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The intracluster medium (ICM) today is comprised largely of hot gas with clouds of cooler gas of unknown origin and lifespan. We analyse the evolution of cool gas (temperatures |$\lesssim 10^{4.5}$|  K) in the ICM of 352 galaxy clusters from the TNG-Cluster simulations, with present-day mass |$\sim 10^{14.3-15.4}\, {\rm M}_\odot$|⁠. We follow the main progenitors of these clusters over the past |$\sim 13$| billion years (since |$z\lesssim 7$|⁠) and find that, according to TNG-Cluster, the cool ICM mass increases with redshift at fixed cluster mass, implying that this cooler past of the ICM is due to more than just halo growth. The cool cluster gas at |$z\lesssim 0.5$| is mostly located in and around satellite galaxies, while at |$z\gtrsim 2$| cool gas can also accrete via filaments from the intergalactic medium. Lower-mass and higher-redshift clusters are more susceptible to cooling. The cool ICM mass correlates with the number of gaseous satellites and inversely with the central supermassive black hole (SMBH) mass. The average number of gaseous satellites decreases since |$z=2$|⁠ , correlating with the decline in the cool ICM mass over cosmic time, suggesting a link between the two. Concurrently, kinetic SMBH feedback shifts the ICM temperature distribution, decreasing the cool ICM mass inside–out. At |$z\approx 0.5$|⁠ , the predicted Mg  ii column densities are in the ballpark of recent observations, where satellites and other haloes contribute significantly to the total Mg  ii column density. Suggestively, a non-negligible amount of the ICM cool gas forms stars in situ at early times, reaching |$\sim 10^{2}\, {\rm M}_\odot \, {\rm ~yr^{-1}}$| and an H  |$\alpha$| surface brightness of |$\sim 10^{-17}\, {\rm ~erg\, s^{-1}\, cm^{-2}\, arcsec^{-2}}$| at |$z\approx 2$|⁠ , detectable with Euclid and JWST. [ABSTRACT FROM AUTHOR]

We present the 'Cosmological Jellyfish' project – a citizen-science classification program to identify jellyfish (JF) galaxies within the IllustrisTNG cosmological simulations. JF are satellite galaxies that exhibit long trailing gas features – 'tails' – extending from their stellar body. Their distinctive morphology arises due to ram-pressure stripping (RPS) as they move through the background gaseous medium. Using the TNG50 and TNG100 simulations, we construct a sample of |$\sim 80\, 000$| satellite galaxies spanning an unprecedented range of stellar masses, |$10^{8.3}{-}10^{12.3}\, \mathrm{M_\odot }$|⁠ , and host masses, |$M_\mathrm{200,c}=10^{10.4}{-}10^{14.6}\, \mathrm{M_\odot }$| back to z  = 2. Based on this sample, |$\sim 90\, 000$| galaxy images were presented to volunteers in the citizen-science Zooniverse platform, who were asked to determine whether the galaxy image resembles a JF. Based on volunteer votes, each galaxy received a score determining if it is a JF or not. This paper describes the project, the inspected satellite sample, the methodology, and the classification process that resulted in a data set of 5307 visually identified JF galaxies. We find that JF is common in nearly all group- and cluster-sized systems, with the JF fraction increasing with host mass and decreasing with satellite stellar mass. We highlight JF galaxies in three relatively unexplored regimes: low-mass hosts of |$M_\mathrm{200,c}\sim 10^{11.5}{-}10^{13}\, \mathrm{M_\odot }$|⁠ , radial positions within hosts exceeding the virial radius R 200, c, and at high redshift up to z  = 2. The full data set of our JF scores is publicly available and can be used to select and study JF galaxies in the IllustrisTNG simulations. [ABSTRACT FROM AUTHOR]