# Questions tagged [infinity-categories]

The infinity-categories tag has no usage guidance.

257
questions

**3**

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**1**answer

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### Proving a Kan-like condition for functors to model categories?

I've been trying to prove this version of the Kan condition for a project that I'm thinking about, and I'm pretty stuck. My experience asking questions about this stuff on MO in the past has been ...

**1**

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**1**answer

149 views

### Existence of pointwise Kan extensions in $\infty$-categories

This answer by Emily Riehl mentions a to-be-published proof of the fact that $\infty$-categorical (pointwise) Kan extensions exist when the target category admits (co)limits indexed by the relevant ...

**2**

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**0**answers

279 views

### Is there such a field as applied $\infty$-category theory?

It seems that applied category theory has exploded in popularity in recent years.
My question is simple: had there been any work using $\infty$-category theory in applications?
Edit: By ...

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**0**answers

79 views

### Simplicial sets of categories as models for $(\infty,1)$-categories [duplicate]

Disclaimer: I am not knowledgeable at all about the subject of this question, so my apologies if I say something incorrect or too imprecise.
In my understanding, there are several models for $(\infty,...

**6**

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**3**answers

342 views

### Definition of $E_n$-modules for an $E_n$-algebra

The category $Mod^{E_n}_A(\mathcal{C})$ of $E_n$-modules for an $E_n$-algebra in a symmetric monoidal $\infty$-category $\mathcal{C}$ is defined in Lurie's Higher Algebra as a special case of a more ...

**8**

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**1**answer

413 views

### Proj construction in derived algebraic geometry

The question
My question is easy to state:
Is there a Proj construction in derived geometry, that produces a derived stack from a “graded derived algebra”?
Given the vagueness of the question, you’...

**5**

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**0**answers

98 views

### CoCartesian vs. locally CoCartesian fibrations

Say $\pi: C\to J$ is an inner fibration of $\infty$-categories. Then "morally", $\pi$ corresponds to a diagram indexed by $J$ in the "category of categories with correspondences", and if $\pi$ is ...

**7**

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**1**answer

408 views

### Reference request: The unit of an adjunction of $\infty$-categories in the sense of Riehl-Verity is a unit in the sense of Lurie

I'm looking for a reference (or proof) for the statement given in the title: that when we have an adjunction between quasicategories in the sense of Riehl and Verity (defined e.g. in Section 4 of ...

**1**

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**1**answer

172 views

### Is the pushforward of a closed immersion of spectral Deligne-Mumford stacks conservative?

Let $ X \hookrightarrow Y$ be a closed immersion of (connective) spectral Deligne-Mumford stacks, is $ i_* : Qcoh(X) \rightarrow Qcoh(Y)$ conservative? Somehow I couldn't find the statement in SAG...

**15**

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**1**answer

521 views

### Natural examples of $(\infty,n)$-categories for large $n$

In Higher Topos Theory, Lurie argues that the coherence diagrams for fully weak $n$-categories with $n>2$ are 'so complicated as to be essentially unusable', before mentioning that several dramatic ...

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49 views

### Computing weak operadic colimits as colimits

I am trying to reduce the computation of weak operadic colimits to colimits. Let me introduct some notation. Let $q:C^{\otimes} \to N(Fin_*)$ be a symmetric monoidal category.
Let $p: K \to C^{\...

**4**

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**1**answer

112 views

### Inner fibrations are Kan fibrations on Map sets

Firstly, a bit of notation. Let $C$ be a simplicial set. We define, for $x,y \in C$ vertices in $C$
$$Map(x,y) = \{x\}\times_{\Delta^{\{0\}}}Map_{sSet}(\Delta^1,C) \times_{\Delta^{\{1\}}} \{y\} $$
...

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75 views

### Straightening of a marked simplicial set

Let $X$ be in $Set_{\Delta}$, let $\phi=id:\mathbb{C}X\rightarrow \mathbb{C}X$ be the map of simplicial categories over which we want to straighten. Assuming that $X$ is a quasi-category, how can one ...

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42 views

### Some properness condition in simplicial sets

Suppose that $C \to D$ is a trivial Kan fibration, and $D' \to D$ is an equivalence of simplicial sets. Let $C'$ be their pullback. Is it true that $C' \to D'$ is an equivalence?
Recall that a ...

**10**

votes

**1**answer

540 views

### What does the homotopy coherent nerve do to spaces of enriched functors?

Suppose I have two fibrant simplicially enriched categories $C$ and $D$. Then I can consider the collection of simplicial functors $sFun(C,D)$ which, if I recall correctly, has the structure of a ...