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).KeywordsNatural TransformationFree AlgebraCompact Hausdorff SpaceLeft AdjointForgetful FunctorThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.