Adequate triple

Definition 13.1.2 ([Barwick (2017)]). An adequate triple \((C,C_L,C_R)\) consists of an \(\infty \)-category \(C\) equipped with two wide subcategories \(C_L,C_R \subseteq C\) satisfying the condition that for every morphism \(l\colon X \to Y\) in \(C_L\) and \(r\colon Y' \to Y\) in \(C_R\), there exists a pullback square of the form

and we have \(l' \in C_L\) and \(r' \in C_R\). We refer to \(C_L\) as the class of backwards morphisms and to \(C_R\) as the class of forward morphisms.

Given another adequate triple \((D,D_L,D_R)\), a morphism of adequate triples is a functor \(F\colon C \to D\) that sends \(C_L\) to \(D_L\) and \(C_R\) to \(D_R\), and sends every pullback square of the form (13.1), with \(l \in C_L\) and \(r \in C_R\), to a pullback square in \(D\). We denote by \[ \AdTrip \quad \subseteq \quad \Fun (\pullback ,\Cat _{\infty }) \] the non-full subcategory spanned by those diagrams \(C_L \hookrightarrow C \hookleftarrow C_R\) in \(\Cat _{\infty }\) that correspond to adequate triples \((C,C_L,C_R)\) and those morphisms that correspond to morphisms of adequate triples.