Fibration of simplicial sets

In mathematics, especially in homotopy theory, a left fibration of simplicial sets is a map that has the right lifting property with respect to the horn inclusions ${\mathrm{\Lambda }}_i^n\subset {\mathrm{\Delta }}^n,0\le i<n$. A right fibration is one with the right lifting property with respect to the horn inclusions ${\mathrm{\Lambda }}_i^n\subset {\mathrm{\Delta }}^n,0<i\le n$. 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.

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.
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