  An omnipresent feature of the multi-phase ``cosmic web'' is that warm/hot
(>$10^5$ K) ionized gas pervades it. This gas constitutes a relevant
contribution to the overall universal matter budget across multiple scales,
from the several tens of Mpc-scale IGM filaments, to the Mpc ICM, all the way
down to the CGM surrounding individual galaxies, on scales from ~1 kpc up to
their respective virial radii (~100 kpc). The study of the hot baryonic
component of cosmic matter density represents a powerful means for constraining
the intertwined evolution of galactic populations and large-scale cosmological
structures, for tracing the matter assembly in the Universe and its thermal
history. To this end, the SZ effect provides the ideal observational tool for
measurements out to the beginnings of structure formation. The SZ effect is
caused by the scattering of the photons from the cosmic microwave background
off the hot electrons embedded within cosmic structures, and provides a
redshift-independent perspective on the thermal and kinematic properties of the
warm/hot gas. Still, current and future (sub)mm facilities have been providing
only a partial view of the SZ Universe due to any combination of: limited
angular resolution, spectral coverage, field of view, spatial dynamic range,
sensitivity. In this paper, we motivate the development of a wide-field,
broad-band, multi-chroic continuum instrument for the Atacama Large Aperture
Submillimeter Telescope (AtLAST) by identifying the scientific drivers that
will deepen our understanding of the complex thermal evolution of cosmic
structures. On a technical side, this will necessarily require efficient
multi-wavelength mapping of the SZ signal with an unprecedented spatial dynamic
range (from arcsecond to degree scales) and we employ theoretical forecasts to
determine the key instrumental constraints for achieving our goals. [abridged]
