BC fibration

Definition 15.1.2. Let \((C,C_L,C_R)\) be an adequate triple. A functor \(p\colon E \to C\) is called a Beck–Chevalley fibration with respect to \((C_L,C_R)\) if the following three conditions are satisfied:

(1)

Every morphism in \(C_L\) admits \(p\)-cartesian lifts with given target.

(2)

Every morphism in \(C_R\) admits \(p\)-cocartesian lifts with given source.

(3)

In a commutative square

whose image in \(C\) is a pullback square with \(p(\tilde l)\in C_L\) and \(p(\tilde r)\in C_R\), assume that \(\tilde r\) is \(p\)-cocartesian and \(\tilde l'\) is \(p\)-cartesian. Then \(\tilde r'\) is \(p\)-cocartesian if and only if \(\tilde l\) is \(p\)-cartesian.