Categories and Monads in HOL-Omega

We consider HOL-Omega, a recent extension by Homeier to the type system of the HOL theorem prover. We describe how it permits an implementation of categories, where we model the objects of a category as the types of HOL-Omega. This gives a “light-weight” implementation of categories into a theorem prover: we explore how this is sufficient for easily proving some non-trivial results of category theory, although some other results cannot be expressed in HOL-Omega in this way. We illustrate the use of HOL-Omega by first proving the multiple characterisations of an adjoint pair of functors. Then we consider compound monads, where we consider distributive laws and the conditions for compatibility of monads. We show how in HOL-Omega we can express the concept of a monad in the Kleisli category of another monad, and use this to obtain easy proof of some results about distributive laws, including a new result simplifying the conditions of Barr & Wells for compatibility of monads.