BECK'S THEOREM CHARACTERIZING

SOFI GJING JOVANOVSKA

Abstract. In this paper, I will construct a proof of Beck's Theorem char- acterizing T -algebras. Suppose we have an adjoint pair of F and G between categories C and D. It determines a T on C. We can associate a T - to the monad, and Beck's Theorem demonstrates when the catego- ry of T -algebras is equivalent to the D. We will arrive at this result by rst de ning categories, and a few relevant concepts and theorems that will be useful for proving our result; these will include natural tranformations, adjoints, monads and more.

Contents 1. Introduction 1 2. Categories 2 3. Adjoints 4 3.1. Isomorphisms and Natural Transformations 4 3.2. Adjoints 5 3.3. Triangle Identities 5 4. Monads and Algebras 7 4.1. Monads and Adjoints 7 4.2. Algebras for a monad 8 4.3. The Comparison with Algebras 9 4.4. 11 4.5. Beck's Threorem 12 Acknowledgments 15 References 15

1. Introduction Beck's Theorem characterizing algebras is one direction of Beck's Monadicity Theorem. Beck's Monadicity Theorem is most useful in studying adjoint pairs of functors. Adjunction is a type of relation between two functors that has some very important properties, such as preservation of limits or colimits, which, unfortunate- ly, we will not touch upon in this paper. However, we need to know that it is indeed a topic of interest, and therefore worth studying. Here, we state Beck's Monadicity Theorem. Since this is an overview, all the technical terms will be de ned later.

Date: AUGUST 29, 2014. 1 2 SOFI GJING JOVANOVSKA

Theorem 1.1. A U : D ! C is monadic if and only if the following hold: (i) U has a left adjoint (ii) U re ects isomorphisms (iii) C has and U preserves all coequalizers of U-split pairs

Thus we see that Beck's Monadicity Theorem does two things. First, it shows when a functor gives rise to a monad; and secondly, if a functor U is monadic (a functor U is monadic if it determines a monad and if its domain category is equivalent to the category of algebras over that monad), what the of the left adjoint of U, say F , is like. In this paper, we will only prove an equivalent statement of the theorem in the direction assuming F is monadic. We will focus on the construction of the proof, and we will introduce all de nitions and theorems needed for it. This paper falls short in examples, as examples in tend t be very long. I will only provide the short ones. Readers can consult the materials this paper is based on: the two published books, Steve Awodey's Category Theory and 's Categories for the Working Mathematician for further explorations, and other topics that are omitted. Sometimes the de nition and the statement of theorems are taken faithfully from the sources, in other instances I have rephrased them, if I consider necessary. The proofs are reconstructed with the intention of making them more understandable for the beginners in this area. As one might have noticed, Mac Lane's book titled itself a literature for professional mathematicians, and its proof sometimes need simpli cation. My paper aims to extend those in the way that I nd most suitable through my experience of reading them. The paper is divided into a few sections. In section 2, we will rst introduce categories from scratch, so that the readers have a good idea of the objects we are dealing with. In section 3, we will de ne some concepts regarding categories, such as isomorphism, coequalizers, and natural transformations that will enable us to de ne adjunctions. It is unfortunate that I am only able to include the most relevant ones. Notions such as duality, products, and limits are such essential aspects of this subject that one may not claim to know category theory without knowing these, but they will not be included, and readers are strongly encouraged to research on those, [2] is a good source for this. Section 4 is the main part pf the paper, in which I will introduce monads, T - algebras, and their relation to adjoints. Some imporatant results include: every adjoint pair gives rise to a monad, every monad is de ned by its T -algebras, and eventually, Beck's Theorem.

