The simplicial-stuff tag has no wiki summary.

**6**

votes

**1**answer

785 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 ...

**34**

votes

**5**answers

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 ...

**17**

votes

**6**answers

1k 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 ...

**9**

votes

**3**answers

660 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 ...

**17**

votes

**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 ...

**7**

votes

**1**answer

365 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

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 ...

**8**

votes

**1**answer

299 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 ...

**13**

votes

**3**answers

513 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 ...

**5**

votes

**2**answers

287 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 ...

**3**

votes

**1**answer

348 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 ...