The simplicial-stuff tag has no usage guidance.

**6**

votes

**2**answers

439 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 ...

**50**

votes

**5**answers

7k 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 ...

**6**

votes

**1**answer

992 views

### Homotopy pullbacks of simplicial spaces, and Bousfield-Friedlander

Let $X_\bullet \longrightarrow Y_\bullet \longleftarrow Z_\bullet$ be a diagram of simplicial spaces (=bisimplicial sets, if you like).
On p. 14-9 of these notes there is an example which shows that ...

**10**

votes

**2**answers

546 views

### Is the geometric realization of a level-wise weak equivalence a weak equivalence?

For the purposes of this question a topological space will mean a compactly generated weak Hausdorff space, though I am actually somewhat flexible on what category of topological spaces we use. I ...

**4**

votes

**1**answer

477 views

### When is homotopy orbit space weakly equivalent to orbit space, other than situation of free action?

Let $M$ be a closed symmetric monoidal model category. Let $X$ be a cofibrant object (it can also be fibrant if you like) and let $\Sigma_n$ act on $X^{\otimes n}$ by permuting the factors (note that ...

**30**

votes

**7**answers

3k views

### Simplicial objects

How should one think about simplicial objects in a category versus actual objects in that category? For example, both for intuition and for practical purposes, what's the difference between a ...

**18**

votes

**6**answers

2k views

### A canonical and categorical construction for geometric realization

There is a very intimate connection between categories, simplicial sets, and topological spaces. On one hand, simplicial sets are the presheaf category on the category $\Delta$ and $\Delta$ is a ...

**19**

votes

**2**answers

650 views

### How many simplicial complexes on n vertices up to homotopy equivalence?

Fix a number $n$, and define $\gamma(n)$ to be the number of simplicial complexes on $n$ unlabeled vertices up to homotopy equivalence. It is unlikely that an explicit formula exists, but what is ...

**11**

votes

**3**answers

1k views

### What are the fibrant objects in the injective model structure?

If C is a small category, we can consider the category of simplicial presheaves on C. This is a model category in two natural ways which are compatible with the usual model structure on simplicial ...

**18**

votes

**2**answers

802 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 ...

**10**

votes

**1**answer

535 views

### Three questions on $\operatorname{hocolim}$

I posted this on math.stack.exchange but didn't get a helpful response, so please let me try it here.
Let $D$ be a small category and $F:D\to sSets$ a functor.
There is a bisimplicial set indicated ...

**10**

votes

**1**answer

624 views

### Why does the internal singular simplicial space realize to the same thing as the discrete singular simplicial set?

There are two version of the singular simplicial space of a topological space $X$, one discrete and one internal. At least if X is nice, both of them have homotopy equivalent geometric realizations ...

**9**

votes

**2**answers

2k 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 ...

**10**

votes

**1**answer

654 views

### Why does the singular simplicial space geometrically realize to the original space?

I have seen it claimed that (for compactly generated Hausdorff spaces) the geometric realization of the singular (internal) simplicial space is homotopy equivalent to the original space. I know how to ...

**3**

votes

**2**answers

891 views

### Mayer-Vietoris homotopy groups sequence of a pull-back of a fibration

This must be an elementary question: could somebody tell me a reference for the Mayer-Vietoris homotopy groups sequence of a pull-back of a fibration?
I'm working in the category of pointed ...

**16**

votes

**3**answers

1k views

### Testing simplicial complexes for shellability

Question
Are there efficient algorithms to check if a finite simplicial complex defined in terms of its maximal facets is shellable?
By efficient here I am willing to consider anything with ...

**12**

votes

**2**answers

526 views

### Exponentiation in finite simplicial sets

A finite simplicial set is a simplicial set having only a finite number of non degenerate simplicies. My question is: if $A$ and $B$ are finite simplicial sets, does this imply that the simplicial set ...

**12**

votes

**1**answer

914 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 ...

**8**

votes

**1**answer

333 views

### Reducing the simplices in the nerve of a category with an object with trivial endomorphism monoid

Let $C$ be a category with an object $X$ such that there are no non-trivial endomorphisms $X\rightarrow X$. Consider a simplex $\sigma$ of the nerve $NC$ of $C$. It is just a string of composable ...

**7**

votes

**1**answer

490 views

### Can we invert barycentric subdivision?

With apologies to fellow algebraic topologists, I confess that I have no idea how to answer this innocent-looking question:
(1) Let's say we know that a finite simplicial complex $S$ is the ...

**5**

votes

**1**answer

307 views

### Descent properties of spaces

I am trying to make sense of what is written in Rezk's draft http://www.math.uiuc.edu/~rezk/i-hate-the-pi-star-kan-condition.pdf
In particular, I am referring to Proposition 2.3, which is there stated ...

**4**

votes

**0**answers

196 views

### Does the fat realization of simplicial spaces commute with finite limits up to homotopy?

I'm willing to work in the category of compactly generated Hausdorff spaces.
The ordinary geometric realization of a simplicial space commutes with taking pullbacks and preserves the terminal object, ...

**3**

votes

**3**answers

565 views

### Does the nerve functor (resp. fundamental groupoid functor) preserve homotopy colimits (resp. homotopy limits)?

Let $\pi _1:SS\to Grpd$ denote the fundamental groupoid functor, from simplicial sets to groupoids, and let $N:Grpd\to SS$ denote the nerve functor. Then $\pi _1$ is left adjoint to $N.$
On ...

**10**

votes

**3**answers

574 views

### References for the “nerve of an algebraic variety”

Let's do algebraic geometry over an arbitrary base ring $k$.
I've frequently seen a definition of the algebraic $n$-simplex, as follows:
$$\Delta^n = ...

**9**

votes

**2**answers

438 views

### Simplicial replacements in smoothing theory

As far as I can tell, ever since Milnor's Microbundles and differentiable structures (1961) paper, whenever people talk about $Diff(\mathbb R^n)$ or $PL(\mathbb R^n)$ or $Homeo(\mathbb R^n)$, they ...

**5**

votes

**2**answers

180 views

### What is this construction using iterated face maps of semisimplicial sets?

Let $X$ be a semisimplicial set (face maps but no degeneracy maps). Fix a positive integer $k$. Let $Y_n$ be $X_{(n+1)k}$ and then define $\partial^Y_i:Y_n\to Y_{n-1}$ by
$$\partial^Y_i = ...

**5**

votes

**1**answer

415 views

### Is every finite subcomplex of a contractible simplicial complex contained in a finite contractible subcomplex?

The question is as in the title:
Is every finite subcomplex of a contractible simplicial complex $K$ contained in a finite contractible subcomplex of $K$? What if we are allowed to take ...

**4**

votes

**1**answer

262 views

### Is there a class of simplicial sets whose weak homotopy type is preserved by symmetrization?

We have the categories, $S$, of simplicial sets and $SS$, of symmetric simplicial sets (whose simplices are unordered). There are functors:
$H:S\to SS$ forgetting the ordering on simplices and
...

**2**

votes

**2**answers

440 views

### Finite simplicial sets

A finite simplicial set is a simplicial set having only a finite number of non degenerate simplicies. It is not hard to show that every finite simplicial set has only a finite number of simplicies in ...