The simplicial-stuff tag has no usage guidance.

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**1**answer

442 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 ...

**8**

votes

**1**answer

634 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 ...

**12**

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**3**answers

696 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

363 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

427 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**

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**1**answer

466 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 ...

**4**

votes

**1**answer

420 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**

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**1**answer

330 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 ...

**2**

votes

**1**answer

355 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**

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**1**answer

209 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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**0**answers

120 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 ...

**11**

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**0**answers

496 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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votes

**2**answers

609 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

301 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**

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**1**answer

256 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

269 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**

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**1**answer

222 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**

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**0**answers

206 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**

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**0**answers

188 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**

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**0**answers

251 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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**2**answers

681 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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**0**answers

194 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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**4**answers

498 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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votes

**2**answers

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 ...

**30**

votes

**1**answer

13k views

### 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 ...

**11**

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**1**answer

422 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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votes

**1**answer

497 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). ...

**4**

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**1**answer

348 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 ...

**13**

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**1**answer

477 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 ...

**12**

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**1**answer

881 views

### 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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**2**answers

362 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 ...

**1**

vote

**1**answer

159 views

### Is there a linear embedding of a simplical 3-complex in R^6?

I've heard that there always is an embedding in $R^7$ (can someone provide a reference for that?) and this number cannot be lowered in general. But I'm interested in a somewhat special case, namely: ...

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vote

**1**answer

568 views

### Nerves of simplicial objects in categories/Waldhausen's S-construction

Is there a good nerve-like functor from simplicial objects in categories to simplicial sets which takes level-wise equivalences of categories to weak equivalences?
To give this some context, I'd ...

**11**

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**1**answer

850 views

### Is the simplicial completion of a localizer always a bousfield localization of the injective model structure?

Background
Recall (from Cisinski's Astérisque volume 308) that given a small category $A$, we define an $A$-localizer to be a class $W$ of morphisms of $\mathrm{Psh}(A)$ satisfying the following ...

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votes

**2**answers

764 views

### Gluing of manifolds and the Hausdorff condition.

Hi!
I apologize in advance if this question is better fit for http://math.stackexchange.com/.
Out of curiosity I'm interested in a particular case of the problem of what properties of a manifold is ...

**2**

votes

**1**answer

226 views

### Induced pretopologies on sSet

Recall that the geometric realisation functor $| - |: sSet \to Top$ preserves products (choosing $Top = k Space$ or similar). Thus any given singleton Grothendieck pretopology on $Top$ gives rise to a ...

**1**

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**1**answer

322 views

### Do bistellar flips preserve shellability?

I notice there is a strong connection between shellability of simplicial complexes and bistellar flips on these complexes; in particular, adding in a new facet of a shelling induces a bistellar flip ...

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votes

**2**answers

2k views

### What is the homotopy theory of categories?

I've heard that Grothendieck, in his letter "Pursuing Stacks," wanted to find alternative models for the classical homotopy category of CW complexes and continuous maps (up to homotopy), and one of ...

**2**

votes

**0**answers

304 views

### simplicial deRham complex and model category structure

To every simplicial manifold is associated its simplicial deRham complex.
Is there any literature that discusses explicitly to which extent this classical construction, regarded as a (contravariant) ...

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votes

**2**answers

520 views

### Collinear vertices and definition of k-simplex

On page 120 of his Basic Topology, Armstrong defines the $k$-simplex in $\mathbb{E}^n$ with verices $v_0,\ldots,v_k$ to be complex hull of said vertices. (A similar definition is given on Wikipedia).
...

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**2**answers

313 views

### Simplicial presheaves that are colimits of themselves?

Suppose $C$ is a small category and $X_{\bullet}$ is a simplicial object in $C$. In particular, by composing with Yoneda $$y:C \to Set^{C^{op}}$$ $y(X)_{\bullet}$ is a simplicial presheaf. I believe ...

**3**

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**1**answer

567 views

### Question on the interpretation of a presheaf category as a co-completion

The category of presheaves $Pre(C)$ on a small category $C$ is the category of functors $C^{op}\to Sets$. Since the category of sets is co-complete and every presheaf is a colimit of representable ...

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**5**answers

6k views

### Is there a high-concept explanation for why “simplicial” leads to “homotopy-theoretic”?

My (limited) understanding is that simplicial methods tend to be used whenever you want some kind of nontrivial homotopy theory -- for instance, to get a nice model structure, you use simplicial sets ...

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**4**answers

918 views

### Degeneracies for semi-simplicial Kan complexes

By a semi-simplicial set I mean a simplicial set without degeneracies. In such a thing we can define horns as usual, and thereby "semi-simplicial Kan complexes" which have a filler for every horn. ...

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**1**answer

1k views

### Analogue of simplicial sets

This question is prompted by this one (and some of the comments that it drew).
Simplicial complex is to ordered simplicial complex as $X$ is to simplicial set. The question is about $X$.
Let ...

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**2**answers

763 views

### Is there a discrete Cerf theory?

Towards the end of the 1990's, Robin Forman developed a discrete version of Morse theory, which concerns maps from a simplicial complex to $\mathbb{R}$ satisfying a combinatorial analogue to the ...

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**2**answers

573 views

### Historical Question about Simplicial Sets

I have a pretty easy historical question about simplicial sets. Unless I am mistaken, simplicial sets first came out of topology, explicitly from combinatorial topology and the study of simplicial ...

**8**

votes

**1**answer

384 views

### delooping under Dold-Kan and simplicial delooping

What maps of simplicial sets exist between
the image under the Dold-Kan correspondence of a chain complex shifted up in degree
and the image under the right adjoint to simplicial looping of the ...

**4**

votes

**1**answer

244 views

### Artin-Mazur codiagonal preserves Kan objects?

The Artin-Mazur codiagonal $\nabla:ssSet \to sSet$ is right adjoint to the total decalage functor $Dec:sSet \to ssSet$. The total decalage functor is defined to be precomposition with the ordinal sum ...