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.

202 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
13
votes
0answers
536 views

Bousfield–Kan spectral sequence for homotopy colimits

Let $\mathcal{J}$ be a small category and let $X : \mathcal{J} \to \mathbf{sSet}$ be a diagram. We define its homotopy colimit $\newcommand{\hocolim}{\mathop{\mathrm{ho}{\varinjlim}}}\hocolim X$ ...
10
votes
0answers
156 views

A theorem of Kan regarding fibrant replacement

Recall that there is an adjunction $$\mathrm{Sd} \dashv \mathrm{Ex} : \mathbf{sSet} \to \mathbf{sSet}$$ where $\mathrm{Sd} (\Delta^n)$ is the first barycentric subdivision of $\Delta^n$. There is a ...
8
votes
0answers
280 views

Andre-Quillen Homology of the cuspidal curve $k[x,y]/(x^2 - y^3)$

I was wondering if I am in the right track here. Let $A := k[x,y]/(x^2 - y^3)$, the cuspidal curve. Obviously this isn't etale or smooth over $k$ so its cotangent complex is not contractible. Now, I ...
7
votes
0answers
60 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 ...
7
votes
0answers
126 views

Characterising singular homology among a more general class of cosimplicial spaces

Is there a way to characterise (up to isomorphism) the simplicial spaces $F: \Delta \to \underline{\text{Top}}$ with $F( \underline{n}) \subset \mathbb{R}^{n+1}$ compact and $F(\underline{0})$ a point?...
7
votes
0answers
242 views

What is a copresheaf on a “precategory”?

Let $\mathcal{S}$ be a category with pullbacks. A precategory $\mathbb{C}$ in the sense of Borceux and Janelidze [Galois Theories, §7.2] comprises three objects $C_0, C_1, C_2$ in $\mathcal{S}$ and ...
6
votes
0answers
151 views

Unique representation of a degenerate simplex

I recently learned about simplicial sets, and now I'm studying some basic properties. I've learned that every degenerate simplex is a degeneracy of a unique non-degenerate (nice) simplex. However, I ...
5
votes
0answers
80 views

Relating Two Notions of Homology

I know of two ways things called "homology" are constructed in mathematics, and would like to know how, if at all, these two types of constructions are one in the same. To keep things simple, I'll ...
5
votes
0answers
263 views

Pushout diagram of the $n$-skeleton of a simplicial set

In Goerss, Jardine book "Simplicial Homotopy Theory" at the very beginning of the book, where they prove that given a simplicial set $X$ the geometric realization is a CW-complex, there is a pushout ...
5
votes
0answers
188 views

Converting a $1$-connected simplicial complex into a $1$-reduced simplicial set

There is a natural way to convert a simplicial complex $C$ into an "equivalent" simplicial set $S$: after ordering the vertices, the simplices in $C$ correspond exactly to the non-degenerate simplices ...
5
votes
0answers
61 views

Replacing faces of a cube in a quasicategory

I have a question on quasicategories which seems to be unavoidably cubical, and which I haven't been able to locate any information about. Suppose I have a cube $U:\mathbf{2}^3:\to C$ in a ...
5
votes
0answers
78 views

Why aren't *weak* test categories enough?

Short version : What is the motivation behind the local condition in the definition of a test category ? Why do we want the slice categories $A/a$ to be weak test ? Long version : I start with ...
5
votes
0answers
340 views

Two definitions of equivariant sheaves

Let $G$ be a topological group. Here are two definitions of $G$-equivariant sheaves on a $G$-space $X$. (a) Define an $G$-equivariant sheaf by a sheaf $F$ (étalé space) equipped with a $G$-action ...
4
votes
0answers
77 views

Filling a map $X\times\Lambda^n_k\rightarrow Y$ to a map $X\times\Delta^n\rightarrow Y$

I've come across the following claim here, in lemma (1.15): if $X$ and $Y$ are simplicial sets and $Y$ is a Kan complex, then for any $k\leq n$ and $\varphi:X\times\Lambda^n_k\rightarrow Y$ there is a ...
4
votes
0answers
70 views

Higher homotopical information in racks and quandles

A quandle is defined to be a set $Q$ with two binary operations $\star,\bar\star\colon\ Q\times Q\to Q$ for which the following axioms hold. Q1. a $\star$ a = a Q2. (a $\star$ b) $\bar\star$ b = (a $\...
4
votes
0answers
64 views

Cohomology of Classifying Space/Simplicial Manifold

Given a simplicial manifold $\,X^{\mathbf{\cdot}}$ (say a classifying spae $BG$ of a Lie group $G$) we have a differential given by $d_n^*=\sum_i (-1)^id^*_{n,i}\,,$ acting on functions $f_n:X^n\to A\,...
4
votes
0answers
57 views

Intuitive meaning for Kan fibration

Other than the formal definition of Kan fibration (https://en.wikipedia.org/wiki/Kan_fibration), is there any intuitive meaning of Kan fibration? Thanks.
4
votes
1answer
207 views

What is a “cubical map” between cubical complexes?

