Relative Monad

A monad is an endofunctor with unit and multiplication; a relative monad keeps the unit-and-extension shape but drops the demand that the underlying map be an endofunctor, by measuring everything relative to a chosen functor .

Fix a functor . Following Altenkirch, Chapman & Uustalu (“Monads Need Not Be Endofunctors”), a relative monad along is an object-map together with a unit and an extension operator sending to , subject to the monad-style laws , -style unit conditions, and associativity of . When and this is an ordinary monad; in general need not be an endofunctor, since its source and target are related only through .

Example. The home of Arrows

Arrows, Hughes’ generalisation of monads for effectful computation, are awkward to present as monads, because the natural candidate is a Yoneda-style embedding that lands in a functor category far too big. Taking to be the Yoneda embedding fixes this. A relative monad along it is defined only on representables, so an Arrow becomes a relative monad rather than a monad on an oversized ambient category.

Remark. Strength is what fills a Kleisli category

Commutative tensorial strength is the structure that carries products and coproducts into a Kleisli category. A strong monad has a monoidal Kleisli category, so effectful maps can be paired side by side. In a probabilistic setting this is what makes a Kleisli category of a probability monad monoidal by tensoring independent computations. A relative monad must supply the same strength if its Kleisli-like category is to be monoidal too.