  This paper investigates the hierarchy of baryon physics assembly bias
relations obtained from state-of-the-art hydrodynamic simulations with respect
to the underlying cosmic web spanned by the dark matter field. Using the Bias
Assignment Method (BAM) we find that non-local bias plays a central role. We
classify the cosmic web based on the invariants of the curvature tensor defined
not only by the gravitational potential, but especially by the over-density, as
small scale clustering becomes important in this context. First, the gas
density bias relation can be directly mapped onto the dark matter density field
to high precision exploiting the strong correlation between them. In a second
step, the neutral hydrogen is mapped based on the dark matter and the gas
density fields. Finally, the temperature is mapped based on the previous
quantities. This permits us to statistically reconstruct the baryon properties
within the same simulated volume finding percent-precision in the two-point
statistics and compatible results in the three-point statistics, in general
within 1-$\sigma$, with respect to the reference simulation (with 5 to 6 orders
of magnitude less computing time). This paves the path to establish the best
set-up for the construction of mocks probing the intergalactic medium for the
generation of such key ingredients in the statistical analysis of large
forthcoming missions such as DESI, Euclid, J-PAS and WEAVE.
