# All Questions

Tagged with higher-category-theory homotopy-theory

135
questions

**2**

votes

**1**answer

200 views

### Fibrant objects in Bousfield localization of homotopy pullback closure of Nisnevich hypercovers

Let $M$ be a model topos and $S$ a set of morphisms, there exists a set of morphism $\bar{S}$ which is generated by the $S$-local equivalences which is closed under homotopy pullbacks in $M$. Suppose ...

**7**

votes

**1**answer

388 views

### Categorical Significance of Fibrations

It is well known that the category $\text{Set}$ classifies covering spaces among $1$-categories. That is, for each topological space $X$, there is an equivalence of categories $[ \Pi (X) , \text{Set}]...

**1**

vote

**0**answers

145 views

### Comparing cohomology using homotopy fibre

I have a question, which might be very basic, but I don't know enough topology to answer.
Suppose you have a map of topological spaces (or homotopy types) $f : X \to Y$, with homotopy fibre given by ...

**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

273 views

### Are there universal homological functors?

There is a bifunctor $H: Stab^{op} \times Ab \to Top$ where $H(C,A)$ is the space of homological functors $C \to A$. Is this bifunctor left or right representable?
That is, for each small abelian ...

**5**

votes

**1**answer

305 views

### If the Tate construction vanishes for all trivial $G$-actions, then does it vanish for all $G$-actions?

Let $\mathcal{C}$ be a semiadditive $\infty$-category, complete and cocomplete, and let $G$ be a finite group. Then for any $X \in Fun(BG,\mathcal{C})$, there is a norm map $N_X: X_G \to X^G$. For ...

**3**

votes

**0**answers

103 views

### Descent for the cotangent complex along faithfully flat SCRs

By Theorem 3.1 of Bhatt-Morrow-Scholze II (https://arxiv.org/pdf/1802.03261.pdf), we know that for $R$ a commutative ring, $\wedge^{i}L_{(-)/R}$ satisfies descent for faithfully flat maps $A \...

**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 ...

**0**

votes

**1**answer

84 views

### A characterization of maps that are homotopic relative to $A$ over $S$

Let $$\begin{array}{ccccccccc}
A & \rightarrow & X \\
i\downarrow & & \downarrow p \\
B & \xrightarrow{v} & S
\end{array} $$ be a commutative diagram of simplicial sets, with ...

**5**

votes

**1**answer

388 views

### Understanding the proof of Proposition 1.2.13.8 in Lurie's Higher Topos Theory

In the first chapter of Lurie's HTT, Proposition 1.2.13.8 shows that the functors out of over (infinity)-categories reflect infinity-colimits.
For the life of me I cannot follow the proof.
Can ...

**26**

votes

**1**answer

810 views

### Deligne's doubt about Voevodsky's Univalent Foundations

In a recent lecture at the Vladimir Voevodsky's Memorial Conference, Deligne wondered whether "the axioms used are strong enough for the intended purpose." (See his remark from roughly minute 40 in ...

**5**

votes

**1**answer

171 views

### Homotopy groups of Diffeomorphisms of punctured d-dim ball

Let $\mathbb{D}_n^d$ be the $d$-dimensional unit ball with $n$ punctures. I am interested in the groups of orientation-preserving diffeomorphisms $Diff(\mathbb{D}_n^d)$ that fix the punctures. In ...

**4**

votes

**1**answer

206 views

### A finite Whitehead Theorem for $\infty$-topos

Let's consider in an $\infty$-topos, we have an object $X$ of homotopy dimension $\leq n$ (in the sense of Lurie HTT), let $f: A\to B$ be an $n$-equivalence morphism. Can we conclude that $f$ induces ...

**7**

votes

**2**answers

413 views

### What does it mean to say the first Goodwillie derivative of $TC$ is $THH$?

A paradox:
Goodwillie calculus considers only finitary functors.
$TC$ isn't finitary.
Yet in some sense $\partial(TC) = \partial(K) = THH$ is the crux of the Dundas-Goodwillie-McCarthy theorem.
(...

**4**

votes

**0**answers

103 views

### Minimal infinity categories in the Segal space picture

There is a well-known notion of a minimal Kan complex (see Goerss/Jardine's book) which is generalized to a minimal quasi-category in these notes by Joyal
www.math.uchicago.edu/~may/IMA/Joyal.pdf
(...