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Here are some things I know of that looks like this: I* have argued here that it is possible to extend the notion of strict $\infty$-category to a notion of $P$-category for any poset $P$. At this po …

Start from a finite simplicial set $K$ which is homotopicaly equivalent to a Kan complex $X$. Then by applying a finite number of pushout of outer horn inclusion to $K$, you can build homotopy equiva …

I will see an $(\infty,\infty)$-category as a functor $\Theta^{op} \to \text{Space}$ that satisfies the usual Segal condition, i.e., i.e. preserve the pushouts encoding the various type of composition …

There has been a lots of approaches to the notion of n-categorical diagram and n-categorical pasting diagram: Street : "Parity complexes" Power : " An n-categorical pasting theorem" Johnson : "The …

An important distinction in the "set case" is that there are two things you can refer to when talking about "the category of groupoids": the 1-category of groupoids. the 2-category of groupoids. It …

If you don't mind, I'll talk about strinct $\infty$-categories, but weak $(\infty,n)$-category to avoid discussing the 'problem' regarding the non uniqueness of the meaning of $(\infty,\infty)$-catego …

He is applying 5.4.2.13 to $C$ and not to $D$: Because $C^\kappa$ is essentially small, and each $F(c)$ is $\lambda_c$-compact for some $\lambda$; there is a $\kappa = \sup_{c \in C^\kappa} \lambda_c$ …

Using Street's "one type" definition of strict $\infty$-category one can see that the concept of "strict $P$-category" makes sense not justs for any ordinals $P$ but in fact for any posets $P$ (though …

I just thought (or maybe remember) a neat proof of this fact. It involve ideas I worked on a few years ago but never published - but that's short enough so that I can explain the key ideas on MO. Let …

In short there isn't: the problem is that if you just have globular sets - and if you want $k$-cells to model $k$-arrows following the globular structure - then globular sets have no way of expressing …

In $1$-category theory, a well pointed endofunctor of a category $C$, is an endofunctor $F:C \rightarrow C$ endowed with a natural transformation $\sigma : Id \rightarrow F$ such that the two natural …

This is just an expended version of the comment. The answer to the question as asked is no. The problem is that for any ($\infty$-)category $J$ the category $D_J$ of functors $J \to Cat_\infty$ that a …

$\newcommand\Set{\mathrm{Set}}$The naïve definition of sheaves is very well behaved if you look at functoriality in the $f_*$ direction: Of course, you are going to need to assume that $\mathcal{A}$ h …

Regarding (1) : A) Every model category has an associated $\infty$-category, obtained for example by taking the Dwyer-Kan localization at the class of all equivalence, (But there are other more expli …

As far as I'm aware, no such conditions is known - The paper of Anel and Lejay is the closest to an answer available in the litterature. So, this is not an answer to the question, but more of an expan …