The simplicial-stuff tag has no wiki summary.

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### Coskeleta of simplicial principal bundles

Let $X_{\bullet}$ be a simplicial topological space. There is a truncation functor $tr^n \colon Fun(\Delta^{op}, Top) \to Fun(\Delta_n^{op},Top)$ (where $\Delta_n$ is the full subcategory of $\Delta$ ...

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**1**answer

175 views

### Simplicial complex made of central idempotents of an algebra

Let $A$ be an algebra, say over $\mathbb{C}$ and finite-dimensional, but not necessary semisimple. I have the strong feeling, which I would like to prove and use, about the following rather natural ...

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105 views

### Do bistellar flips in a simplicial sphere preserve facet connectivity?

Define two facets of a simplicial sphere as k-connected if there are at least k disjoint edge-vertex paths between them in their facet-ridge graphs.
Do bistellar flips preserve facet k-connectivity?
...

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**2**answers

648 views

### Does the classification diagram localize a category with weak equivalences?

Let $(C,W)$ be a category equipped with a subcategory of weak equivalences. Its "classification diagram" or "bisimplicial nerve" $N(C,W)$ is a bisimplicial set, for which $N(C,W)_n$ is the nerve of ...

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454 views

### About a statement in Jardine and Goerss “Simplicial Homotopy Theory”

Hello,
This probably just technical, but anyway:
In "Simplicial Homotopy Theory" by Goerss and Jardine, chap. III, par. 2, after cor. 2.12, they describe a model structure on $Ch^{+}$, the category ...

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215 views

### good covers and simplicial maps

Let $X$ be a paracompact topological space and choose a good cover $U_i$ of $X$. Remember that a good cover is one that consists of open subsets, such that each set $U_i$ is contractible and all ...

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545 views

### simplicial spaces without degeneracies

Suppose I have a simplicial space $X_{\bullet}$ without degeneracies (sometimes called semi-simplicial space or incomplete simplicial space). There still is a geometric realization $\lVert X \rVert$ ...

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**1**answer

190 views

### local model structure on simplicial presheaves

Hello,
Let $\mathcal{C}$ be a (small) category equipped with a Grothendieck pretopology.
Let $sPSh(\mathcal{C})$ be the category of simplicial presheaves on $\mathcal{C}$, together with its ...

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325 views

### Cartesian cubes and groupoids

Given a groupoid $G,$ one can consider the canonical epimorphism $$G_0 \to G.$$ Since it is an epimorphism in the $2$-topos of groupoids, $G$ is the weak colimit of the corresponding Cech diagram ...

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620 views

### How to compute homology of symmetric products of complexes?

First, I would appreciate references on the notion of derived symmetric powers of perfect modules over various kinds of derived commutative algebras (say cdgas in characteristic zero, simplicial ...

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313 views

### Where can I see the proof that the homology groups of the Moore Complex of a simplicial group coincide with the homotopy groups of its geometric realization?

I'd like to know where to find it since it's very used in the articles of Loday and others.

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221 views

### When does $M \otimes_{A} \pi_{0}(A) \simeq 0$ imply $M \simeq 0$?

Let $A$ be a simplicial commutative ring over a field $k$ of characteristic zero (or a cdga
in non-positive degrees with differential of degree -1). Let $M$ be a perfect $A$ module. If necessary, ...

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**0**answers

114 views

### Simplicial morphisms from degenerate simplices

Frequently I read the following statement on morphisms of simplicial sets:
Suppose $X$ and $Y$ are simplicial sets, $f: X \rightarrow Y$ is a simplicial morphism and we write $X_n$ as well as $Y_n$ ...

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259 views

### Simplicial approximation for simplicial spaces

Given two simplicial topological spaces $X_{\bullet}$ and $Y_{\bullet}$ (i.e. a simplicial object in Top) and a continuous map between their geometric realizations $f \colon \lvert X_{\bullet} \rvert ...

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387 views

### (Co)homological characterization of homotopy pullbacks

For a commutative square of spaces (of manifolds, or of simplicial sets):
$$S=\left(\begin{array}{ccc} A & \to & B \newline \downarrow & & \downarrow \newline C & \to & ...

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315 views

### Asymptotics for the number of triangulations of a manifold M

In Gromov's talk at the Clay Math Research from 23:23 to 25:55 Gromov says (slightly paraphrased)
I want to emphasize a problem which
comes from mathematical physics which
is unsolved which is ...

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**1**answer

298 views

### Coboundary map on the cochain complex of abelian cosimplicial groups?

Maybe I'm looking at the wrong places, but I can't find a definition of
the coboundary map on the cochain complex of abelian cosimplicial groups.
What I have in mind is something similar to the ...

