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In recent years, algebraic studies of the differential calculus and integralcalculus in the forms of differential algebra and Rota-Baxter algebra have beenmerged together to reflect the close relationship between the two calculithrough the First Fundamental Theorem of Calculus. In this paper we study thisrelationship from a categorical point of view in the context of distributivelaws which can be tracked back to the distributive law of multiplication overaddition. The monad giving Rota-Baxter algebras and the comonad givingdifferential algebras are constructed. Then a mixed distributive law of themonad over the comonad is established. As a consequence, we obtain monads andcomonads giving the composite structures of differential and Rota-Baxteralgebras.

Monads and distributive laws for Rota-Baxter and differential algebras