Lyman Alpha Absorption in The Cosmic Web Dick Bond, CITA Large scale structure at high redshift is described in terms of the Cosmic Web picture for $\{S,\Lambda,O,H\}$CDM models: how galactic-scale ``peak-patches'', filaments and membranes create an interconnected intergalactic medium. The ideas are applied to Ly$\alpha$ forest simulations of ``shear-field patches'', which use smooth particle hydrodynamics for the gas and tree$P^3M$ for the gravitational forces due to dark matter and gas. The response to changes in power spectrum shape and amplitude, and in cosmological parameters, is described: with galaxy-scale {\it rms} fluctuations $\sim 1$ at $z=3$ and a UV choice motivated by proximity effect observations, the results agree well with HI and HeII column density data, and $\Omega_b {\rm h}^2$ derived from UV rescaling is in good agreement with big bang nucleosynthesis estimates. Clouds are brought back into the Lyman alpha cloud story: the importance of $\sim 30 \kms$ halos along filaments is shown, an aspect missed by lower resolution simulations. The signal from the intercloud medium is dominant for $N_{HI} < 10^{14} {\rm cm}^{-2}$, with contributions from filaments with overdensities $\sim 5 - 10$ and, at lower column depths, from membranes connecting the filaments with overdensities $\sim 2$.