We extend categorical semantics of monadic programming to reversible
computing, by considering monoidal closed dagger categories: the dagger gives
reversibility, whereas closure gives higher-order expressivity. We demonstrate
that Frobenius monads model the appropriate notion of coherence between the
dagger and closure by reinforcing Cayley's theorem; by proving that effectful
computations (Kleisli morphisms) are reversible precisely when the monad is
Frobenius; by characterizing the largest reversible subcategory of
Eilenberg-Moore algebras; and by identifying the latter algebras as
measurements in our leading example of quantum computing. Strong Frobenius
monads are characterized internally by Frobenius monoids.