Members of this chat room may be interested in the talk I gave yesterday at the Masaryk University Algebra Seminar. In it, I introduce a new notion of double ∞-category, which I motivate by way of a new model structure on the category of double categories. Video, slides, and abstract may be found on the seminar webpage.

@AlexanderCampbell Yeah thanks!! I was supposed to come but I slept through it by accident

@SaalHardali I know nothing, and thanks! I have a feeling that the correct thing to look at, as @PiotrPstrągowski suggest, is a simplicial object in which all the objects are X and the maps are identity except the last face map in each degree, which should be the idempotent. It seem that such a diagram can be constructed out of the idempotent datum and that if the idempotent is an equivalence it gives back X.

@AlexanderCampbell Is this supposed to also provide a way of seeing the fully faithfulness of the Sq functor of Gaitsgory-Rozenblyum?

@EdoardoLanari I don't think it will make that fully faithfulness any easier to prove.

I asked Rozenblyum at MSRI if he knew how to prove that fully faithfulness, and he said he could probably prove it if someone put a gun to his head. As much as I would like to see a proof, I don't think I'll go that far.

@S.carmeli I just realized my argument only works if my diagram itself has products (it doesn't help that the target category has products). In other words this argument only show that if a category has all geometric realizations then it has colimits for diagrams which have products and a weakly terminal object.

@AlexanderCampbell haha yeah we heard that sentence pretty often

@AlexanderCampbell Thanks!

@SaalHardali Hey, I don't have your e-mail. If you want to send me one so I can send you some scheduling info, that would be helpful!