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.

References

• Ch. 2 of Lurie's Higher Topos Theory.
• Lurie, J. (2009). "Lecture 9 of Algebraic K-Theory and Manifold Topology (Math 281)" (PDF).