  Virgo is the nearest galaxy cluster; it is thus ideal for studies of galaxy
evolution in dense environments in the local Universe. It is embedded in a
complex filamentary network of galaxies and groups, which represents the
skeleton of the large scale Laniakea supercluster. Here we assemble a
comprehensive catalog of galaxies extending up to ~12 virial radii in
projection from Virgo to revisit the Cosmic Web structure around it. This work
is the foundation of a series of papers that will investigate the
multi-wavelength properties of galaxies in the Cosmic Web around Virgo. We
match spectroscopically confirmed sources from several databases and surveys
including HyperLeda, NASA Sloan Atlas, NED, and ALFALFA. The sample consists of
~7000 galaxies. By exploiting a tomographic approach, we identify 13 filaments,
spanning several Mpc in length. Long > 17 Mpc/h filaments, tend to be thin (< 1
Mpc/h in radius) and with a low density contrast (< 5), while shorter filaments
show a larger scatter in their structural properties. Overall, we find that
filaments are a transitioning environment between the field and cluster in
terms of local densities, galaxy morphologies, and fraction of barred galaxies.
Denser filaments have a higher fraction of early type galaxies, suggesting that
the morphology-density relation is already in place in the filaments, before
galaxies fall into the cluster itself. We release the full catalog of galaxies
around Virgo and their associated properties.