2. Categories In this section, we introduce the idea of a category, and a few relevant concepts, including equalizers and adjoints. Intuitively a category is a collection of mathe- matical objects with a set of maps between the objects of the category. The objects of a category can be groups, topological spaces, etc. Thus one useful way a category can be understood as a type of mathematical objects. Of course this is not the most inclusive way of thinking about categories, as a category can consist of only one object; but such cases are not particularly useful in exploring the relations between types of mathematical objects, such as that between monads and algebras, or eld extensions and groups, as those who are familiar with Galois Theory may recall. BECK'S THEOREM CHARACTERIZING ALGEBRAS 3

De nition 2.1. A category satis es the following axioms:  It contains a family of objects, which are denoted in this paper with italic capital letters A, B, C ... The collection of all objects is denoted C0.  It contains a family of arrows, which are denoted with small italic letters f, g, h ... The collection of all arrows is denoted C1. These arrows should be understood as maps between object of the category. Therefore, for each f, there are two objects A and B in the category that exist as the domain and codomain of f, denoted as dom(f ), cod(f ), such that f : A ! B:

 For any two arrows f : A ! B and g : B ! C, the composite of f and g, g  f : A ! C is also an arrow in the category.  For any object A, there exists an identity arrow 1A such that 1A : A ! A.  Composition is associative, meaning for any three composable arrows f; g and h in the category: h  (g  f) = (h  g)  f:

 For a map f : A ! B, f  1A = f = 1B  f. Immediately, we can see a lot of familiar mathematical objects satisfying the axioms of categories. Here is a short list of examples of objects and arrows of categories, respectively:  Grp: Groups and homomorphisms  Vect: Vector spaces and linear maps  Top: Topological spaces and continuous maps  Pos: Partially ordered sets and monotone functions One natural question arises whether there can be a category whose objects are categories. The answer is yes, though the objects of the category Cat need to be restricted to small categories. We will touch upon a brief explanation for this later, as we need to rst de ne the maps between categories, for Cat to be de ned at all. De nition 2.2. A functor F : C ! D between categories C and D is a mapping from objects to objects, and arrows to arrows, such that domains, codomains, identity and composition are preserved, or  F (f : A ! B) = F (f): F (A) ! F (B),  F (1A) = 1F (A) ,  F (g  f) = F (g)  F (f). It is simple to check that a functor satis es the axioms of arrows in categories. Therefore Cat is a well-de ned category. And eventually, let us distinguish among three types of categories: large, small, and locally small. We impose a restriction on the categories we talk about in this paper, namely, all categories concerned are locally small, as when a category is \too large", we encounter diculty with conventional set theory, which is the foundation of the approach to category theory in this paper.

De nition 2.3. A category C is called small if both the collection C0 and C1 are sets. Otherwise, C is large. 4 SOFI GJING JOVANOVSKA

De nition 2.4. A category C is called locally small if for all objects X, Y in C the collection HomC(X;Y ) = ff 2 C1 j f : X ! Yg is a set (called a hom-set). All nite categories are small. On the other hand, common categories Pos, Top, and Grp are all large and locally small.

3. Adjoints In this section, we aim to de ne adjunct pair of functors. In a way, an adjunct pair can be juxatposed with an inverse pair of maps, except it is a weaker notion. In this section we will introduce a few relevant concepts that enable us to de ne adjoints. 3.1. Isomorphisms and Natural Transformations. De nition 3.1. For any category C, an arrow f : A ! B is an isomorphism if there is an arrow g : B ! A in C such that g  f = 1A and f  g = 1B. We say A is isomorphic to B, written as A = B. We are familiar with various notions of isomorphisms for di erent structured sets, such as isomorphisms of groups, or topological spaces (called homeomorphisms), and the way they are de ned in each area of mathematics. It is easy to check that this de nition is equivalent to all the de tions of isomorphism we have seen so far, and is, in fact, universally applicable to all categories of mathematical objects. De nition 3.2. For categories C, D and functors F;G : C ! D; a # : F ! G is a family of arrows in D (# : FC ! GC) ; C C2C0 0 such that for any f : C ! C in C, one has #C0  F (f) = G(f)  #C , that is, the following diagram commutes:

