Fibrations of Simplicial sets Hello,
Maybe it is too vague a question, but I would like to ask if anybody could say some explanatory words about the importance (for infinity category study) of studying all the kinds of fibrations there are for simplicial sets (there are more than eight: right, left, mid, Kan, trivial, and so on..). In general, it might be believed that there are few "important" notions, so why do we have so many fibrations, all of which are studied?
Thank you
 A: I'm not an expert, but, here is my understanding. Right-fibrations are important because they are the infinity-version of a category fibered in groupoids (that is an infinity-category fibered in infinity-groupoids). In particular, given an infinity-category $C$,there is a model structure on $sSet/C$, called the contravariant model structure, such that the fibrant and cofibrant objects are precisely the right-fibrations over $S$, and this model structure is Quillen-equivalent (through a generalization of the Grothendieck construction) to the projective model-structure of simplicial presheaves over $w(C)$, where $w(C)$ is a simplicial category and $w$ is the left-adjoint to the homotopy-coherent nerve. Both of these (simplicial) model categories model $Fun(C^{op},\infty-Gpd)$- the infinity-categeory of "weak presheaves in infinity groupoids". So, the upshot is, right fibrations are the infinity-analogue of Grothendieck fibrations in groupoids and provide a model for weak presheaves. This presheaf infinity-topos is the starting point for higher topos theory; infinity topoi are just left-exact (accessible) localizations of such presheaf-infinity categories.
Now, dually, left-fibrations are a model for "infinity-categories COfibered in infinity-groupoids".
The next step, is Cartesian-fibrations. Cartesian-fibrations are the infinity-version of categories fibered in categories (not necessarily groupoids), i.e. they are "infinity categories fibered in infinity-categories". Nearly everything above goes through again, except we need to work with marked-simplicial sets, where we "mark the cartesian-edges".
Again, dually, CoCartesian fibrations model "infinity categories cofibered in infinity-categories". You may wonder why we need both notions? In fact, we need both notions TOGETHER in order to define adjunctions. An adjunction between two infinity-categories $C$ and $D$ is a functor $K \to \Delta[1]$ which is simultaneously a Cartesian-fibration and a CoCartesian fibration, together with Joyal-equivalences $K_{0} \cong C$ and $K_{1} \cong D$. This definition is a generalization to the infinity-world of a characterization of adjunctions using cographs.
Now, Kan-fibrations are a relic of homotopy theory. They are the fibrations on the Quillen-model structure on simplicial sets. In a similar spirit, categorical fibrations are the fibrations in the Joyal-model structure on simplicial sets. Other than that, they are not that well behaved; they don't really play a role in infinity-category theory.
Finally, inner fibrations, as far as I know, are only used in defining Cartesian fibrations. That is, a Cartesian fibration is defined to be an inner fibration satisfying extra properties.
I hope this helps.
