Is there a name for a model category that is cofibrantly generated by (I, J) but where I and J do NOT permit the small object argument? In particular, I do not require to have any cellularization. All it means is that fibrations have right lifting property with J and trivial fibrations have right lifting property with I.
@JeremyHahn to be honest, we only studied the nonperiodic version you get by slowly trying to deform Z x BU into Z rather than the periodic one that goes * -> QCP^infty -> ... -> Z x BU, though some of the formal steps are the same
In HTT, in the introduction to section 6.5.3 Lurie mentions that for a small infinity-site C, the hypersheafification of a presheaf F on C is obtained by the formula colim_{U} lim F(V) where U ranges over the hypercoverings in the presheaf category. I know that in the world of simplicial (pre)sheaves this is essentially the generalized Verdier hypercovering theorem, but I wonder whether there is a more direct proof of this statement available that does not use simplicial sheaves?
