# Tagged Questions

The simplicial-stuff tag has no usage guidance.

**2**

votes

**2**answers

1k views

### The Area of Spherical Polygons

I am interested in finding a canonical general expression for the area of a spherical polygon in $\mathbb{S}^2$ knowing the side lengths of the polygon and a bound on the internal angles (we can ...

**6**

votes

**2**answers

436 views

### Necessary conditions for cofibrancy in global projective model structure on simplicial presheaves

Consider the global projective model category
of simplicial presheaves on some category
(the category of smooth manifolds is particularly interesting to me).
In Section 9.1 of Dugger's paper ...

**4**

votes

**0**answers

229 views

### Extending the vertex-facet correspondence from Δ to Θ

Recall that in the $n$-simplex $\Delta[n]$, we have a combinatorially crucial bijection between facets, (codimension $1$ faces) and vertices, where the $i$th face of a simplex corresponds to the full ...

**2**

votes

**0**answers

220 views

### On small generators of an infinity category

Suppose that $\mathcal{C}$ is an $\infty$-category with pullbacks and small coproducts. Suppose that $\mathcal{D}$ is a small subcategory for every object $C,$ the canonical map ...

**1**

vote

**1**answer

111 views

### Generalizing the internal angle of a graph in $\mathbb{E}^2$ to $\mathbb{S}^2$

I am currently working on research involving packing problems and am finding myself needing the tools from Combinatorial Geometry (in particular, I've been reading Pach and Agarwal's book on the ...

**7**

votes

**1**answer

766 views

### Derived functors of symmetric powers

What do the derived functors of the symmetric powers look like? I understand that this is related to the homology of the symmetric groups, but I don't know a reference for that.
Namely, I'm ...

**4**

votes

**1**answer

294 views

### Generalizing the circle packing theorem to 3-dimensions

The circle packing theorem (Koebe–Andreev–Thurston theorem) states that every finite planar graph is the nerve of some disk packing in the plane, where the nerve of a packing $P$ is a graph $G=(V,E)$, ...

**7**

votes

**1**answer

272 views

### What is the homotopy type of a free simplicial ring?

Is there a good description of the homotopy type of a free simplicial ring (or simplicial $R$-algebra) on a given simplicial set, in terms of the homotopy type of that simplicial set?
(This is mostly ...

**8**

votes

**2**answers

660 views

### Connective spectra versus simplicial abelian groups - very basic question

Hello,
I have very general , "introductory" questions (It is quite hard for me to seek for specific things in the algebraic topology literature).
I guess that connective spectra have a model ...

**5**

votes

**2**answers

616 views

### Reference request - CDGA vs. sAlg in char. 0

Hello,
Are the model categories of simplicial commutative algebras over $k$ and that of commutative differential graded algebras (in negative cohmological dimension) Quillen equivalent in char. 0 (or ...

**3**

votes

**1**answer

153 views

### 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 $\Delta$ ...

**2**

votes

**1**answer

177 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 ...

**1**

vote

**0**answers

112 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?
...

**13**

votes

**3**answers

953 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 ...

**1**

vote

**2**answers

523 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 ...

**4**

votes

**0**answers

236 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 ...

**6**

votes

**2**answers

668 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 $\lVert X \rVert$ ...

**2**

votes

**1**answer

207 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 ...

**3**

votes

**2**answers

337 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 ...

**8**

votes

**2**answers

775 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 ...

**1**

vote

**2**answers

361 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.

**3**

votes

**2**answers

224 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, ...

**1**

vote

**0**answers

121 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 $Y_n$ ...

**3**

votes

**2**answers

279 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 ...

**5**

votes

**4**answers

408 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 & ...

**14**

votes

**0**answers

338 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 ...

**1**

vote

**1**answer

403 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 ...

**4**

votes

**1**answer

377 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:
...

**4**

votes

**1**answer

513 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 ...

**9**

votes

**1**answer

740 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 ...

**14**

votes

**3**answers

781 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 ...

**6**

votes

**1**answer

391 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 ...

**3**

votes

**1**answer

446 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, ...

**3**

votes

**1**answer

472 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 ...

**5**

votes

**1**answer

470 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 ...

**4**

votes

**1**answer

359 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 ...

**3**

votes

**1**answer

362 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 ...

**1**

vote

**1**answer

213 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 ...

**1**

vote

**0**answers

121 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 ...

**13**

votes

**0**answers

549 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 ...

**2**

votes

**2**answers

646 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? ...

**1**

vote

**1**answer

309 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 ...

**1**

vote

**1**answer

262 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 ...

**2**

votes

**1**answer

284 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 $F:C\to D$ ...

**1**

vote

**1**answer

223 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?

**4**

votes

**0**answers

209 views

### 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 ...

**1**

vote

**0**answers

190 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 ...

**12**

votes

**2**answers

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 ...

**5**

votes

**0**answers

255 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 ...

**8**

votes

**2**answers

742 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 ...