Fibration of simplicial sets explained

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

Λ ^{n} _{i} \subset \Delta ^{n,} 0 \le i < n . A right fibration is one with the right lifting property with respect to the horn inclusions Λ ^{n} _{i} \subset \Delta ^{n,} 0 < i \le n . 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.^[2]

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.
• Web site: J. . Lurie . Lecture 9 of Algebraic K-Theory and Manifold Topology (Math 281) . 2009.

Notes and References

1. Raptis . George . 2010 . Homotopy theory of posets . Homology, Homotopy and Applications . EN . 12 . 2 . 211–230 . 10.4310/HHA.2010.v12.n2.a7 . 1532-0081. free .
2. Beke . Tibor . 2008 . Fibrations of simplicial sets . math.CT . 0810.4960 .