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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 …

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 …

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 …

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 …

This has not been done, and there are good reasons for it: While $sSet$-enriched categories are indeed very good to easily get examples of $\infty$-categories, they are very bad at understanding what …

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 …

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 …

There is a bit of notation to be careful about here: $\mathcal{X}_{\leqslant 1}$ is often used to denote the full subcategory of $\mathcal{X}$ of set-truncated object. For example if $\mathcal{X}$ is …

A pushout $B \coprod_A C$ will almost never have all coproduct. the problem is that objects in $B \coprod_A C$ are all either objects of $B$ or objects of $C$, so if $B \coprod_A C$ has coproduct, it …

Probably not the best one can do, and what follows might be a bit 'overkill', but it answer the question about dependency on universe, and it is a nice argument. Also if you know how the proof of th …

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 …

I just found an example, so I thought it would be good to post it here, but if anyone knows other examples, or a more general way to construct some I would be interested to see them as well. This exa …

While it can be obtain formally using the adjoint functor theorem as mentioned by Marc Hoyois in the comment. There are several explicit constructions. First, the localization functor which send a sm …