Lifting property

Summary

A morphism $i$ has the left lifting property with respect to a morphism $p$ when every suitable commutative square has a diagonal filler. The square has $i$ on its left side and $p$ on its right side. Equivalently, $p$ has the right lifting property with respect to $i$. Lifting properties define fibrations and cofibrations in homotopy theory and in model categories, and they describe many classes of maps by a single test. When the filler is always unique, $i$ and $p$ are called orthogonal.