• Saunders Mac Lane
Chapter
Part of the Graduate Texts in Mathematics book series (GTM, volume 5)

## 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 τSet, 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〉: SetAlg τ resides in the category Set; indeed, the composite T = GF is a functor SetSet , 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〉:XA 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).

## Keywords

Natural Transformation Free Algebra Compact Hausdorff Space Left Adjoint Forgetful Functor
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## Notes

2. Thus, about 1965, it became urgent to decide how to characterize the category of algebras over a monad. Linton [1966] treated the case for monads in Set, and then Beck established his theorem (unpublished, but presented at a conference in 1966 ).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 [1971], 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. [1969]).Google Scholar
3. The description of algebras by monads is closely related to another description by algebraic theories (Lawvere [1963], described in Pareigis [1970]).Google Scholar