  The ribbon cocycle invariant is defined by means of a partition function
using ternary cohomology of self-distributive structures (TSD) and colorings of
ribbon diagrams of a framed link, following the same paradigm introduced by
Carter, Jelsovsky, Kamada, Langfor and Saito in Transactions of the American
Mathematical Society 2003;355(10):3947-89, for the quandle cocycle invariant.
In this article we show that the ribbon cocycle invariant is a quantum
invariant. We do so by constructing a ribbon category from a TSD set whose
twisting and braiding morphisms entail a given TSD $2$-cocycle. Then we show
that the quantum invariant naturally associated to this braided category
coincides with the cocycle invariant. We generalize this construction to
symmetric monoidal categories and provide classes of examples obtained from
Hopf monoids and Lie algebras. We further introduce examples from
Hopf-Frobenius algebras, objects studied in quantum computing.

Geometric Topology;Quantum Algebra