Fibration of simplicial sets

In mathematics, especially in homotopy theory,[1] a left fibration of simplicial sets is a map that has the right lifting property with respect to the horn inclusions ${\displaystyle \Lambda _{i}^{n}\subset \Delta ^{n},0\leq i<n}$.[2] A right fibration is one with the right lifting property with respect to the horn inclusions ${\displaystyle \Lambda _{i}^{n}\subset \Delta ^{n},0<i\leq n}$.[2] A Kan fibration is one with the right lifting property with respect to every horn inclusion; hence, a Kan fibration is both a left and right fibration.[3]

On the other hand, a left fibration is a coCartesian fibration and a right fibration a Cartesian fibration. In particular, category fibered in groupoids over another category is a special case of a right fibration of simplicial sets in the ∞-category setup.