Admissibility structure

Let \(C\) be a category with finite limits. An admissibility structure is a wide subcategory \(C^{\ad} \subseteq C\) of admissible morphisms satisfying the following properties:

• \(C^{\ad}\) is closed under base change in \(C\);

• \(C^{\ad}\) is left cancellable: given morphisms \(f\colon X \to Y\) and \(g\colon Y \to Z\), if \(g\) and \(gf\) are admissible morphisms, then so is \(f\);

• admissible morphisms are closed under retracts.