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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 …

I don't really see a way to give a single answer here, these are 7 different questions (well actually a lot more than 7 if we count all the different flavours of cubes, and of the other ones, probably …

It is known that the category of finite non-empty set is a test category, in particular there exists a model structure on the category of presheaf on $Fin_+$ whose cofibrations are the monomorphisms a …

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 …

Here is a counter example in the general case: Take $E = [1] = 0 \to 1 $. $D = \{0\} \coprod \{1\} $ and $C = \{1\}$. where all maps are weak equivalence. The lax-pullback is $\{id:1 \to 1\}$, and the …

I would think about this question in this way: If you have a construction that violates the equivalence principle then either (A) it is a strictified or simplified version of something that is compati …

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 …

Here is an argument for the 1-categorical version that essentially bypass the use of internal site and should be much easier to generalize to the $\infty$-categorical case. ( I mean you can still see …

Yes, this is possible. The following is a classical result of the theory of quasi-categories (You'll find it in the early part of Lurie's Higher topos theory or in Joyal notes on quasi-categories - w …

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 …

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 …

Well, it is easy to give definitions, the problem is finding the "right" one. Here "right" can mean that gives the correct notion up to homotopy (many definition will be equivalent) but also it can me …

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 …

That is indeed a problem, and one could argue that this is part of the reason why these questions are difficult. But I feel that in practice that has never been a strong obstruction. I guess, one way …

Let me clarify a bit what I meant in my comment on how the notion of bilinearity will depends on "how commutative" are $A$, $B$ and $C$, and this is one way to define a hierarchy of notion of bilinear …