Characterizing freely adjoining K-filtered colimits as K-continuous presheaves

Question

Note: This question has a 1-categorical and an $\infty$-categorical versions. I am interested in the $\infty$-categorical one so this is the version that I write below, but an answer for the 1-categorical version would be interesting too.

Given an $\infty$-category $\mathcal{C}$ and a collection of simplciial sets $\mathcal{I}$, there is a notion of "freely adjoining to $\mathcal{C}$ all colimits of shape $\mathcal{I}$". This is the full subcategory $P_{\mathcal{I}}(\mathcal{C})$ of all presheaves on $\mathcal{C}$ generated from the image of the Yoneda by $\mathcal{I}$-colimts (HTT 5.3.6). Now, assume that $\mathcal{I}$ is not any collection, but the collection of all $I$, such that $I$-colimits commute in spaces with all $K$-limits for all $K$ in some collection $\mathcal{K}$. e.g.

1) $\mathcal{K}$ is finite simplicial sets and $\mathcal{I}$ is filtered simplicial sets.

2) $\mathcal{K}$ is finite discrete simplicial sets and $\mathcal{I}$ is sifted simplicial sets.

Assuming $\mathcal{C}$ admits all $K^{op}$-colimits for all $K\in \mathcal{K}$, in these two examples it is known that $P_{\mathcal{I}}(\mathcal{C})$ can be described as the full subcategory $P^{\mathcal{K}}(\mathcal{C})$ of the $\infty$-category of presheaves spanned by the ones which preserve $\mathcal{K}$-limits (for (1) HTT 5.3.5.4 and for (2) HTT 5.5.8.15). The proofs of these two facts seem to be quite different, so I am wondering

Question: Under the above conditions on $\mathcal{I}$, $\mathcal{K}$ and $\mathcal{C}$, does $P_{\mathcal{I}}(\mathcal{C})$ always coincide with $P^{\mathcal{K}}(\mathcal{C})$?

It is easy to see that we have an inclusion of $P_{\mathcal{I}}(\mathcal{C})$ into $P^{\mathcal{K}}(\mathcal{C})$, but the converse seems more complicated. I also have a creepy feeling that the words "sound doctrine" may come up...

Answer 1

For the 1-categorical case, it seems to be indeed a question of soundness. More precisely, the condition that $P_{\cal I}({\cal C}) = P^{{\cal K}}({\cal C})$ is equivalent to the condition that every ${\cal K}$-limit preserving presheaf $F:{\cal C}^{op} \to {\rm Set}$ is an ${\cal I}$-colimit of representables, or, in this case, a ${\cal K}$-filtered colimit of representables. Looking in this paper we see that this indeed holds whenever ${\cal K}$ is sound (theorem 2.4), and that furthermore, this being the case appears to be the main motivation for the definition of soundness. For example, it seems to me that $P_{\cal I}({\cal C})$ doesn't contain the terminal presheaf in the non-sound case described in Example 2.3 (vi) of loc.cit.

I would say that it's quite likely that the $\infty$-categorical case will behave similarly, once one defines soundness appropriately (maybe replace "connected" with "weakly contractible" in Definition 2.2 of loc.cit)?