Let $\kappa$ be a regular cardinal. It seems reasonable to introduce the following definition:

> **Definition.** A simplicial set $K$ is *$\kappa$-sifted* if, for every set $E$ with $\lvert E\rvert<\kappa$, the diagonal functor $K\to K^E$ is cofinal.

It follows from [[Lurie: Higher Topos Theory](https://www.math.ias.edu/~lurie/papers/HTT.pdf), Prop 5.3.1.22] that every $\kappa$-filtered $\infty$-category is $\kappa$-sifted. On the other hand, the category ${\mathbf\Delta}^{\operatorname{op}}$ seems to be $\kappa$-sifted since the product of any $\kappa$'s simplices is still weakly contractible.

I wonder:

1. Is there any study of this notion in the literature?
2. To what extent, $\kappa$-sifted colimits = $\kappa$-filtered colimits + geometric realizations? For example, let $F\colon\mathcal C\to\mathcal D$ be a functor. Under what mild assumptions on $\mathcal C$ and $\mathcal D$, it is true that, if $F$ preserves geometric realizations and is $\kappa$-accessible, then it preserves $\kappa$-sifted colimits? I am aware of the following references when $\kappa=\aleph_0$:

    a. Suppose that $\mathcal C$ has small colimits. Then it is true, as proved in [[Joyal: On Logoi](https://ncatlab.org/nlab/files/JoyalOnLogoi2008.pdf), §33.24] or [[Lurie: Higher Topos Theory](https://www.math.ias.edu/~lurie/papers/HTT.pdf), Cor 5.5.8.17].

    b. In the 1-category situation, and suppose that $\mathcal C$ has small limits, it is true, as proved in [[Adámek–Rosický–Vitale, On Algebraically Exact Categories and Essential Localizations of Varieties](https://www.sciencedirect.com/science/article/pii/S0021869300985776)].

However, I am more interested in the situation that $\kappa>\aleph_0$, and hopefully it is simpler than the case that $\kappa=\aleph_0$.