# 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.

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### Simplicial set represented by an (unordered) set

Let $X$ be a (finite if you want) set and form the simplicial set $F^{\bullet}(X)$ with
$$
F^{n}(X) = \mathrm{Hom}_{\mathrm{set}} ([n], X)
$$
where the right hand side denotes arbitrary maps of sets (...

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votes

**0**answers

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### How exactly to adapt Brown's collapse from monoids to algebras?

In The Geometry of Rewriting Systems (Springerlink behind paywall), Kenneth Brown describes a method to collapse the bar resolution of a monoid. Roughly:
Given a simplicial set $X$ equipped with a ...

**5**

votes

**1**answer

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### pair of injective morphisms of simplicial groups

Let $X$ and $Y$ be two simplicial groups such that there exists $f:X\rightarrow Y$ and $g:Y\rightarrow X $ two injective morphisms of simplicial groups. Suppose that $X$ is contractible, does it ...

**5**

votes

**1**answer

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### Simplicial Objects in Additive Categories

I am looking for a reference, preferably as elementary as possible, for the following statement.
Let $X_{m,n}$ be a bi-simplicial object in an additive category $\mathcal{A}$. Then the complex $|X_{...

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vote

**0**answers

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### What is the normalized complex of a simplicial set with a monoid action?

This question is a follow up to this question I posted on Math.SE. I will make this question self-contained, though.
In a certain point on the paper The Geometry of Rewriting Systems, Kenneth Brown ...

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votes

**1**answer

320 views

### A few questions while reading Higher Topos Theory

I am now reading Higher Topos Theory. I have recently met the following questions that I am not able to figure out and I am here to look for some answer or help.
First, in Lemma 2.2.3.6, while ...

**4**

votes

**1**answer

169 views

### Thomason fibrant replacement and nerve of a localization

The Thomason model structure on the category $\mathrm{Cat}$ of small categories is transferred along the right adjoint of the adjunction $$\tau_1 \circ \mathrm{Sd}^2 \colon s\mathrm{Set} \...

**12**

votes

**1**answer

318 views

### Turning simplicial complexes into simplicial sets without ordering the vertices

Given an abstract simplicial complex $K$, one can make a simplicial
set $X(K)$ with $n$-simplices given by sequences $(x_0, \ldots, x_n)$
such that $\{x_0, x_1, \ldots, x_n\}$ is a simplex of $K$. The ...

**5**

votes

**1**answer

130 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

65 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 ...

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### 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 ...

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### 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

488 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

60 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 ...