We have investigated the Y adsorption and YN thin film formation on the GaN(0001)-2×2 surfaces using first principles total energy calculations within the density functional theory. Ga-rich conditions are modeled with a Ga-bilayer terminated GaN(0001)-2×2 surface. N-rich, and intermediate growth conditions were studied using an ideally GaN bilayer terminated GaN(0001)-2×2 surface. On the ideally terminated surface, when the Y atom is constrained on top of the surface, the Y on a second-layer nitrogen atom (T4-site) is the most favorable structure. However, when the Y atom migrates into the first Ga monolayer, the displaced Ga ad-atom occupies a site bonded with three Ga atoms of the first monolayer (T4-2-site) as the most stable structure. Under Ga-rich conditions the Y atom occupies a site on top of a second-monolayer Ga (T4-site) as the most stable structure. Nevertheless, in the energetically most favorable configuration, the Y atom replaces a third inner layer Ga atom bonding to N atoms and forming an YN pair. When the coverage is increased to a full Y monolayer, three different YN configurations are formed. Under Ga-rich conditions an YN bilayer is formed underneath a Ga bilayer, in the intermediate conditions an YN bilayer is formed under a Ga-T4 layer, and under N-rich conditions a w-YN bilayer is formed on top of the ideal surface. The stable configurations density of states shows that the metallic characteristic is preserved.