# FC C / GC

F f Gf   FC0 / GC0: #C0 And if a natural transformation is an isomorphism in Fun(C; D) (the category of functors from C to D), we call it a natural isomorphism. Example 3.3. Let M(X) be the free with the generating set X. We can de ne a natural transformation  : 1Sets ! UM, where 1Sets is the identity functor in the category of sets, and U : Mon ! Sets is the on the monoid (meaning it forgets the structure, or operation, of the monoid), such that each X : X ! UM(X) is given by the \insertion of generators" { taking every element x to itself as a word. Then we can construct the following diagram, which demonstrates its naturality.  X X / UM(X)

f UM(f)   Y / UM(Y ) y BECK'S THEOREM CHARACTERIZING ALGEBRAS 5

3.2. Adjoints. De nition 3.4. Given two categories C and D and two functors F : C ! D and U : D ! C, F is the left adjoint of U (written F a U) if there exists a natural isomorphism between hom-sets  1  : HomD(FC;D) = HomC(C;UD):  1 The map  determines two maps (1FC ) = C : C ! UFC, and  (1UD) = D, called unit and counit of the adjunction, respectively.  We denote the adjoint pair (F; G; ; ; ), where F a U,  : HomD(FC;D) = HomC(C;UD), and  and  are the unit and the counit, respectively. In the case that the isomorphism between the hom-sets is unimportant, we omit it in the notation, and write (F; G; ; ). De nition 3.5. Given two adjoint pairs of functors (F; G; ; ; ): C ! D, and (F 0;G0; 0; 0; 0): C0 ! D0, a map of adjunctions from the rst to the second pair is a pair of functors K : D ! D0 and L : C ! C0 such that the following digram of functors D G / C F / D

(3.6) K L K    D0 / C0 / D0 G0 F 0 commutes, and such that the diagram of hom-sets and adjunctions

 Hom(FC;D) / Hom(C; GD)

K L   (3.7) Hom(KFC;KD) Hom(LC; LGD)

= =   0 Hom(F 0LC; KD) / Hom(LC; G0KD) commutes for all objects C 2 C and D 2 D. Here the functors K and L are restricted FC ! D and C ! GD, respectively. 3.3. Triangle Identities. Given a pair of F : C o / D :U with unit and counit

 : 1C ! U  F

 : F  U ! 1D;  1 we have  : HomD(FC;D) = HomC(C;UD):  . Thus, for any f : FC ! D, (f) = U(f)   gives a map from C to UD. 1 And similarly,  (g) = D  F (g) gives a map from FC to D. These properties gives rise to the triangle identities. First, look at 1UD : UD ! 1 UD. Recall from the de nition that D =  (1UD). Therefore

(3.8) 1UD = (D) = U(D)  UD: 6 SOFI GJING JOVANOVSKA

And under the same logic, 1 (3.9) 1FC =  (C ) = FC  F (C ): We see that the triangle identities are equivalent to that the following diagrams commute.  UD UD / UFUD

UD 1UD $  UD

F  FC C / FUFC

FC 1FC $  FC Theorem 3.10. Given two categories C and D and two functors F : C ! D and U : D ! C, and natural transformations:  : 1C ! U  F

 : F  U ! 1D; then F a U if and only if the triangle identities (3.8) and (3.9) hold. Proof. The proof for one direction is done through the construction of the triangle identities above. Therefore we only need to prove that if the triangle identities hold, then F a U. And to prove this, we need show that there is a natural isomorphism  between the hom-sets, i.e.,  : HomD(FC;D) = HomC(C;UD): ; or equivalently, for any f 2 HomD(FC;D) and g 2 HomC(C;UD), such that ( (g)) = g and ((f)) = f:

We can de ne elements from HomD(FC;D) and HomC(C;UD) by

(f) = U(f)  C 2 HomC(C;UD); and

(g) = D  F (g) 2 HomD(FC;D): Therefore

( (g)) = (D  F (g))

= U(D  F (g)  C

= U(D)  UF (g)  C

= U(D)  UD  g = g

((f)) = (U(f)  C )

= D  FU(f  C )

= D  FU(f)  F C

= f  FC  F C = f BECK'S THEOREM CHARACTERIZING ALGEBRAS 7

And this isomorphism is natural, as the equations above hold for all compatible objects and arrows in the categories. £ Remark 3.11. Given the triangle identities, we see that  not only determines, but also is determined by the unit and counit  and . Corollary 3.12. Given two pairs of adjuntions (F; G; ; ; ): C ! D, and (F 0;G0; 0; 0; 0): C0 ! D0, and the maps between them, K and L, diagram (3.7) commutes if and only if L = 0L and 0K = K.

