# Questions tagged [higher-category-theory]

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

**8**

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95 views

### Where is it shown that homotopy sheaves form a higher stack?

Many references on infinity categories etc. advertise that one application is that it's an appropriate setting to glue (the appropriate replacement for) derived categories of sheaves.
What's the ...

**3**

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**0**answers

64 views

### Counit map for compactly generated categories

Any compactly generated presentable stable $\infty$-category $C$ is known to be dualizable (with respect to Lurie's tensor product), so there is a coevaluation map:
$$Sp \to C \otimes C^{dual}.$$
...

**4**

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117 views

### Examples of Lurie tensor product computations

I am interested in examples of computing the Lurie tensor product.
For example, if $A$ and $B$ are connective DG algebras (over $\mathbb{Z}$, say), then I think there is an equivalence $A\text{-mod} \...

**2**

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127 views

### Derived $\infty$-category of sheaves and $\infty$-category of sheaves taking values in derived $\infty$-category

I am trying to understand the essential image of the following functor. Given a scheme $X$, we consider the corresponding small Zariski site $X_{zar}$. For a commutative ring $\Lambda$, let $\mathcal ...

**5**

votes

**1**answer

185 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**

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86 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 \...

**5**

votes

**1**answer

304 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**

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133 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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402 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**

votes

**1**answer

270 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**

votes

**1**answer

82 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**

votes

**1**answer

190 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}^\...

**4**

votes

**0**answers

74 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**

votes

**1**answer

370 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

711 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 ...