The deep and extended sky surveys performed in the last two decades have definitely shown low- and intermediate-redshift galaxies, galaxy groups and clusters to distribute in an intricately connected web-like pattern. This cosmic web features nodes, filaments, walls and voids, along which gas and galaxy density decreases, with voids being the least populated regions of the local universe. Recent studies of galaxy properties in these different cosmic features have revealed the existence of a segregation between star-forming and quiescent galaxies, with the former being located farther away from the filaments. Also, among the star-forming population the median specific star-formation rate increases with distance from the filaments, as if the cosmic web were able to quench galaxy star-formation activity. In this talk I will present how we physically quantified this cosmic-web quenching, by investigating the dependence of the mean stellar properties of SDSS central galaxies with distance from the nodes, filaments and walls identified in SDSS. In doing so we controlled for galaxy stellar mass and local density (field vs group environment). Our results show that centrals closer to either nodes, walls, or filaments are on average older, metal richer, and alpha-enhanced compared to their equal mass peers at larger distances. The amplitude of these gradients appear independent of the galaxy local environment. Our analysis thus supports a cosmic web quenching that rises from nurture effects (e.g. ram pressure stripping and strangulation) and/or nature effects linked to the intrinsic properties of the cosmic web.