Unveiling the faint ultraviolet Universe.

With this paper we participate to the call for ideas issued by the European Space Agency to define the Science Program and plan for space missions from 2035 to 2050. In particular we present five science cases where major advancements can be achieved thanks to space-based spectroscopic observations at ultraviolet (UV) wavelengths. We discuss the possibility to (1) unveil the large-scale structures and cosmic web in emission at redshift <~1.7; (2) study the exchange of baryons between galaxies and their surroundings to understand the contribution of the circumgalactic gas to the evolution and angular-momentum build-up of galaxies; (3) constrain the efficiency of ram-pressure stripping in removing gas from galaxies and its role in quenching star formation; (4) characterize the progenitor population of core-collapse supernovae to reveal the explosion mechanisms of stars; (5) target accreting white dwarfs in globular clusters to determine their evolution and fate. These science themes can be addressed thanks to UV (wavelength range lambda ~ 90 - 350 nm) observations carried out with a panoramic integral field spectrograph (field of view ~ 1 x 1 arcmin^2 ), and medium spectral (R = 4000) and spatial (~ 1" - 3") resolution. Such a UV-optimized instrument will be unique in the coming years, when most of the new large facilities such as the Extremely Large Telescope and the James Webb Space Telescope are optimized for infrared wavelengths.