  The past decade has seen an explosion of discoveries and new insights into
the diffuse gas within galaxies, galaxy clusters, and the filaments composing
the Cosmic Web. A new decade will bring fresh opportunities to further this
progress towards developing a comprehensive view of the composition, thermal
state, and physical processes of diffuse gas in the Universe. Ultraviolet (UV)
spectroscopy, probing diffuse 10^4-10^6 K gas at high spectral resolution, is
uniquely poised to (1) witness environmental galaxy quenching processes in
action, such as strangulation and tidal- and ram-pressure stripping, (2)
directly account for the baryon content of galaxy clusters in the cold-warm
(T<10^6 K) gas, (3) determine the phase structure and kinematics of gas
participating in the equilibrium-regulating exchange of energy at the cores of
galaxy clusters, and (4) map cold streams and filaments of the Cosmic Web that
feed galaxies and clusters. With a substantial UV undertaking beyond the Hubble
Space Telescope, all of the above would be achievable over the entire epoch of
galaxy cluster formation. Such capabilities, coupled with already-planned
advancements at other wavelengths, will transform extragalactic astronomy by
revealing the dominant formation and growth mechanisms of gaseous halos over
the mass spectrum, settling the debate between early- and late-time metal
enrichment scenarios, and revealing how the ecosystems in which galaxies reside
ultimately facilitate their demise.
