# Questions tagged [simplicial-stuff]

For questions about simplicial sets, simplicial (co)algebras and simplicial objects in other categories; geometric realization, Dold-Kan correspondence, simplicial resolutions etc.

**3**

votes

**1**answer

53 views

### Topological realisation of a stack (explicit description)

Let $X$ be an Artin stack over $k=\mathbf{C}$. I've heard that $X$ has a topological realisation, but I've not been able to find an explicit decription.
My first guess would be: take a smooth cover $...

**1**

vote

**0**answers

59 views

### A homotopy problem for morphisms of dg-algebras

Let $\mathfrak g$ be a real finite-dimensional Lie algebra, and suppose we are given two morphisms of dg-algebras $f,g: (C^\bullet(\mathfrak g),d_{CE}) \to (\Omega^\bullet(\Delta^n),d)$, such that ...

**6**

votes

**0**answers

120 views

### Does Ex^∞ send homotopy inverse limits of ∞-categories to homotopy inverse limits of spaces

Kan's functor $\operatorname{Ex}^{\infty}$ plays the role of total localization or groupoid completion in the theory of $\infty$-categories; specifically, it can be viewed as the left-adjoint of the ...

**6**

votes

**0**answers

95 views

### Extension of sheaves of $\infty$-algebras

Let
$(\mathcal{C},\tau)$ be a site, whose Grothendieck topology is $\tau$
$F : \mathcal{C}\to D(k)$ a sheaf of $\infty$-$k$-algebras, with $k$ a ring and $D(k)$ the derived category of $k$-modules.
...

**12**

votes

**2**answers

468 views

### Why is Kan's $Ex^\infty$ functor useful?

I've always heard that Kan's $Ex^\infty$ functor has important theoretical applications, but the only one I know is to show that the Kan-Quillen model structure is right proper. What else is it useful ...

**2**

votes

**0**answers

45 views

### Simplicial models for mapping spaces of filtered maps

Let $S$ and $K$ be simplicial sets with $K$ Kan. Given simplicial sets $S$ and $K$, we let $SIMP(S,K)$ be the internal hom in the category of simplicial sets. Hence $SIMP(S,K)$ is Kan.
Suppose that ${...

**3**

votes

**1**answer

59 views

### Simplicial models for fibrations between mapping spaces

Let $S,T$ and $K$ be simplicial sets with $K$ Kan. Given simplicial sets $S$ and $K$, we let $SIMP(S,K)$ be the internal hom in the category of simplicial sets. We let $|S|$ denote the geometric ...

**4**

votes

**1**answer

142 views

### Monad, algebras and reflexive coequalizer

Suppose we have an adjunction of categories $F:M\leftrightarrows N:U$. We define the associated (co)monad $G=F\circ U$. For any object $x\in N$ we define the simplicial resolution of $x$ given by
$$
...

**6**

votes

**1**answer

153 views

### Explicit chain homotopy for the Alexander-Whitney, Eilenberg-Zilber pair

Let $A$ and $B$ be simplicial abelian groups, and let $N_\ast(-)$ denote the normalized chain complex functor. Let
$$AW_{A,B}\colon N_\ast(A\otimes B)
\longrightarrow N_\ast(A)\otimes N_\ast(B)$$
and
...

**3**

votes

**0**answers

97 views

### On the existence of nice hypercovers

Let $\mathcal{C}$ be a site and $X$ a sheaf of sets on $\mathcal{C}$.
Then there exists a hypercover $K_{\bullet}$ of $X$ such that $K_n$ is a coproduct of representable presheaves on $\mathcal{C}$ ...

**2**

votes

**0**answers

36 views

### Contiguity for simplicial maps between simplicial sets

I begin by recalling the definition of contiguous simplicial maps between abstract simplicial complexes:
Definition. Two simplicial maps $\varphi,\psi\colon K \to L$ are said to be contiguous if for ...

**6**

votes

**1**answer

125 views

### Simplicial localization of the cofibrant-fibrant objects

Let $M$ be a model category. I don't assume that $M$ has functorial factorizations or that $M$ is simplicial. Write $M^{c}$ (respectively, $M^{cf}$) for the full subcategory of $M$ on the cofibrant ...

**7**

votes

**1**answer

236 views

### Does the cubical nerve preserve weak equivalences of simplicial sets?

The finite cartesian powers of $\Delta[1]$ form a cubical object in simplicial sets, inducing a "cubical nerve" functor $N_\Box: sSet \to Set^{\Box^{op}}$.
$N_\Box$ is a right Quillen equivalence, ...

**3**

votes

**0**answers

169 views

### Is there a natural homotopy inverse to the map $∣Sing(X)∣\rightarrow X$

Let $Sing:C \rightarrow SSet$ be the functor sending a CW complex to its singular complex (a simplicial set).
Let $∣-∣: SSet \rightarrow C$ be the geometric realization functor.
For every $X$ we ...

**8**

votes

**1**answer

172 views

### Simplicially enriched cartesian closed categories

In this question I asked whether for a complete and cocomplete cartesian closed category $V$, there can be a complete and cocomplete $V$-category $C$ (with powers and copowers) whose underlying ...