Australian Category Seminar

A category parametrized by a category C was taken to merely be a functor $\partial \colon A \to C$. Categorical matters to do with limits were claimed to do with cartesian cones in A. The existence of enough cartesian cones includes the fibration condition and completeness of fibres with respect to stable limits. Monics also have to do with limits and so we get a notion of parametric monic in A. This leads to a natural notion of wellpoweredness for parametrized categories.