is the mass of air that's keeping this warm air near the surface in its place a warm or cold mass of air
Surface air is warmed by the sun heating the surface, which in turn heats the air by conduction.
Warm (less dense) air is prevented from rising on a synoptic scale by a huge mass of cool, dense, air at tropopause level trying to sink onto it. This is the driver of the whole system.
But on a local scale air will rise unless, in rising and cooling at 2*/1000', it encounters air that is warmer & less dense than itself. Perhaps the air above is 2.5*/1000' cooler, so for every 1000' of height rise the surface air becomes 0.5 degrees cooler than the new air it encounters. Being relatively cooler and relatively denser, the rising air immediately tends to sink again. It is kept in its place by warm air immediately above it. But this warm air isn't driving the whole system; it is part of the subsidence & stable airmass. You have to go very high to find the subsiding, denser, air.
What causes high level air to descend, driving a surface high pressure system? It could be Hadley effects (see "Hadley cell") for example. But there are many other effects...