# Questions tagged [higher-category-theory]

For questions involving one or more categorical dimensions, or involving homotopy coherent categorical structures.

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### 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**

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127 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 ...

**8**

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363 views

### Visualization and new geometry in higher stacks (soft question)

I am trying to develop a geometrical intuition for "higher spaces", i.e. both in the sense of higher dimensional spaces (more than three dimensions) and in the sense of abstractions beyond manifolds ...

**3**

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**1**answer

263 views

### Definition A.3.1.5 of Higher Topos Theory

I am now reading the book Higher Topos Theory. In A.3.1.5, it gives the definition
of a $\mathbf{S}$-enriched model category, where $\mathbf{S}$ is a monoidal model category. But in the book model ...

**0**

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**1**answer

81 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 ...

**8**

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**1**answer

164 views

### Weighted (co)limits as adjunctions

It's well known that a category $\mathcal{C}$ having (conical) limits/colimits of shape $\mathcal{D}$ is equivalent to the diagonal functor $\Delta^\mathcal{D}_\mathcal{C}:\mathcal{C}\to\mathcal{C}^\...

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67 views

### Natural transformation of $A_\infty$-functors lifted from homology

Suppose you have two $A_\infty$-functors $\mathcal{F,G}:\mathcal{A}\longrightarrow \mathcal{B}$ which descend to $F,G:A \longrightarrow B$ in homology (here $A=H^0(\mathcal{A})$ and same for $\mathcal{...

**5**

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**1**answer

359 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**

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**1**answer

699 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 ...

**10**

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**3**answers

784 views

### Higher $\infty$-categories

Is there a reason we consider $\infty$-categories to be the $\omega^{th}$ step in the 2-internalization inside Cat (or enrichment over Cat if you prefer)* process made invertible above some finite ...

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**1**answer

186 views

### Smash product and the integers in a Grothendieck $(\infty, 1)$-topos

Let $\mathcal{H}$ be a Grothendieck $(\infty,1)$-topos. According to this page in nlab, for any $X \in \mathcal{H}$, the suspension object $\Sigma X$ is homotopy equivalent to the smash product $B \...

**5**

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**1**answer

106 views

### Inverting a suspension object in a stable monoidal category

Suppose we are given a cocomplete closed symmetric monoidal stable $(\infty,1)$-category $\mathcal{C}$ with suspension $\Sigma$, and let $X \in \mathcal{C}$ be dualizable. I'd like to create a new ...

**5**

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**1**answer

148 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 ...

**5**

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**0**answers

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### Infinity categorical analogue of 2-dimensional monad theory

I'm wondering whether there is an infinity categorical analogue to the results of Two-dimensional monad theory. For the most part, I'm interested in the relation between strict functors of infinity ...

**5**

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68 views

### Compact Generation of Co-Module Categories

Let $\mathcal{C}$ be a compactly generated stable $\infty$-category, linear over a field of characteristic $0$ (i.e., so that it is in particular a dg-category). Let $A$ be a co-monad acting on $\...