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**1**answer

313 views

### Extend Alexander-Whitney and Eilenberg-Zilber map to n-fold tensor products

See the definition of the Alexander-Whitney transformation:
http://ncatlab.org/nlab/show/Alexander-Whitney+map
and the Eilenberg-Zilber transformation:
...

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**1**answer

391 views

### Cech nerve as homotopy colimit?

Given a category $\mathcal{C}$ with a notion of covering $\{ U_{i} \rightarrow X \}$ for an object $X$ (say $\mathcal{C}$ is a Grothendieck site), we can form the Cech nerve
$$ \cdots ...

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**1**answer

578 views

### Computing homotopy (co)limits in a nice simplicial model category?

I'm trying to learn about and compute homotopy (co)limits. Specifically, if $\mathcal{C}$ is some Grothendieck site and $\mathcal{P}$ the simplicial model category of simplicial presheaves (say with ...

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626 views

### Necessity of hypercovers for sheaf condition for simplicial sheaves

I'm trying to understand where the definition of simplicial sheaf on a space/site comes from.
For a presheaf $F$ of sets on a topological space $X$, the sheaf condition can be viewed as saying that ...

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**1**answer

348 views

### Detecting equivalences of (infinity) categories by nerves

I have two questions:
Is there a way to tell if a functor $F:C \to D$ between two small categories is an equivalence in terms of the map $$N(F):N(C) \to N(D)$$ between simplicial sets? More ...

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403 views

### Simplicial presheaves enriched, tensored and cotensored over simplicial sets correctly?

Let $\mathcal{C}$ be a category and $\mathcal{P}=Functors(\mathcal{C}^{op},\mathcal{S})$ the category of simplicial presheaves, where $\mathcal{S}=sSet$. I want $\mathcal{P}$ to be enriched, tensored, ...

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461 views

### What is the chromatic number of the graph whose vertices dimension $n-2$ subsimplicies of $\Delta^n$ and an edge between two vertices is given if the two associated $n-2$ vertices are contained in the same $n-1$ subsimplex?

Let us first remark that all of this takes place on the boundary of $\Delta^n$. The question that I wanted to solve that led to the question in the title is as follows: Let ...

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**1**answer

394 views

### Persistent homology of Markovian dynamical systems

Consider a dynamical system $(T,X)$ that admits a Markov partition $\mathcal{M}$ (e.g., an Anosov map), and consider the corresponding 0-1 transition matrix $A$. It is commonplace to study information ...

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**1**answer

312 views

### Writing an infinity groupoid as a colimit of sets

If we are given a simplicial set $X:\Delta^{op} \to Set$, we may regard it as a \emph{simplicial} simplicial set, i.e. a bisimplicial set by composing with the "constant" inclusion $Set \to ...

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**1**answer

343 views

### Extending the definition of “pure of dimension n” from simplicial complexes to simplicial sets?

Recall that a (combinatorial) simplicial complex $X$ is said to be $n$-dimensional if it contains at least one face of dimension $n$ and no faces of dimension $n+1$. Further, an $n$-dimensional ...

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**1**answer

208 views

### On Lifts in Kan Simplicial Sets

In a Kan simplicial set $X_\bullet$ we have the lifting property, that is for
any $n$-tupel $\left(x_0,\ldots,\hat{x_j},\ldots,x_n \right)$ of $(n-1)$-simplices $x_k \in X_{n-1}$ with ...

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119 views

### Certain (pre) simplicial sets have torsion-free homology?

Suppose $X$ is a pre-simplicial set defined on a finite vertex set $\{v_1, v_2, \ldots, v_k\}$ recursively as follows:
Let $X_1$ have a single vertex $\{v_1\}$.
For $i > 1$, $X_i$ is obtained ...

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457 views

### Does the category of strict $2$-categories together with Dwyer-Kan equivalences provide a model for $(\infty,1)$-categories?

The question is the title.
In what follows, all $2$-categories and $2$-functors will be strict. Let $2-Cat$ denote the categories whose objects are $2$-categories and whose morphisms are ...

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577 views

### Examples of Simplicial Groupoids in Nature

For me, a simplicial groupoid is a simplicial object in ${\mathbf{Grpd}}$. I am more general than Goerss-Jardine in this definition.
Do you have examples simplicial groupoids that occur in nature? ...

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296 views

### Is there a dual notion for the Nerve functor?

Say in the most classical case, we probe a topological space $X$ by the n-simplices $\Delta^n$ by using the nerve functor $Hom_{Top}(-,X)$. Is another functor $Hom_{Top}(X,-)$ of any use, or is there ...

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252 views

### The boundary map of Kan simplicial sets

I don't know whether or not the following is a research level question,
but since it is concerned with simplicial sets and since they
are very popular these days, I think this is the right place to ...

