Monads and Algebras
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 Transformation Free Algebra Compact Hausdorff Space Left Adjoint Forgetful Functor
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- The recognition of the power and simplicity of the use of monads and comonads came quite slowly, and started from their use in homological algebra (see § VII.6). Mac Lane  mentioned in passing (his § 3) that all the standard resolutions could be obtained from universal arrows (i.e., from adjunctions). Then Godement  systematized these resolutions by using standard constructions (comonads). P. J. Huber , starting from. Huber , starting from “homotopy theory” in the Eckmann-Hilton sense, explored the examples of derived functors which can be defined by comonads and then in  studied the resulting functorial simplicial resolutions for more general abelian categories. Then Hilton (and others) raised the question as to whether any monad arises from an adjunction. Two independent answers appeared: Kleisli’s constructions in  of the “free algebra” realization and the decisive construction by Eilenberg-Moore  of the category of algebras for a monad. Stimulated by this description of the algebras, Barr-Beck in  showed how the resolutions derived from monads and comonads can be used even in non-abelian categories — obtaining the surprising result that the free group monad in Set does lead to the standard cohomology of groups. Subsequent developments in this direction are sumarized in their paper [ 1969 ].Google Scholar
- Thus, about 1965, it became urgent to decide how to characterize the category of algebras over a monad. Linton  treated the case for monads in Set, and then Beck established his theorem (unpublished, but presented at a conference in 1966 ). The absolute “coequalizer” form of the theorem, due to Paré , made possible Paré’s elegant proof (§ 9) that compact Hausdorff spaces are monadic. Many other developments in this direction are summarized in Manes’ thesis (cf. ).Google Scholar
- The description of algebras by monads is closely related to another description by algebraic theories (Lawvere , described in Pareigis ).Google Scholar