What is a natural definition of a cubical map between cubical complexes? What is its geometric realization? I found some definitions, such as here or here, where a cubical map between cubical ...
4
votes
0answers
192 views

Tensoring a connective chain complex with a simplicial set

Let $\mathrm{Ch}_{\geq 0}(R)$ be the category of chain complexes of $R$-modules concentrated in nonnegative degrees, equipped with the projective model structure. By a general theorem about model ...
4
votes
0answers
375 views

Kan's loop group construction

I'm looking for a good place to read about the loop group construction $G : \mathbf{sSet_0} \to \mathbf{sGrp}$ taking a reduced simplicial set $X$ and producing a simplicial group $GX$. I would also ...
4
votes
0answers
112 views

Converting a space to a simplicial set

Given a compact manifold and a Morse function, we obtain a convenient cellular decomposition. But suppose I have a (finite dimensional, possibly singular) space $X$ that is not a manifold and I wish ...
4
votes
0answers
99 views

Morita theory for simplicial rings

My question is the following: is there an analog of Morita theorem in the simplicial setting? I mean, we can define two simplicial rings $A,B$ to be simplicially Morita equivalent is the categories ...
4
votes
0answers
299 views

Algebraic Morse theory

In 2005, prof. Emil Skoldberg developed a theory, similar to Forman's Discrete Morse Theory, but suited for arbitrary based chain complexes, in his Morse Theory from an algebraic viewpoint. I'm going ...
4
votes
0answers
215 views

What is an “absolute, equational pushout”?

I was leafing through Joyal and Tierney's Notes on simplicial homotopy theory. In the first few lines of the section on the skeleton of a simplicial set they display the simplicial identities as a ...
3
votes
0answers
36 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 ...
3
votes
0answers
57 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 ...
3
votes
0answers
81 views

Homology is homotopy invariant, fake proof with simplicial methods?

Here is a maybe false proof that I came up with that homology of topological spaces is homotopy invariant. I'm thinking that it is indeed fake because why hasn't anyone else come up with this much ...
3
votes
0answers
48 views

Generalization of singular homology of topological spaces to varieties over $k$.

To define the singular homology of topological spaces we define a sequence of topological spaces $\Delta^n$ and maps $r_i:\Delta^n \rightarrow \Delta^{n+1}$. I'm kinda skipping a lot of details ...
3
votes
0answers
41 views

Reproving a standard result about singular homology with methods from model categories

If $A \rightarrow X$ is a cofibration of topological spaces then the induced map $H_i(X,A)\rightarrow H_i(X/A,*)$ is an isomorphism. I am aware of many proofs of this fact but none use methods from ...
3
votes
0answers
105 views

Reedy model structure, simplicial sets and model categories

I have two questions, the first one is just wether the following statement is true or not? Is there a reference for this? The second one is much more open, it's basically what restricitions do we ...
3
votes
0answers
97 views

Kan extension along any functor implies adjoint exists

I trying to decide whether the existence of a, say, a left Kan extension along $i: C \to D$ for any functor $F: C \to E$ implies that $i$ has a right adjoint. I have proven the converse, and I know ...
3
votes
0answers
45 views

Showing that $\pi_n(X,v)$ satisfies inverse axiom.

Given a fibrant simplicial set $X$ (has lifting condition with all horns) and a vertex $v \in X_0$. I want to show that the simplicial homotopy group $\pi_n(X,v)$ satisfies the inverse axiom. ...
3
votes
1answer
125 views

Why does Spivak define the realization functor from fuzzy simplicial sets to extended pseudo metric spaces the way he does?

In Spivak's paper on metric realization of fuzzy simplicial sets, he sends a fuzzy $n$-simplex of strength $a$ to the set $$ \{(x_0,x_1,\dots,x_n) \in \mathbb{R^{n+1}} |x_0+x_1+\dots+x_n = -\lg(a) \} $...
3
votes
0answers
30 views

Triangulation of fiber bundles

Let us suppose we are given a fiber bundle $(E,B,F,p)$ where all spaces involved are triangulable and compact. Assume we choose a triangulation for the base B. I believe it is possible to give E a ...
3
votes
0answers
107 views

Theorem 2.1.2.2 Higher Topos Theory

At the page 74 of HTT, there is the following theorem Let $S$ be a simplicial set, $\mathcal{C}$ a simplicial category, and $\phi: \mathfrak{C}[S] \rightarrow \mathcal{C}^{op}$ a simplicial functor....
3
votes
0answers
39 views