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**1**answer

257 views

### Is the nerve of a category a fully faithfully functor up to homotopy?

Let $F,G:C\to D$ be naturally isomorphic functors. Taking the nerve, is $NF,NG:NC\to ND$ homotopy equivalent? Conversely, given a simplicial map $f:NC\to ND$, does there exists a functor $F:C\to D$ ...

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221 views

### Is the nerve of a groupoid an EM space?

Let $G$ be a connected groupoid. Is the nerve $BG$ a $K(\pi, 1)$, and if so, is there a groupoid homomorphism $f:G\to \pi$ that induces the homotopy equivalence?

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### Is the class of inner-anodyne morphisms right-cancellative with respect to the of the class of monomorphisms?

Recall:
Given a category $A$, and two classes of morphisms $S,S'$, we say that $S$ is right-cancellative with respect to $S'$ if for any pair of maps $f\in S, g\in S'$ such that $gf$ is defined, we ...

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187 views

### Proof for a simplicial morphism

Sorry for the title but I can't come up with another one...
...
Suppose X and Y are simplicial sets
and $f : X \rightarrow Y$ is a map
such that:
1.) f maps dimensions the right way,
that is ...

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2k views

### What is a symmetric monoidal $(\infty,n)$-category?

This question arose from reading Jacob Lurie's "Classification of topological field theories" paper. In that paper, he uses complete $n$-fold Segal spaces as a model for $(\infty,n)$-categories, but ...

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247 views

### Left adjoints to several inclusions of homotopy simplicial model categories

The left adjoint to the inclusion $sGrp\hookrightarrow sPtSet$ of the category of simplicial groups into the category of pointed simplicial sets is homotopy equivalent to the loop suspension functor ...

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655 views

### Do simplicial objects in a Topos form a model category?

Sometimes people say "If you don't like the word 'topos', just think the category of Sets", but I'm not sure to what extent this analogy holds.
The real question here is, do simplicial object in a ...

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192 views

### The Mayer-Viertoris exact sequence as a (Zariski) descent spectral sequence.

For certain 'spaces' $U,V$ (they are certain Henselizations of subvarieties) I would like to compute (certain etale) cohomology of $U\cup V$ in terms of the corresponding cohomology of the diagram ...

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491 views

### Decomposition of inner horns

Hi guys, I have a question. Prove or disproof the statement:
Any inner horn $\Lambda[n,k],0< k< n $ admits a filtration $\mathbf{n}<\cdots<\Lambda[n,k]$, such that each step is filling an ...

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1k views

### Semi-simplicial versus simplicial sets (and simplicial categories)

Hi,
Let's denote by "semi-simplicial set" a simplicial set without degeneracies, i.e. a contravariant functor from the category $\Delta_{inj}$ of finite linearly ordered sets and order preserving ...

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### If I want to study Jacob Lurie's books “Higher Topoi Theory”, “Derived AG”, what prerequisites should I have?

I've been told that it's important to know modern physics, Differential Geometry and Algebraic Topology for understanding higher structures. Is there any other prerequisite for understanding Lurie's ...

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**1**answer

414 views

### The weak equivalences in the covariant model structure

Let $S$ be a simplicial set. Recall that there is a model structure, called the covariant model structure (see HTT ch. 2 and this question), on $\mathbf{SSet}/S$ such that:
The cofibrations are the ...

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**1**answer

494 views

### What does the “category” of $(\infty,1)$ category look like.

One knows that in higher category theory, the category of $(\infty,n-1)$ categories is naturally an $(\infty,n)$ category ,(I use the word category to mean category in the correct weakened sense). ...

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341 views

### A “join” of ω-categorical simplices

Recall that the category of level trees $\mathcal{T}$ is defined to be the category $[\mathbb{N}^{op},\Delta_a]$, where $\Delta_a$ is the skeleton of the category of finite possibly empty linearly ...

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**1**answer

448 views

### When can a contractible 2-complex be embedded in R^3?

Let $X$ be a contractible 2-dimensional simplicial complex. Are there nice necessary and sufficient conditions for $X$ to be embeddable in $\mathbb R^3$? Clearly it is necessary that the link of ...

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### The reflexive free-category comonad-resolution is a cofibrant replacement of the discrete simplicial category associated with an ordinary category in the Bergner model structure on the category of small simplicial categories?

Let $X$ be the category of reflexive quivers, and let $Cat$ be the category of small categories. There exists an evident forgetful functor $U:Cat\to X$ sending a category $A$ to its underlying ...

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354 views

### $N$-step simplicial complexes

Recently, answering a question here, Dror Bar-Natan observed that «way too often two-step complexes have a natural extension to become many-step complexes». By such a thing I mean (and I think Dror ...