  In 1966, John Isbell introduced a construction on categories which he termed
the "couple category" but which has since come to be known as the Isbell
envelope. The Isbell envelope, which combines the ideas of contravariant and
covariant presheaves, has found applications in category theory, logic, and
differential geometry. We clarify its meaning by exhibiting the assignation
sending a locally small category to its Isbell envelope as the action on
objects of a pseudomonad on the 2-category of locally small categories; this is
the Isbell monad of the title. We characterise the pseudoalgebras of the Isbell
monad as categories equipped with a cylinder factorisation system; this notion,
which appears to be new, is an extension of Freyd and Kelly's notion of
factorisation system from orthogonal classes of arrows to orthogonal classes of
cocones and cones.
