  The equilibrium properties of a weakly interacting atomic Bose gas across the
Berezinskii-Kosterlitz-Thouless (BKT) and Bose-Einstein condensation (BEC)
phase transitions are numerically investigated through a dimensionality
crossover from two to three dimensions. The crossover is realised by confining
the gas in an experimentally feasible hybridised trap which provides
homogeneity along the planar xy-directions through a box potential in tandem
with a harmonic transverse potential along the transverse z-direction. The
dimensionality is modified by varying the frequency of the harmonic trap from
tight to loose transverse trapping. Our findings, based on a stochastic
(projected) Gross-Pitaevskii equation, showcase a continuous shift in the
character of the phase transition from BKT to BEC, and a monotonic increase of
the identified critical temperature as a function of dimensionality, with the
strongest variation exhibited for small chemical potential values up to
approximately twice the transverse confining potential
