Monads and Algebras

Abstract

In this chapter we will examine more closely the relation between universal algebra and adjoint functors. For each type τ of algebras (§V.6), we have the category AIg _τ of all algebras of the given type, the forgetful functor G: Alg _τ→Set, and its left adjoint F, which assigns to each set S the free algebra FS of type τ generated by elements of S. A trace of this adjunction 〈F,G,φ〉: Set ⇀Alg _τ resides in the category Set; indeed, the composite T = GF is a functor Set →Set , which assigns to each set S the set of all elements of its corresponding free algebra. Moreover, this functor T is equipped with certain natural transformations which give it a monoid-like structure, called a “monad”. The remarkable part is then that the whole category Alg _τ can be reconstructed from this monad in Set. Another principal result is a theorem due to Beck, which describes exactly those categories A with adjunctions 〈 F, G, φ〉:X⇀A which can be so reconstructed from a monad T in the base category X. It then turns out that algebras in this last sense are so general as to include the compact Hausdorff spaces (§ 9).