# All Questions

Tagged with higher-category-theory simplicial-stuff

86
questions

**7**

votes

**2**answers

281 views

### What is an example of a quasicategory with an outer 4-horn which has no filler?

A quasicategory has fillers for all inner horns $\Lambda^i[n]$ where $n\geq 2$ and $0<i<n$, but it need not have fillers for $i=0$ (or $i=n)$. In particular, for $n=2$ and $n=3$ there are easy ...

**4**

votes

**0**answers

200 views

### A compendium of weak factorization systems on $sSet$

A (weak) factorization system on a category $\mathcal{C}$ consists of a pair of classes of morphisms in the category $(L,R)$ satisfying
Every morphism $f:x \to y \in \mathcal{C}$ can be factored (...

**5**

votes

**1**answer

416 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 ...

**6**

votes

**0**answers

142 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 ...

**3**

votes

**0**answers

136 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 ...

**4**

votes

**2**answers

200 views

### Proof that a quasicategory is equivalent to the homotopy coherent nerve of a simplicial category

On the abstract of a paper by Emily Riehl and Dominic Verity, it is stated that
Every quasicategory arises as a the homotopy coherent nerve of a simplicial category up to equivalence.
Where can ...

**6**

votes

**1**answer

409 views

### Need help understanding comment in Higher Topos Theory

I am having trouble understanding this part in Lurie's Higher Topos Theory. This can be found in section 2.4.4.4 right after Lemma 2.4.4.1.
Lemma 2.4.4.1. Let $p : \mathcal{C} \rightarrow \mathcal{...

**2**

votes

**1**answer

81 views

### Factorization of a map from a contractible Kan complex through a Kan complex

Suppose we are given a contractible Kan complex $S$ and a map of simplicial set $f : S \rightarrow T$. Under what conditions can we say that $f$ factors through the largest Kan complex $Z$ contained ...

**5**

votes

**1**answer

316 views

### Remark 2.4.1.4 Higher Topos Theory

In HTT, given a inner fibration $p : X \rightarrow S$ of simplicial, an edge $f : x \rightarrow y$ of the simplicial set $X$ is said to be a $p$-Cartesian if the induced map
$$ X_{/f} \rightarrow ...

**9**

votes

**1**answer

540 views

### Higher Topos Theory Theorem 2.2.5.3

The following question is found in the proof of Theorem 2.2.5.3 of HTT but since it can be understood in a more general context I will just ask it without stating the theorem.
We have a trivial Kan ...

**7**

votes

**0**answers

171 views

### Dugger and Spivak's combinatorial proof of Joyal's isofibration theorem, a fortiori?

In what follows, let $E^n = \operatorname{cosk}_0(\Delta^n)$
Joyal's isofibration theorem says precisely
An inner fibration $p:X\to Y$ of quasi-categories is a fibration for the Joyal model ...

**6**

votes

**2**answers

645 views

### Theorem 2.1.2.2 Higher Topos Theory

At the page 74 of HTT, there is the following theorem
Let $S$ be a simplicial set, $\mathcal{C}$ a simplicial category, and $\phi: \mathfrak{C}[S] \rightarrow \mathcal{C}^{op}$ a simplicial functor....

**11**

votes

**0**answers

204 views

### Higher homotopical information in racks and quandles

A quandle is defined to be a set $Q$ with two binary operations $\star,\bar\star\colon\ Q\times Q\to Q$ for which the following axioms hold.
Q1. a $\star$ a = a
Q2. (a $\star$ b) $\bar\star$ b = (a $...

**3**

votes

**1**answer

138 views

### Cellularity of anodyne extensions?

Are the anodyne extensions of simplicial sets always relative cell complexes of horn inclusions? (i.e. there is no need to consider retracts)
If not, is there a known counterexample?
Similarly, does ...

**9**

votes

**0**answers

254 views

### Homotopy pullbacks of simplicial sets; Joyal vs Kan-Quillen model structures

I am interested in comparison of homotopy pullback squares in the category of simplicial sets with respect to Joyal' model structure and Quillen's one.
Suppose we are given a (strict) pullback square
...