4. Monads and Algebras 4.1. Monads and Adjoints. Given the diagram of a pair of adjoint functors: F : C o / D :U one observes that the composite of the adjoint pair is an endofunctor. So one natural thing to think about is whether the converse holds. Given any endofunctor T = U  F : C ! C, is T always a composite of an adjoint pair F a U to and from another category D? And how can we nd the category D and the unit and counit? To answer this, we rst need to introduce monads. Then we will see that monads are special endofunctors that give rise to adjoint pairs. De nition 4.1. A monad on a category C consists of an endofunctor T : C ! C and two natural transformations

 : 1C ! T  : T 2 ! T denoted as T = (T; ; ), such that

  T =   T 

  T = 1T =   T  Or, equivalently, that the following diagrams commute

T  T 3 / T 2

T    T 2 / T 

 T  T T / T 2 o T

 = =  ~ T Theorem 4.2. Every adjoint pair of functors F a U such that U : D ! C, with unit  : UF ! 1C and counit  : 1D ! FU gives rise to a monad (T; ; ) on C with (4.3) T = U  F : C ! C

(4.4)  : 1 ! T

2 (4.5)  = UT : T ! T 8 SOFI GJING JOVANOVSKA

Proof. It is easy to check that (4.3), (4.4), and (4.5) hold from the triangle identities. £ 4.2. Algebras for a monad. As we have introduced a monad, which is an endo- fuctor, one might think that a monad is precisely the type of endofunctor that rises from an adjoint pair. And the answer is yes, and we can actually construct two di erent pairs of adjunctions. In the following part of this paper, we will focus on the pair of adjunct to and from a category CT called the Eilenberg-Moore Category of T -algebras. De nition 4.6. Given a monad (T; ; ), a T algebra,(C; h) 2 CT, is a pair consisting of an object C 2 C (the underlying object of the algebra) and an arrow h : TC ! C of C (called the structure of the algebra), such that the following diagrams commute, with the rst being the associative law, the second the unit law. T T 2C C / TC

C h   TC / C h

 C C / TC

h 1 !  C A morphism f :(C; h) ! (D; g) of T-algebras is an arrow f : C ! D of C such that the diagram below commuttes.

TC h / C

T f f   TD / D g Theorem 4.7. Every monad is de ned by its T -algebra. That is to say, if (T; ; ) is a monad on C, then the collection of all T -algebras and their morphisms form a category CT. And there is an adjunction (F T ;GT ; T ; T ): C ! CT in which the functors GT and F T are given by the respective assigments (C; h) / C ;

f f   (D; g) / D

/ C (T C; C ) ;

f T f   / D (T D; D) where T =  and T (C; h) = h for each T -algebra (C; h). The monad de ned on C by this is the given monad (T; ; ). BECK'S THEOREM CHARACTERIZING ALGEBRAS 9

Proof. We need to check rst that the T -algebras form a category, and secondly, the adjunction is well-de ned. Given the de nition of T -algebras gives us associativity and unit laws for free, we only need to check the composition axiom for it to satisfy all requirements for a category. Suppose f :(C; h) ! (D; k) and g :(D; k) ! (B; l) are two arrows in CT, then their composite g  f :(C; h) ! (B; l) is self-evidently an arrow in CT. Now we show that the adjunction is well-de ned. From the construction of GT : C ! CT, it is clear that GT is the forgetful functor that simply \forgets" the structure imposed on the set. While (T C; C ) is a T -algebra (recall that by de nition, C : T (TC) ! TC), called the free T -algebra on C. Hence by mapping T T each C 7! (T C; C ), we see that there is actually a fuctor F : C ! C . Then T T T for any C 2 C0, the composition of fucntors G F C = G (T C; C ) = TC, so, T T T G F = T and  =  : 1C ! T is a natural transformation. As for the other T T composition F G (C; h) = (T C; C ), while by de nition, the map h : TC ! C is a morphism (T C; C ) ! (C; h) of T -algebras, therefore resulting transformation T T T (C;h) = h : F G (C; h) ! (C; h) is a natural transformation. And so we have the triangle identities:

