I think I have a pretty good *intuitive* understanding of most types of [fibrations of quasicategories](https://ncatlab.org/nlab/show/fibrations+of+quasi-categories):


* a (trivial) Kan fibration is a bundle of (contractible) spaces with equivalent fibers,
* a left/right fibration is a bundle of spaces with covariant/contravariant functors between fibers,
* a (co)Cartesian fibration is the same as left/right but now the fibers are $\infty$-categories,
* an inner fibration is bundle of $\infty$-categories with correspondences between fibers.


One major exception is the class of categorical fibrations. I know they are the fibrations in the Joyal model structure on sSet but that description isn't very illuminating to me. I feel this is problematic since categorical fibrations are central to the theory of $\infty$-operads, which I am trying to learn at the moment.


What would be the best way to describe categorical fibrations in a similarly intuitive way?



Intuition for categorical fibrations?
Categorical fibrations are not particularly meaningful in their own right. Luckily, there is a characterization in the most interesting case, of categorical fibrations $p:Q\to R$ between quasicategories. Namely such a map $p$ is nothing more than an inner fibration and an *isofibration*, that is, it is weakly orthogonal to the inclusion of either endpoint into $E[1]$, the nerve of the isomorphism category. Alternative characterizations are that the restriction of $p$ to the cores is a Kan fibration, which might be the most intuitive description, or that $p$ induces an isofibration on homotopy categories. This approach to categorical fibrations is used throughout Riehl and Verity's work and can also be found in Rezk's notes "Stuff about Quasicategories."


It may be worth remarking here that every functor of $\infty$-categories is equivalent to an inner fibration with the same codomain, so that the most invariant and conceptual way of thinking about a categorical fibration may be simply as an isofibration, full stop. This is in close analogy to the canonical model structure on the category of small categories.


