The simplicial-stuff tag has no wiki summary.

**21**

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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 ...

**9**

votes

**1**answer

374 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 ...

**6**

votes

**1**answer

478 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**

votes

**1**answer

325 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

402 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 ...

**10**

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**1**answer

771 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

339 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**

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**1**answer

153 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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**2**answers

527 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 ...

**9**

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**1**answer

730 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 ...

**4**

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97 views

### What is known about the number of permissible simplicial complexes given the number of k-cells?

Motivation: I am working on a problem that reduces to finding simplicial complexes given some data (details are unnecessary), but all I have managed to wrangle from my input is the number of cells of ...

**8**

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**2**answers

702 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

222 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 ...

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**1**answer

284 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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1k 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**

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**0**answers

287 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) ...

**0**

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**2**answers

436 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

306 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

440 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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4k 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 ...

**10**

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**4**answers

768 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. ...

**16**

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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 ...

**17**

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**2**answers

708 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

531 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 ...

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**1**answer

346 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**

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**1**answer

234 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 ...

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**2**answers

485 views

### Inner hom and geometric realization.

I would like to prove the following fact, which I learned from a previous MO question.
Let $S_\cdot,T_\cdot\in\mathbf{sSET}$ be simplicial sets, and assume that $T_\cdot$ is Kan. Then there is a ...

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**1**answer

438 views

### Nerve: Groupoids-> Kan Complexes. Nerve: Bicategories w. adjoints -> ?

If you take the nerve of a groupoid, you get a Kan complex.
Question:
Take a bicategory that has adjoints for 1-morphisms, which is one notion of 'weak' groupoid (if all 2-morphisms are ...

**3**

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**0**answers

136 views

### Is the homotopy of a primitively generated Hopf algebra still primitively generated?

Let $A=\oplus A_n$ be a primitively generated graded Hopf algebra, where each $A_n$ is a simplicial group. This allows us to define the homotopy group $\pi_*(A)$.
Question: is the graded Hopf algebra ...

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**2**answers

326 views

### Algorithm that decreases the size of the simplicial triangulation

Hello!
Let X be a topological space. We are considering only abstract simplicial complexes, i.e. a finite list of finite lists of integers. Is there any algorithm (more or less efficient?), that ...

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**1**answer

374 views

### Does this “flipping lexicographic” ordering have a standard name?

I’ve run into the following straightforward variant of lexicographic ordering, and am wondering if it has a standard name. I’ve been calling it the flipping lexicographic ordering, for evident ...

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292 views

### A combinatorial approximation functor sSet->qCat

Let $sSet_J$ denote the category of simplicial sets equipped with the Joyal model structure. Simply by the fact that $sSet_J$ is locally presentable and its class of anodynes ($\neq \mathbf{Cof} \cap ...

**3**

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**2**answers

326 views

### Model categories of simplicial objects

If $\mathcal{C}$ is a category, then surely the category of simplicial
objects $s\mathcal{C}$ is not automatically a model category. What conditions
must $\mathcal{C}$ satisfy in order for ...

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2k views

### History of classifying spaces

Where did the idea and formal definition of the "classifying space of a (small) category" first appear?
Added: As Andy Putman noted below, the "classical" early reference for this is G. Segal's ...

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**0**answers

58 views

### maps of oo-stacks inducing monomorphisms on all homotopy sheaves

What is known about factoring morphisms of simplicial (pre)sheaves/$\infty$-sheaves through maps that induce monomorphisms on all homotopy sheaves? Is there any useful theory for that?

**5**

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**1**answer

945 views

### Geometric Realization of a Simplicial Category

Let $S:\varDelta^{op}\to (cat)$ be a functor where the category on the right is the category whose objects are categories with cofibrations and morphisms are exact functors(from Waldhausen's paper, ...

**3**

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**1**answer

190 views

### Classifying space of variant on category of simplices

This question might have an easy answer, but my research is far from the region of topology that makes use of classifying spaces of categories, so I can't find it.
For (possibly infinite) integers $0 ...

