Left derivable cocartesian fibration

Definition 20.1.10. Let \(p\colon C \to S\) be a cocartesian fibration. Assume that for every object \(s \in S\) we are given a class \(W_s\) of morphisms in the fiber \(C_s := p^{-1}(s)\), and let \(\gamma _s\colon C_s \to C_s[W_s^{-1}]\) denote the localization functor. We say that \(p\) is left derivable with respect to \(\{W_s\}_{s \in S}\) if the following two conditions are satisfied:

(1)

For every morphism \(f\colon s \to t\) in \(S\), consider the cocartesian transport functor \(f_!\colon C_s \to C_t\). Then \(\gamma _t \circ f_!\colon C_s \to C_t[W_t^{-1}]\) admits an absolute left derived functor \[ \bL f_!\colon C_s[W_s^{-1}] \to C_t[W_t^{-1}]. \]

(2)

For two composable morphisms \(s \xrightarrow {f} t \xrightarrow {g} u\) in \(S\), the canonical comparison map constructed below \[ \bL g_! \circ \bL f_! \to \bL (g \circ f)_! \] is a natural isomorphism.