# Questions tagged [higher-category-theory]

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

735
questions

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### The model category structure on $\mathbf{TMon}$

I ask this question here since I asked it here on Math.SE, and got no answers after a week of a bounty offer.
I am trying to understand the homotopy colimit of a diagram of topological monoids, and ...

**3**

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**1**answer

135 views

### 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 ...

**3**

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47 views

### Concerning the definition of a 2-crossed module

Question:
Is there some generalization of the definition of crossed-module which appropriately fits into the holonomy-considerations I am interested in and has, as an example, the generalization of ...

**2**

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**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 ...

**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 ...

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97 views

### Classifying Objects for Fibrations Defined by a Lifting Property

I have been studying weak factorization systems for their use in model categories. I am trying to use these to abstract away from a common phenomenon underlying fibrations. In brief, it seems as ...

**2**

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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,...

**4**

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112 views

### Derived weight filtration on motivic Galois representations

Thanks to modern techniques (such as the pro-etale site), we can now understand etale (co)homology of varieties and motives as "genuinely" derived (e.g. DG) Galois-equivariant objects. I'm looking for ...

**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 ...

**7**

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**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}]...

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91 views

### Tensor schemes “with relations”

In "The geometry of tensor calculus I," Joyal and Street introduce the notion of a tensor scheme, a set of abstract objects, together with formal morphisms between words in this set. They then prove ...

**5**

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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 ...

**6**

votes

**1**answer

157 views

### Tricategorical coherence

Why does coherence begin to matter at the tricategorical level?
It is well known that every weak $2$-category is equivalent to a strict $2$-category, with the equivalence essentially given by (unless ...

**15**

votes

**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 ...

**4**

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**1**answer

92 views

### Coherence theorem for tetracategories, weak $n$-categories

Is there a coherence theorem/conjecture for tetracategories (weak $4$-categories)?
Todd Trimble mentions in his notes on tetracategories that his pasting definitions are essentially unambiguous due ...

**3**

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**1**answer

332 views

### Composition map in $\infty$-categories

Let $\mathcal{C}$ be an $\infty$-category, and let $u:x\rightarrow y$ be an edge. It seems reasonable to say that: The map $Map_{\mathbb{C}[\mathcal{C}]}(a,x)\rightarrow Map_{\mathbb{C}[\mathcal{C}]}(...

**2**

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**1**answer

121 views

### On the Group Structure of Morphism Set of a Strict 2-Group

The standard definition of a strict 2-group says that it is a strict monoidal category in which every morphism is invertible and each object has a strict inverse.
Also it is a well known fact that a ...

**10**

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**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 ...

**1**

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**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 ...

**3**

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58 views

### On cofibrations of simplicially enriched categories

Let $\mathbb{C}$ be an strict 2-category and denote by $C$ is underlying 1-category viewed as as a 2-category only having identity 2-cells.
We have a canonical inclusion functor ,
$$i: C \...

**13**

votes

**2**answers

500 views

### Describing fiber products in stable $\infty$-categories

Let $f\colon X \rightarrow Z$ and $g\colon Y \rightarrow Z$ be two morphisms in a stable infinity category $\mathcal{C}$. How does one show that the $\infty$-categorical fiber product $X \times_Z Y$ ...

**7**

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**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 ...

**10**

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236 views

### When is Fun(X,C) comonadic over C with respect to the colimit functor?

Because I'm primarily interested in this question from the point of view of $\infty$-categories (in this case, modeled by quasicategories), I'll ask this question using that terminology. In particular,...

**5**

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**1**answer

200 views

### Are semisimplicial hypercoverings in a hypercomplete $\infty$-topos effective?

A hypercovering in an $\infty$-topos $E$ is a simplicial object $U \in {\rm Fun}(\Delta^{\rm op},E)$ such that each map $U_n \to ({\rm cosk}_{n-1} U)_n$ is an effective epimorphism. Theorem 6.5.3.12 ...

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132 views

### 2-categorical Yoneda embedding and limits

Does the 2-categorical Yoneda embedding defined in A.2 Section 5 of `A study in derived algebraic geometry' (Gaitsgory and Rozenblyum, version of 2018-11-14) preserve limits for any $(\infty,2)$-...

**10**

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**2**answers

857 views

### Results relying on higher derived algebraic geometry

Are there any results in number theory or algebraic geometry whose statement does not involve either higher categories or any derived structures but whose most natural (known) proof uses derived $n$-...

**7**

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**1**answer

1k views

### Anabelian geometry ~ higher category theory

Note: I'm worried this question might be taken as controversial, because it relates to Shinichi Mochizuki’s work on the abc conjecture. However, my question has nothing to do with the correctness of ...

**3**

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**1**answer

203 views

### Geometry of 2-arrows

It is frequently said that one of the contributions of Grothendieck to geometry was to systematically think about the properties of morphisms, as opposed to the properties of spaces themselves (the ...

**4**

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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 (...