Abstract
A categorical bimonoid consists of a monoid and a comonoid which act homomorphically on one another. In applications bimonoids are typically called bialgebras or Hopf algebras. The definitions are given at a level suitable to computer science applications and examples are included. The elements of the theory of graded bialgebras are developed and we provide a universal recursion theorem for morphisms between algebras and between coalgebras. With this theory we explicate refinements of concurrent programs and give a treatment of traces. The concluding section characterizes all coalgebras which interact with the match algebra to form a bialgebra and characterizes all algebras which interact with the fork (diagonal) coalgebra to form a bialgebra.
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