Adjunctions and Monads

This chapter presents two essential and closely related notions: adjunctions and monads; each adjunction generates a monad, each monad generates an adjunction. Every module on a ring R is an additive group; but conversely, is there a best R-module associated with an additive group? This is the idea of a pair of adjoint functors: two interdependent constructions, between two categories, in both directions. And what is a real vector space? An elegant way to grasp the spirit of that notion is to say: this is a set in which real linear combinations make sense. Like in a monoid, this approach presents an associative composition: a linear combination of linear combinations yields a linear combination. This provides an example of a monad on the category of sets: given a set A, one considers the set T(A) of all formal real linear combinations of elements of A; a vector space is a set A provided with an action of T(A), a set in which every formal linear combination has been given a value. Of course, a monad can be defined over every category, not just Set. The Beck criterion characterizes those categories which are categories of algebras for a monad.