definition of a $\kappa$-small simplicial set

Question

Let $\kappa$ be a regular cardinal. A set $X$ is $\kappa$-small, if $|X|<\kappa$.

What does it mean for a simplicial set $X\colon \Delta^{op}\to Sets$ to be $\kappa$-small.

I can imagine at least two possible definitions:

1. $\forall n\geq 0:X_n$ is a $\kappa$-small set,
2. The set $\bigsqcup_{n\geq 0} X_n$ is a small set

These are different for the regular cardinal $|\mathbb{N}|$.

Lurie mentions $\kappa$-small simplicial sets in HTT but I can't find a definition there.

Answer 1

There are lots of reasonable definitions which only differ for $\kappa = \aleph_0$. However, I think what Lurie means is this:

A $\kappa$-small simplicial set is a simplicial set with $< \kappa$ non-degenerate simplices.

This is not equivalent to either of your definitions. For instance, if $\mathcal{C}$ is the category freely generated by an idempotent endomorphism, then $N (\mathcal{C})$ is degreewise $\aleph_0$-small but not $\aleph_0$-small. Indeed, one can infer that Lurie means for "$\aleph_0$-small simplicial set" to be a synonym of "finite simplicial set" from the proof of (Corollary 4.4.2.4 and) Proposition 4.4.2.6 in [Higher topos theory].

Incidentally, a simplicial set is $\kappa$-small if and only if it is $\kappa$-compact.