CABool is monadic over almost all categories

Abstract

We give a description of the Eilenberg-Moore category of algebras induced by an adjunction F ⊣ U : Set op → C and prove that, under mild conditions, U is weakly monadic. This enables us to sketch another proof of the well-known fact that CABool is equivalent to Set op, which justifies our title. Next, we characterize the category of algebras Top T , T being the monad induced in Top by the Sierpiński space, in terms of the topology of the spaces underlying T -algebras, concluding that Top T is, up to isomorphism, and so CABool is, up to equivalence, a reflective subcategory of Top. Finally, we show that the conclusion still holds for the skeleton of each category of algebras C T , whenever T is induced by a C -object A satisfying some conditions, if C has no unnatural isomorphisms between any two powers of A, in the sense of V. Trnková.