Monads are the view of adjunctions which we get by looking at them from one end, generalising closure operations from Section 3.7. They describe (singlesorted) finitary equational theories, but also characterise many apparently unalgebraic categories as categories of infinitary algebras.
Let F\dashv U be an adjunction and put M = U·F. Then the natural transformation m = e·F (the multiplication) makes the following diagrams commute, by the triangle laws (Definition 7.2.1) and naturality of e.
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DEFINITION 7.5.1 A triple consisting of a functor M:S® S and natural transformations h:\id_{S}® M and m:M^{2}® M that obey these laws is called a monad.
REMARK 7.5.2 Algebras for functors have arisen in informatics, and algebras for monads in categorical algebra, so the uncritical reader of abstract accounts of these separate topics is in danger of making inappropriate valuejudgements. Functors and monads can both be used to code the same (free) theory, but in different ways:
In particular, MX is much bigger than TX, and may be a proper class if, for example, T = P.
We shall only consider singlesorted theories (as before, for manysorted theories we must work in Set^{S} ), and start with a free algebraic theory in the sense of Section 6.1, presented as a functor T. Let \LeftAdjoint_{0}X be the free Talgebra on X, with inclusion h_{X}:X® \Monad_{0}X, where \Monad_{0} = U·\LeftAdjoint_{0}. Then (\Monad_{0},h,m) is a monad, for a certain natural transformation m.
For example, let T = {1} +()^{2} and X = {a,b}. Then \Monad_{0}X contains


A monad is needed to code laws. For a Talgebra \ev_{A}:TA® A, the map e_{A}:\Monad_{0}A® A constructed in Example 7.4.6 evaluates terms of arbitrary depth, whereas \ev_{A} only handles those of depth 1. The more comprehensive structure is needed to test whether A satisfies the laws of an equational theory of which T is the signature, as these laws may be between terms of any depth. But even for a free theory, e_{A} must satisfy certain conditions if it is given by some \ev_{A}.
The Kleisli and EilenbergMoore categories All monads arise from adjunctions, in fact in two different ways, both found in 1965.
PROPOSITION 7.5.3 Let (M,h,m) be a monad on a category S.
DEFINITION 7.5.4 An adjunction F\dashv U (or just the functor U) for which the functor E is a weak equivalence is said to be monadic.
Beck's theorem The importance of monadic adjunctions lies in the fact that universal constructions with algebras may be computed for the carriers, so long as the two structures commute. Jon Beck's theorem was presented at a conference in 1966 but he never wrote it up for publication.
PROPOSITION 7.5.6 The forgetful functor Mod(M,h,m)® Set creates all small limits (Definition 4.5.10), and whatever colimits M preserves.
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PROOF: The structure map for the (co)limit algebra is the mediator a to lim_{i}\typeA_{i} or from \colim_{i}M\typeA_{i} shown dotted. Similarly the EilenbergMoore equations h;a = id and Ma;a = m;a hold because both sides are mediators to lim_{i}\typeA_{i} or from \colim_{i}M\typeA_{i} or \colim_{i}M^{2}\typeA_{i}. []
This property is characteristic, as we can see from a special case:
REMARK 7.5.7 [Bob Paré] Consider contractible coequalisers (Exercise 5.2). Such coequalisers exist in any category S where idempotents split, and they are preserved by all functors out of S, in particular by M.
An algebra for the monad (by definition) makes the two squares below commute and the rows identities (recall that Z indicates a naturality square). It is a contractible coequaliser.
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The contraction h_{MA} is not a homomorphism: the coequaliser diagram for the algebras only becomes contractible when we apply the forgetful functor U. Such a parallel pair of homomorphisms \nearrow ,r:C\rightrightarrows B for which there is some Smap \:B®C with \;\nearrow = \id_{B} and \nearrow ;\;r = r;\;r is called a Ucontractible coequaliser.
Notice in particular that the structure map of any algebra a:FA® A is the coequaliser of Fa and m_{A} not only in S but also in the category of algebras and homomorphisms, so it is a self presentation.
EXAMPLE 7.5.8 For the discrete\dashv points adjunction (Theorem 7.4.1), the maps Fa and m_{A} are both the identity on the underlying set of A equipped with the discrete topology, so the original topology on the space A is not recovered as a coequaliser.
THEOREM 7.5.9 [Jon Beck] Let F\dashv U between categories in which idempotents split (Exercise 4.16). Then the following are equivalent:
The comparison functor is full and faithful (and in particular reflects invertibility) iff every e_{A} is a selfpresentation.
