  Translating notions and results from category theory to the theory of
computability models of Longley and Normann, we introduce the Grothendieck
computability model and the first-projection-simulation. We prove some basic
properties of the Grothendieck computability model, and we show that the
category of computability models is a type-category, in the sense of Pitts. We
introduce the notion of a fibration and opfibration-simulation, and we show
that the first-projection-simulation is a split opfibration-simulation.
