2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/69636We define the notion of a Kirby element of a ribbon category C (not necessarily semisimple). Kirby elements lead to 3-manifolds invariants. We characterize (in terms of the structure maps of some categorical Hopf algebra) a set of Kirby elements of C which is sufficiently large to recover the known quantum invariants computed from C. The cases of a semisimple ribbon category and of a category of representations are explored in details.45 pagesGeometric TopologyQuantum Algebra57M27; 18D10; 81R50text