Hi, I hope everyone is doing well! quick question: is every quasi-category (Joyal/categorical) weakly equivalent to one for which each non-degenerate simplex is embedded, namely, the representing map of each non-degenerate simplex is injective?

I meant to say is every simplicial set (Joyal/categorically) equivalent to one with this property...

I'm very confused about the proof of HTT 5,4.2.9 (which is basically the fundamental lemma needed for doing anything with accessible categories). This proof uses 5.4.2.8, but it quotes it incorrectly as far as I can tell. 5.4.2.8 as stated doesn't seem to apply in the case he wants.

5,4.2.8 starts: "If $\kappa'\gg\kappa$", then any $\kappa'$-filtered partially ordered set $\mathcal J$ ...". But the proof of 5.4.2.9 wants to apply this to "the case where $\mathcal J$ is the nerve of a $\kappa$-filtered partially ordered set".

$\kappa'\gg \kappa$ does not imply that every $\kappa$-filtered poset is also $\kappa'$-filtered, as far as I know (if anything, it is the reverse case).

@CharlesRezk I think that's right, every kappa-filtered poset should be a kappa'-filtered colimit of kappa'-small kappa-filtered posets

So it's a typo in 5.4.2.8?

One of the fundamental facts about infinity categories is that the restricted yoneda to $\Delta$ is fully faithful. A weaker statement is that the restricted yoneda $Cat_{\infty} \to \mathcal{P}(Cat^{fin}_{0})$ is fully faithful. Where $Cat^{fin}_0$ is just the category of finite posets. This seems to me slighly irreducible fact. Similarly to how spaces are the colimit of their finite subsets. It's also a tautology in the CSS model. Now I have a question about $\infty$-operads....cont.

Similarly to how we have a fully faithful nerve functor $Cat_1 \to Cat_{\infty}$ we have an operadic nerve functor $Op_1 \to Op_{\infty}$. We may restricted to $Op_0$ i.e. operads where multi-mapping spaces are either empty or contractible (these are very close to being forests, except they are not "directed" what ever that means). My question: is there an argument for why $Op_{\infty} \to \mathcal{P}(Op_0)$ is fully faithful that doesn't go through the equivalence with some other model?

I mean an argument inside lurie's model of $\infty$-operads.

I realize that this is almost asking for the equivalence between lurie's model and dendroidal segal spaces.

Which isn't extremely straightforward, so maybe i'm asking for too much

Alternatively I guess one could ask if there argument for a direct equivalence between Lurie's model and DSS that doesn't pass through Barwick's model of segal presheaves on $\Delta_{Fin}$.

The adjunction between them is obvious of course, the question is whether we can prove it is an equivalence without using Barwick's model i guess.

@CharlesRezk That's my guess.