Definition of duality.- Multiplicative categories.- Algebras over a category with multiplication.- Cogebras over a category with multiplication.- Bigebras.- C*-algebras.- A multiplication for the category C*.- The confunctor spec for commutative C*-algebras.- Finite coproducts of commutative C*-algebras with identity.- C*-Cogebras.- The duality of compact semigroups.- Involutive bigebras and their relation to C*-bigebras.- Dualities for compact groups.- Dualities for totally disconnected compact semigroups.- Discrete involutive semigroups and duality.