Conjecturing a Characterization of Joyal Fibrations

The Joyal model structure on simplicial sets has been an important setting for developing the theory of higher category theory. The fibrant objects are the quasi-categories, a model for (∞,1)-categories, which have a straightforward characterization: they are the simplicial sets with lifts of inner horn inclusions. However, so far in the literature there has been no such characterization of the fibrations in terms of a countable set of concrete maps. We give a conjecture of such a characterization, describing a class of ``special horn'' extensions. We use Cisinski's theory to build a model structure whose fibrant objects are simplicial sets with special horn extensions.