  Discontinuous shear-thickening in dense suspensions naturally emerges from
the activation of frictional forces by shear flow in non-Brownian systems close
to jamming. Yet, this physical picture is incomplete as most experiments study
soft colloidal particles subject to thermal fluctuations. To characterise
discontinuous shear-thickening in colloidal suspensions we use computer
simulations to provide a complete description of the competition between
athermal jamming, frictional forces, thermal motion, particle softness, and
shear flow. We intentionally neglect hydrodynamics, electrostatics,
lubrication, and inertia, but can nevertheless achieve quantitative agreement
with experimental findings. In particular, shear-thickening corresponds to a
crossover between frictionless and frictional jamming regimes which is
controlled by thermal fluctuations and particle softness and occurs at a
softness dependent P\'eclet number. We also explore the consequences of our
findings for constant pressure experiments, and critically discuss the reported
emergence of `S-shaped' flow curves. Our work provides the minimal ingredients
to quantitatively interpret a large body of experimental work on discontinuous
shear-thickening in colloidal suspensions.
