  At high redshift, the cosmic web is widely expected to have a significant
impact on the morphologies, dynamics and star formation rates of the galaxies
embedded within it, underscoring the need for a comprehensive study of the
properties of such a filamentary network. With this goal in mind, we perform an
analysis of high-$z$ gas and dark matter (DM) filaments around a Milky Way-like
progenitor simulated with the {\sc ramses} adaptive mesh refinement (AMR) code
from cosmic scales ($\sim$1 Mpc) down to the virial radius of its DM halo host
($\sim$20 kpc at $z=4$). Radial density profiles of both gas and DM filaments
are found to have the same functional form, namely a plummer-like profile
modified to take into account the wall within which these filaments are
embedded. Measurements of the typical filament core radius $r_0$ from the
simulation are consistent with that of isothermal cylinders in hydrostatic
equilibrium. Such an analytic model also predicts a redshift evolution for the
core radius of filaments in fair agreement with the measured value for DM $(r_0
\propto (1+z)^{-3.18\pm 0.28})$. Gas filament cores grow as $(r_0 \propto
(1+z)^{-2.72\pm 0.26})$. In both gas and DM, temperature and vorticity sharply
drop at the edge of filaments, providing an excellent way to constrain the
outer filament radius. When feedback is included the gas temperature and
vorticity fields are strongly perturbed, hindering such a measurement in the
vicinity of the galaxy. However, the core radius of the filaments as measured
from the gas density field is largely unaffected by feedback, and the median
central density is only reduced by about 20%.
