$-algebras in the ...
0
votes
0answers
29 views

Proving that for every element $x\in X_n$ there is a unique $m$, surjection $f\in\Delta(n, m)$, and nondegenerate $y\in X_m$ such that $x=f^*(y)$.

Let $X$ be a simplicial set, and $|X|=X\otimes_\Delta\Delta^{\bullet}$ its geometric realization. We say $x\in X_n$ is degenerate if there is a surjection $f\in\Delta(n,m)$ with $m<n$ and $y\in X_m$...
5
votes
2answers
49 views

Constructing a simplicial map from a diagram

I am trying to read Goerss and Jardine's book (Simplicial Homotopy Theory) and in the proof of Theorem 7.10 (in chapter 1), they claim that there is a simplicial homotopy $\Delta^n\times\Delta^1\to\...
1
vote
0answers
25 views

A lifting problem in $\infty$-categories

Let $\mathcal{C}$ be an $\infty$-category and consider the outer horn inclusion $\Lambda[3]_3 \subset \Delta[3]$. Given a diagram $f:\Lambda[3]_3 \to \mathcal{C}$, if the image of $\Delta^{\{2,3\}}$ ...
3
votes
1answer
44 views

Why two simplices become equal in the colimit

The following is an image of a proof found in Hovey's Model Categories, for which my question concerns just the second paragraph. I do not see the justification for the statement that we can find $\...
2
votes
1answer
59 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 ...

1
2 3 4 5
11