**4**

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**4**answers

2k views

### How to triangulate real projective spaces (as simplicial complexes in Mathematica)?

Hello!
I have written a program in Mathematica 7, which calculates for a (finite abstract) simplicial complex all its homology groups. I would really like to test it on the projective spaces, but ...

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**0**answers

146 views

### What's the best way to test if a sphere is a polytope? (algorithms for the Simplicial Steinitz Problem)

The problem of recognizing whether a simplicial face lattice is polytopal is sometimes called the Steinitz problem.
Sturmfels and Bokowski advanced a set of methods in the late 80s to test whether ...

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420 views

### Historical and terminological questions about Dan Kan's Ex functor and its relation to the classical case of simplicial complexes

Recall that we may define a functor $\xi:\Delta\to \operatorname{Poset}$ sending a simplex $[n]$ to the set of monotone injections $[k]\hookrightarrow [n]$ for $k\geq 0$ (effectively, $k\leq n$ as ...

**2**

votes

**1**answer

212 views

### Local injective model structure for simplicial presheaves

The category of simplicial presheaves on a small Grothendieck site $\mathcal{C}$ can be given a model structure by defining weak equivalences and cofibrations sectionwise. It's called the (global) ...

**4**

votes

**1**answer

229 views

### Compatibility of classifying space with inner-hom?

Let $\mathbf{sTop}$ be the functor category $\mathbf{Top}^{{\mathbf{\Delta}}^{\textit{op}}}$, and let $\mathbf{sCat}$ be the functor category
$\mathbf{Cat}^{{\mathbf{\Delta}}^{\textit{op}}}$, and let ...

**5**

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**1**answer

744 views

### Fibered/cofibered higher categories, relative model structures, slicing, and (∞,2)-category theory

Jacob Lurie defined a model structure on the category of marked simplicial sets sliced over a fixed simplicial set $S$ called the cartesian model structure. (For a definition, see here or HTT ...

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**2**answers

477 views

### From chain complex to simplicial abelian group

In many places, I have seen the slogan that "simplicial abelian group = chain complexes of abelian groups". These same sources usually tell me how to go in one direction. Namely, given a simplicial ...

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740 views

### Hodge star and harmonic simplicial differential forms

Is there a notion of harmonic forms and Hodge theory for Sullivan's piecewise smooth differential forms on a simplicial set?
Let me recall some background.
Hodge Theory on a Riemannian manifold
A ...

**4**

votes

**1**answer

623 views

### Simplicial “universal extensions”, the hammock localization, and Ext

Let $M,B$ be $R$-modules, and suppose we're given an n-extension $E_1\to\dots\to E_n$ of $B$ by $M$, that is, an exact sequence $$0\to M\to E_1\to\dots\to E_n \to B\to 0.$$
A morphism of ...

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**4**answers

1k views

### Model structure on Simplicial Sets without using topological spaces

The category of simplicial sets has a standard model structure, where the weak equivalences are those maps whose geometric realization is a weak homotopy equivalence, the cofibrations are ...

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**3**answers

606 views

### Triangulations of polyhedra

A topologist came to me with this question, but everything I think should work doesn't.
How many triangulations are there of a polyhedron with n vertices?
By a "triangulation" of a polyhedron P we ...

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**1**answer

1k views

### Formally smooth morphisms, the cotangent complex, André-Quillen cohomology, and representability of nilpotent extensions as trivial extensions over a cofibrant replacement

Recall that an $R$-algebra $R\to S$ is called formally smooth (resp. formally unramified resp. formally étale) if given any lifting problem of the form
$$\begin{matrix}
R&\to &T\\
...

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**3**answers

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### What facts in commutative algebra fail miserably for simplicial commutative rings, even up to homotopy?

Simplicial commutative rings are very easy to describe. They're just commutative monoids in the monoidal category of simplicial abelian groups. However, I just realized that a priori, it's not clear ...