Simplicial intersection product

Let $X$ be both a simplicial set and a closed $n$-dimensional manifold. We have a duality isomorphism $H^k(X)\to H_{n-k}(X)$. Furthermore, let $Y,Y'\subseteq X$ be two subcomplexes and closed ...
3
votes
0answers
90 views

Simplicial set morphisms from an inner horn to a nerve are determined on the spine

This is a subject I'm completely new to, so I am a bit wary of my own proofs. I will follow the notations of Charles Rezk, Stuff about quasicategories, as of 5 April 2020. I want to prove the ...
3
votes
0answers
86 views

$\mathrm{\Gamma}$ free group functor of Barratt-Eccles

In the article A free group functor for stable homotopy theory, Barratt and Eccles define for each $X\in\mathsf{sSet}_{\ast}$, the free simplicial monoid $\Gamma^{+}X$. Proposition 6.2 states if ...
3
votes
1answer
150 views

Homology of the simplicial complex obtained from an octahedron by removing 4 faces.

Suppose you have the surface of an octahedron and you remove 4 of the eight faces as follows: If you remove one face then you don't remove all the adyacent faces and so on. You can look at this as ...
3
votes
0answers
34 views

The dimension of $H_i(X_n, \mathbb{Q})$ is linear in $n$ with a bounded nonnegative error, where the error is periodic?

Let $X$ be a finite simplicial complex with a simplicial map to $S^1$. Take covers of $X$ associated to the subgroups $n \mathbb{Z}$ of $\mathbb{Z} = \pi_1(S^1)$. This defines a sequence of spaces $...
3
votes
0answers
117 views

Can some Lie groups ($S^3$ in particular) be converted to simplicial groups?

I'm trying to understand the concept of Kan simplicial sets and simplicial groups in particular. My question is, in a nutshell: What is the relation between the group operation in a simplicial ...
3
votes
0answers
179 views

Cohomology of geometric realization of a simplicial topological space

Let $X$ be a simplicial topological space. We can consider to notions of cohomology of $X$. Denote by $|X|$ geometric realization of $X$. Then we can take just $H^*(|X| )$ (i.e. usual cohomology of $|...
3
votes
0answers
142 views

Fibrant (Kan complex) geometric meaning

A simplicial set $X$ is said to be fibrant (or Kan complex) if it satisfies the following homotopy extension condition for each $i$: Let $x_0,\dots,x_{i-1},x_{i+1},\dots,x_n\in X_{n-1}$ be any ...
3
votes
1answer
147 views

Degeneracies of simplex $y$ which appears as any face of some simplex $x$

Let $K$ be simplicial set and $d_i:K_n\rightarrow K_{n-1}$, $s_i: K_n\rightarrow K_{n+1}$ ($i = 0,...,n$) face and degeneracy maps respectively. Suppose we have some $x\in K_n$ with $d_0x = ... = ...
3
votes
0answers
63 views

Classifying space of resolution of a n-regular hypergraph

Let $G =(V(G),E(G)))$ be a $n$-regular hypergraph. Let $Z_n$ be a cyclic group with generator $a.$ Definition : A resolution of $G$ is finite partially ordered set $C$ with $C_0$ is the set of ...
3
votes
0answers
216 views

How to compute (co)limits of enriched categories?

Let $\mathscr{V}$ be a monoidal category. Let $\mathbf{Cat}_{\mathscr{V}}$ be the category of (small) categories. I would like to know how to compute (co)limits in $\mathbf{Cat}_{\mathscr{V}}$. This ...
3
votes
1answer
89 views

Universality of the Simplex Category. Proving Functoriality of the Map.

Let $B$ be a strict monoidal category, and $\left \langle c,\mu ',\eta ' \right \rangle$ a monoid in $B$. Now suppose we consider the simplex category $\left \langle \triangle ,+,0 \right \rangle$, ...
3
votes
0answers
174 views

Is the projection from the cyclic nerve to the simplicial nerve a cyclic map?

I’m trying to understand Loday’s proof of Theorem 7.3.11 in “Cyclic Homology” (2nd ed 1998). I have a problem right at the beginning where it says: The projection map $\text{proj}:\Gamma_\bullet G \...
3
votes
0answers
229 views

Associativity of the tensor product of dendroidal sets

For any smal category $A$, I shall write $\widehat A$ for the category $[A^{\text op}, \mathbf{Set}]$ of presheaves on $A$, and $y_A\colon A \to \widehat A$ for the Yoneda embedding relative to $A$. ...
3
votes
0answers
86 views

When may the bar resolution be truncated for calculating $\operatorname{hocolim}$?

Suppose $D$ is a small category and $F\colon D\to sSets$ a functor. I want to calculate the simplicial set $$\operatorname{hocolim}F$$ via the bar construction: $\operatorname{hocolim}F$ is the ...

1
2 3 4 5