# 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.

**5**

votes

**0**answers

84 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

460 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

44 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

151 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

95 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

169 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 ...

**4**

votes

**1**answer

87 views

### Subdivision of simplicial sets but not the barycentric one!

Suppose $K$ and $L$ are simplicial sets. When should one consider that $K$ is a subdivision of $L$? I ask with a view to defining some notion of `finer' generalising that of 'finer triangulation' of ...

**3**

votes

**0**answers

114 views

### Why is the Straightening functor the analogue of the Grothendieck construction?

In classical category theory, there is the notion of functor (co)fibered in groupoids. Furthermore, via Grothendieck construction we have an equivalence between peseudo functors into the category of ...

**8**

votes

**0**answers

254 views

### About Kan-Thurston theorem

The Kan-Thurston theorem obsesses me recently, which is proved by Daniel Kan and William Thurston, Every connected space has the homology of a K(π,1), Topology Vol. 15. pp. 253–258, 1976. Later, there ...