PROOF: To show that E is full and faithful, let g:UA® UB in S. There is a unique f with g = Uf iff the top row is a coequaliser in A:
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For essential surjectivity, let a:UFX® X be an algebra. Then Fa and e_{FX} form a Ucontractible pair in A, whose coequaliser A (created by U) gives rise to the algebra E(A) º (X,a). []
PROPOSITION 7.5.11 Every finitary monad on Set, ie for which M preserves filtered colimits, is isomorphic to that given by the free algebras for some singlesorted finitary algebraic theory L, and then Mod(L) is externally locally finitely presentable (Definition 6.6.14(c)).
PROOF: We shall describe the Lawvere theory (Exercise 4.29). The set of kary operationsymbols is Cn^{×}_{L}(k,1) = Mk. The composition

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The multiplication m thereby captures the laws of L ( cf saturation for closure conditions). For a finite set n, the value at n of the monad derived from the theory L is the carrier of the free algebra on n generators, which is Cn^{×}_{L}(n,1) = Mn as required, and similarly for functions between finite sets. Since every set is a filtered colimit of finite(ly presented) sets (Exercise 7.21) and M preserves filtered colimits, M is determined up to unique isomorphism by its values Mn, so it agrees with the monad arising from L. Again since M preserves filtered colimits, U creates them; F preserves finite presentability, and Mod(L) is LFP. []
As we have repeatedly pointed out, infinitary algebraic theories with arbitrary laws present problems when we reject the Axiom of Choice. Fred Linton developed monads as a useful alternative, showing that the symbolic and diagrammatic notions are equivalent in the presence of Choice [Lin69 ]. Monads can give an unexpected algebraic perspective on topology: for example the category of compact Hausdorff spaces is monadic over Set, the left adjoint being the space of ultrafilters. See [ Man76] for this and a monadic treatment of algebra.
REMARK 7.5.12 The infinitary operations of most interest are meets, joins, limits and colimits. Proposition 3.2.7(b) and Theorem 3.9.7 showed in particular how to add joins to a poset, retaining certain specified joins. If a poset A is able to carry an algebra for a joinadding monad, then this structure a is unique, and a\dashv h_{A}; such monads may be recognised by the fact that m_{X}\dashv h_{MX}. They were investigated by Anders Kock [Koc95] and V. Zöbelein. For meet or limit adding monads, the adjunctions are reversed. Alan Day found such a monad over Sp (or Dcpo) whose algebras are continuous lattices and whose homomorphisms preserve Ù and Ú^{} [GHK^{+}80]. This was generalised (to smaller classes of meets) in [Tay90] and [ Sch93].
REMARK 7.5.13 The Transfinite Recursion Theorem 6.7.4 relates the algebras for the covariant powerset monad (complete join semilattices) equipped with an endofunction, to the well founded coalgebras for the functor alone, which we discussed in Example 6.3.3 and Remark 6.7.14 . In fact the coalgebras for any functor which happens to be part of a monad carry partial ``successor'' and ``union'' operations (Exercise 7.45) [CP92, JM95,Tay96b].
REMARK 7.5.14 Eugenio Moggi has argued that certain monads should be regarded as notions of computation [Mog91]. To each type X we associate the type MX of computations (whose ultimate results would be) of type X. For example, with the lift monad (Definition 3.3.7), morphisms G® LiftX are partial (possibly nonterminating) programs. Additional structure, known as a strength, is needed for this, as for parametric recursion in Remark 6.1.6 and Exercise 6.23. Moggi gave a symbolic form (the `` let'' calculus) for this piece of category theory.
REMARK 7.5.15 Jon Beck himself studied monads to unify homological algebra [ BB69]. Applying the functor M repeatedly to an object X, the natural transformations M^{ni1}m_{MiX} provide the boundary operations (Smorphisms) of a simplicial complex (Exercise 4.15). On the other hand the ``ordinals'' M^{n}Æ just mentioned provide an abstract system of simplices (point, interval, triangle, tetrahedron, ...), so we may use maps M^{n}Æ® X to investigate the homotopy of any object X.
Echoing what we said about the utility of adjoints at the end of Section 7.3, whenever you have a construction which yields an object of the same category as its data, it is well worth looking for natural transformations making it a monad. The opportunities for mathematical investigation from such a simple thing are quite striking: the algebras for the monad often turn out to form important categories in their own right, and the iterates of its functor provide detailed information about the abstract topology and recursion theory of the category.