Grothendieck–Verdier duality patterns in quantum algebra

After a brief survey of the basic definitions of Grothendieck– Verdier categories and dualities, I consider in this context dualities introduced earlier in the categories of quadratic algebras and operads, largely motivated by the theory of quantum groups. Finally, I argue that Dubrovin's 'almost duality' in the theory of Frobenius manifolds and quantum cohomology must also fit a (possibly extended) version of Grothendieck–Verdier duality. (paper)