T TC C / TTC

UC 1TC #  TC;

 C C / TC

h 1C !  C: And by Theorem 4.3, F T a GT . T T T T Finally, a monad rises from this adjunction, let T = G F ,  : 1C ! G F , T T T T T T T and T = G  F implies  C = G  (T C; C ) = G C = C , which are the requirements for a monad. £

4.3. The Comparison with Algebras. In this section we prove the Comparison Theorem. If we have an adjunction (F; G; ; ) such that F : C ! D, and the monad T de ned by it, and CT the category of T -algebras de ning the monad, we would like to know how the category CT is realted to D. The Comparison Theorem shows that there is a functor that preserves the adjunction. Theorem 4.8. Let F : C o / D :U with unit and counit:

 : 1C ! U  F

 : F  U ! 1D be a pair of adjoint functors, and T = (GF; ; GF ) be the monad de ned on C. Then there is a unique functor K : D ! CT with GT K = G and KF = F T , such 10 SOFI GJING JOVANOVSKA that the following diagram commutes:

K / T DO CO

(4.9) F G F T GT   / C = C Proof. By the triangle identities for the counit, we have the following diagrams:  GD GD / GF GD

GD 1GD $  GD

GF GF GD GF GD/ GF GD

GD GD   GF GD / GD GD Therefore, for any f : D ! D0 in D, we de ne K by

KD = (GD; GD) 0 Kf = Gf :(GD; GD) ! (GD ; GD0 ): Since  is a natural transformation, Kf commutes with G and so is a morphism of T -algebras. To check that KF = F T and GT K = G, let C 2 C, compute

(4.10) KF (C) = (GF (C); GF (C))

T (4.11) F (C) = (GF (C); GF (C)); and let D 2 D, T T (4.12) G K(D) = G (GD; GD) = GD = G(D): Thus the equalities hold. Finally, we show K is unique. Since KC = (GC; GC ) is a T-algebra, we need to check that its underlying set and the structure morphism are unique. Its underlying set is GC, which is clearly uniquely determined. Therefore, we only need to check the structure morphism h of the T-algebra, which is GC . To do this, we de ne a map between the two adjunct pairs (F; G; ; ) and (F T ;GT ; T ; T ). Since diagram (4.9) commutes, using equations (4.10) and (4.11), T T T T T we get that for any C 2 C, C = G F C = G KFC = GF C = C, so  =  , and furthermore, K : D ! CT and I : C ! C de ne a map between the two adjunct pairs. By proposition (3.9), K = T K. And since we have de ned for any 0 f : D ! D in D, Kf = Gf, K = G, and KD = GD for any D 2 D. On the other hand, Theorem 4.7 shows that T (C;h) = h; so in this case, T KD = T (GD; h) = h. But since K is de ned as KD = T T (GD; GD),  KD =  (GD; GD) = h. Thus GD = h. The structure map is unique. £ BECK'S THEOREM CHARACTERIZING ALGEBRAS 11

4.4. Coequalizers. De nition 4.13. For any category C, given two parallel arrows

f A / B; g a of f and g is an object Q with an arrow q : B ! Q, such that q  f = q  g. Further, given the following diagram such that z  f = z  g, there exists a unique u such that the following diagram commutes.

