Bar categories and star operations
The category of complex vector spaces admits an involutive endofunctor taking each vector space to itself with conjugate scalar multiplication. The bar categories defined here abstract this to a pseudo-involutive endofunctor on a monoidal category; there are some more isomorphisms and some axioms. Categories of modules (or comodules) over a star-Hopf algebra are clear examples; but there are others involving flip-star-, and quasi-star-Hopf algebras. A Tannaka-like result reconstructs a star-Hopf algebra from a (strong) bar category with appropriate fibre functor.
scientific article