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Presentable $\infty$-categories can be understood without every having to think about cardinals. An $\infty$-category is presentable iff it is equivalent to one of the form $\mathcal{P}(C,R)$, where …

As you point out, there are "inclusion" functors $\mathrm{Cat}_n\to \mathrm{Cat}_{n+1}$. These inclusion functors admit both left and right adjoints (in the sense of functors between $(\infty,1)$-cat …

Let $C$ be an $\infty$-category, and $n\geq -1$. The following are equivalent: $C$ is $n$-truncated. The $\infty$-groupoids $\def\Map{\operatorname{Map}}\Map(\Delta^0,C)$ and $\Map(\Delta^1,C)$ are …

An $n$-monomorphism is a map $A\to B$ for which $Map(X,A)\to Map(X,B)$ has all homotopy fibers $n$-truncated, for all $X$; a space (=$\infty$-groupoid) is $n$-truncated if its homotopy groups all vani …

The Yoneda lemma is certainly true for $(\infty,1)$-categories, and I can give you a proof: if you let me choose the model of $(\infty,1)$-categories I use! I'll take topological categories (=catego …

I would also like to know the answer to your question. Since no one has given an answer yet, I'll speculate recklessly and irresponsibly on how this might work (by riffing off of the final paragraph …

According to the abstract of http://arxiv.org/abs/math/0108143 (Schwede & Shipley, Classification of Stable Model Categories), they deal with the case of stable model categories (=stable presentable ( …

Yes. Let $W$ be a complete Segal space, thought of as a simplicial "space" $(W_q)$. The fibrant objects of your model category will be the fibrations $f:X\to W$ such that for each simplicial operato …

The question "can a given homotopy idempotent admit multiple inequivalent coherentifications" ought to be approachable by the standard spectral sequence machinery, so let me try to do that. I'll more …

Here is how I think about this; don't know if it will help. Start with the straightening construction: this takes a map $f\colon X\to S$ of simplicial sets to a simplicial functor $\def\St{\mathrm{St …

I don't think so. Take $n=2$, and consider a map $f\colon b\to a$ between objects of $\mathrm{Fun}(I^2, \mathcal{S})$. If $a$ and $b$ are pullback squares, then any map $f$ between them is relativel …

I think the point is that a map $f\colon A\to B$ in a slice $\mathscr{C}_{/Y}$ is $k$-connected if and only if the underlying map in $\mathscr{C}$ is $k$-connected, and this is supposed to be a formal …