f q A // B / Q : g

z 9u  Z We need to introduce a certain type of coequalizers to prove our nal result. De nition 4.14. A fork in a category C is a diagram

f q A / B / Q g such that q  f = q  g: De nition 4.15. A split fork in C is a fork with two more arrows ABo t o s Q such that the following equations are satis ed: q  f = q  g;

q  s = 1Q;

f  t = 1B; g  t = s  q; or equivalently, such that the following diagram commutes: =

f B t / A / B

q g q    Q / B / Q; s q @ = in which case s and t split the fork, and q is a split morphism with s as its right inverse. Lemma 4.16. In every split fork, q is the coequalizer of f and g. Proof. From the de nition of a split fork already requires that q  f = q  g. We just need to check that for any z : B ! Z, there is a unique u such that z = q  u; equivalently, the diagram in De nition 3.2 commutes. So let u = z  s, using the equations in the de nition of a split fork, we have: u  q = z  s  q = z  g  t = z  f  t = z 12 SOFI GJING JOVANOVSKA so we have existence. To prove uniqueness, pick an arbitrary k : Q ! Z such that z = k  q, then u = z  s = k  q  s = k, thus z is unique. £

De nition 4.17. An absolute coequalizer is a coequalizer that is preserved under all functors. That is, given a category C and the following

f A / B g where q is the coequalizer for f and g, for any F : C ! D, F (q) is the coequalizer of F (f) and F (g). Corollary 4.18. In every split fork, q is an absolute coequalier of f and g. Proof. Since a functor preserves composites and identities, it preserves all equations in the de nition of a split fork, and thus the coequalizer is uniquely determined. £

4.5. Beck's Threorem. Now we prove Beck's Theorem. This theorem proves three equivalent statements under the assumption of the existence of an adjunction and a monad de ned by it. The rst statement as a result is the most interesting one, as it demonstrates when we can know that the category of T -algebra de ning the monad is isomorphic to the category the left adjoint functor maps to. Theorem 4.19. Given a pair of adjoint functors F : C o / D :U with unit and counit:

 : 1C ! U  F

 : F  U ! 1D and the monad de ned by it (T; ; ), and CT the category of T -algebras de ning the monad, the following conditions are equivalent: (i) The (unique) comparison functor K : D ! CT is an isomorphism; (ii) For a parallel pair f, g in D such that Gf, Gg has an absolute coequalizer in C, the functor G : D ! C creates coequalizers for the pair f, g; (iii) For a parallel pair f, g in D such that Gf, Gg has a split coequalizer in C, the functor G : D ! C creates coequalizers for the pair f, g. Proof. We will prove the theorem in the order of (i) ) (ii), (ii) ) (iii), and (iii) ) (i). To show (i) ) (ii), take C;D 2 D0, f; g 2 D1, such that f and g have an absolute coequalizer q

f q C // D / Q; g and consider two maps of T -algebras

f (C; h) / (D; k) : g We want to extend the coequalizer q to the maps for T -algebras. To do so, we need to nd a unique T -algebra structure m : T q ! q on q such that the following BECK'S THEOREM CHARACTERIZING ALGEBRAS 13 diagram commutes:

T f T q TC // TD / TQ T g (4.20) h k m  f   C // D / Q; g q and afterwards, we need to show that the map m is the structure map for (Q; m) as a T -algebra, and further, this construction of m implies that q is a coequalizer in CT. Now, both squares of the left side of the diagram commute by the de nition of T -algebras. And since q is an absolute coequalizer of f and g, T q is a coequalizer of T f and T g as maps of T -algebras. This leaves only to check that the arrow m closes up the diagram. Notice q  k  (T f) = q  k  (T g). Thus the map q  k serves as the map needed in the de nition of a coequalizer illustrated below:

T f T q TC // TD / TQ T g 9m qk !  Q And so such a unique m exists by the denition of a coequalizer. Next, we need to show that m is the unique structure map for Q. The associative law for m (outer square) can be deduced from the associative law of the structure map k (inner square), if we can show that the following diagram commutes:

