Let $\Phi$ be a class of limit diagrams that contains all finite diagrams. Some examples include the classes $\Phi_{\kappa}$ of all diagrams of size bounded by a cardinal $\kappa$... but are there any other examples? To make this more concrete, here are two specific questions I am interested in:

Considering that filtered colimits commute with finite limits, and $\kappa$-filtered colimits commute with $\kappa$-small limits, are there any other limit-commutation classes

*strictly contained in filtered colimits*? (edited)There is a heirarchy of 2-categories of categories having all $\Phi$-limits as $\Phi$ varies (where I should reiterate that $\Phi$ contains all finite diagrams). Is this more than a linear order? (edited)

I'm just asking about ordinary/conical limits here, but more exotic answers/counterexamples could be of interest.

containfinite limits, other than the $\lambda$-small ones. I can't think of any myself--it feels as if basically anything you might add (some discrete infinite categories, some cofiltered ones...) gets you to something that's interchangeable with some $\Phi_\kappa$ for most purposes. $\endgroup$3more comments