  We develop a comprehensive theoretical model for Lyman-alpha emission, from
the scale of individual Lyman-alpha emitters (LAEs) to Lyman-alpha halos
(LAHs), Lyman-alpha blobs (LABs), and Lyman-alpha filaments (LAFs) of the
diffuse cosmic web itself. To do so, we post-process the high-resolution TNG50
cosmological magnetohydrodynamical simulation with a Monte Carlo radiative
transfer method to capture the resonant scattering process of Lyman-alpha
photons. We build an emission model incorporating recombinations and collisions
in diffuse gas, including radiative effects from nearby AGN, as well as
emission sourced by stellar populations. Our treatment includes a physically
motivated dust model, which we empirically calibrate to the observed LAE
luminosity function. We then focus on the observability, and physical origin,
of the $z=2$ Lyman-alpha cosmic web, studying the dominant emission mechanisms
and spatial origins. We find that diffuse Lyman-alpha filaments are, in fact,
illuminated by photons which originate, not from the intergalactic medium
itself, but from within galaxies and their gaseous halos. In our model, this
emission is primarily sourced by intermediate mass halos ($10^{10} -
10^{11}\,$M$_{\odot}$), principally due to collisional excitations in their
circumgalactic media as well as central, young stellar populations.
Observationally, we make predictions for the abundance, area, linear size, and
embedded halo/emitter populations within filaments. Adopting an isophotal
surface brightness threshold of
$10^{-20}\,$erg$\,$s$^{-1}\,$cm$^{-2}\,$arcsec$^{-2}$, we predict a volume
abundance of Lyman-alpha filaments of ${\sim}10^{-3}$ cMpc$^{-3}\,$ for lengths
above $400\,$pkpc. Given sufficiently large survey footprints, detection of the
Lyman-alpha cosmic web is within reach of modern integral field spectrographs,
including MUSE, VIRUS, and KCWI.