2 T m / T Qb =TQ T 2q T q

T 2D T k / TD

q D k m   TD / D k T q q  { "  TQ / Q m Since  is natural, the left trapezoid commutes, and the other three commute by the de nition of m in the earlier diagram. Therefore 2 2 m  T m  T q = m  Q  T q Since q is an absolute coequalizer, T 2q is also an coequalizer, which has a right 2 inverse. So we can cancel T q, and that leaves m  T m = m  Q, which is the associative law for m. Similarly, take the triangle diagram  D D / TD

h 1 !  D; 14 SOFI GJING JOVANOVSKA and construct a larger triangle for Q, and it is routine to show that it commutes using the similar reasoning, which gives unit law for m. Therefore m is the unique structure map for (Q; m). Thirdly, we need to show that q is a coequalizer in CT. Consider a map d : (D; k) ! (W; n) of T -algebras with d  f = d  g. Then f : D ! W is an arrow in C such that d  f = d  g, and q is a coequalizer of f and g. Therefore there is a map d0 : Q ! W such that d = d0q. Following the same logic of the previous proofs, , d0 is a map of T -algebras, and it is unique with d = d0e. Therefore q is a coequalizer in CT. (ii) ) (iii) is straightforward. Every split coequalizer is an absolute coequalizer by Corollary 4.18. So if a pair Gf, Gg has an absolute coequalizer in C gives certain property for f, g, then it is necessarily true for when Gf, Gg has a split coequalizer. So (ii) implies (iii). We will prove (iii) ) (i) though two claims. First, suppose

C T 2C // TC h / C T h is a fork split by   T 2CTCo TC o C C ; then h : TC ! C is a structure map for the T -algebra (C; h): Proof of claim: The fork reads h  C = h  T h, which is also the associative law from h, and the composite h  C = 1 is the unit law for (C; h), and C  TC = 1 and T h  TC = C  h hold because T is a monad. This proves the claim. Let C 2 C, the adjunction (F; G; ;  : C ! D) gives a fork:

F GD  F GF GD // F GD D / D F GD in D which we call the \canonical presentation" of D. Apply G to the fork, we get a split fork in C. Now let (F 0;G0; 0; 0): C ! D be another adjunction which de nes the same monad in C. And let M : D ! D such that MF 0 = F and GM = G0 be a comparision functor. Theorem 4.8 shows the comparison functor is a morphism of adjuntions and therefore satis es M0 = M. The scond claim we will prove is that M is unique. And this will complete the proof from (iii) to (i) Proof of claim: The comparison map satis es F GM = MF 0G0 and M0 = M, we have just shown in the earlier part of the proof that M must carry the canonical presentation of D0 to the canonical presentation of MD0, or equivalently, that there exists a unique k such that the following holds

FG0D0 F GF G0D0 = FG0F 0G0D0 // FG0D0 k / MD0 : FG00 D0 0 On the other hand, k = MD0 = MD0 . Apply G to the fork, we get that

0 GF G0D0 G  GF G0F 0G0D0 / GF G0D0 D0 / G0D0 TG00 D0 is a split fork in C, as C = G0D0. Therefore the choices of k and MD0 are unique as above. Thus M is unique. BECK'S THEOREM CHARACTERIZING ALGEBRAS 15

To show that M is a functor, let f 2 D0 be such that f : D0 ! E0. Then we obtain the diagram below:

FG0F 0G0D0 / FG0D0 k / MD0

FG0F 0G0f FG0f Mf    FG0F 0G0E0 // FG0E0 / ME0: kE Using same logic as the earlier part of the proof (look at diagram (4.19)), since k is a coequalizer, Mf : MD0 ! ME0 is a unique map, and thus M : D0 ! D is a functor. Thus we proved the claim. Now that we have constructed two comparison functors K : D ! CT, and M : CT ! D. The composites MK : D ! D and KM : CT ! CT are then comparisons that are the identities, accoridng to the lemma. We have MK = KM = 1. Therefore K is an isomorphism, and we proved (iii) ) (i). £ Acknowledgments. It is a pleasure to thank my mentors, Henry Chan and Rolf Hoyer, for meeting with me twice per week during the program, suggesting this wonderful topic, reviewing the material I read and explain the parts that I failed to understand, and eventually, reading and editing this paper. I would also like to thank Pater May for organizing this program and providing comments.

References [1] Saunders Mac Lane. Categories for the Working Mathematician. Springer-Verlag Publishing, 1971. [2] Steve Awodey. Category Theory. Oxford University Press. 2010.