# Questions tagged [higher-category-theory]

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

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### 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 ...

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### 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 ...

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### 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 \...

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457 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 $X \times_Z Y$ ...

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### 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 ...

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### 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,...

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### 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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### 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 $(\infty,2)$-...

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806 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 $n$-...

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### 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 ...

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### 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 ...

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### 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 (...

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### A construction for the free $\omega$-category generated by a globular set

The forgetful functor from strict $\omega$-categories to globular sets has a left adjoint. Where can one find an explicit construction for this free functor?

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### 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 ...

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### 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 ...