BECK'S THEOREM CHARACTERIZING ALGEBRAS

Transcription

1 BEK'S THEOREM HARATERIZING ALGEBRAS SOFI GJING JOVANOVSKA Abstract. In this paper, I will construct a proo o Beck's Theorem characterizin T -alebras. Suppose we have an adjoint pair o unctors F and G between cateories and. It determines a monad T on. We can associate a T -alebra to the monad, and Beck's Theorem demonstrates when the cateory o T -alebras is equivalent to the cateory. We will arrive at this result by rst denin cateories, and a ew relevant concepts and theorems that will be useul or provin our result; these will include natural tranormations, adjoints, monads and more. ontents 1. Introduction 1 2. ateories 2 3. Adjoints Isomorphisms and Natural Transormations Adjoints Trianle Identities 5 4. Monads and Alebras Monads and Adjoints Alebras or a monad The omparison with Alebras oequalizers Beck's Threorem 12 Acknowledments 15 Reerences Introduction Beck's Theorem characterizin alebras is one direction o Beck's Monadicity Theorem. Beck's Monadicity Theorem is most useul in studyin adjoint pairs o unctors. Adjunction is a type o relation between two unctors that has some very important properties, such as preservation o limits or colimits, which, unortunately, we will not touch upon in this paper. However, we need to know that it is indeed a topic o interest, and thereore worth studyin. Here, we state Beck's Monadicity Theorem. Since this is an overview, all the technical terms will be dened later. ate: AUGUST 29,

2 2 SOFI GJING JOVANOVSKA Theorem 1.1. A unctor U :! is monadic i and only i the ollowin hold: (i) U has a let adjoint (ii) U reects isomorphisms (iii) has and U preserves all coequalizers o U-split pairs Thus we see that Beck's Monadicity Theorem does two thins. First, it shows when a unctor ives rise to a monad; and secondly, i a unctor U is monadic (a unctor U is monadic i it determines a monad and i its domain cateory is equivalent to the cateory o alebras over that monad), what the imae o the let adjoint o U, say F, is like. In this paper, we will only prove an equivalent statement o the theorem in the direction assumin F is monadic. We will ocus on the construction o the proo, and we will introduce all denitions and theorems needed or it. This paper alls short in examples, as examples in cateory theory tend t be very lon. I will only provide the short ones. Readers can consult the materials this paper is based on: the two published books, Steve Awodey's ateory Theory and Saunders Mac Lane's ateories or the Workin Mathematician or urther explorations, and other topics that are omitted. Sometimes the denition and the statement o theorems are taken aithully rom the sources, in other instances I have rephrased them, i I consider necessary. The proos are reconstructed with the intention o makin them more understandable or the beinners in this area. As one miht have noticed, Mac Lane's book titled itsel a literature or proessional mathematicians, and its proo sometimes need simplication. My paper aims to extend those in the way that I nd most suitable throuh my experience o readin them. The paper is divided into a ew sections. In section 2, we will rst introduce cateories rom scratch, so that the readers have a ood idea o the objects we are dealin with. In section 3, we will dene some concepts reardin cateories, such as isomorphism, coequalizers, and natural transormations that will enable us to dene adjunctions. It is unortunate that I am only able to include the most relevant ones. Notions such as duality, products, and limits are such essential aspects o this subject that one may not claim to know cateory theory without knowin these, but they will not be included, and readers are stronly encouraed to research on those, [2] is a ood source or this. Section 4 is the main part p the paper, in which I will introduce monads, T - alebras, and their relation to adjoints. Some imporatant results include: every adjoint pair ives rise to a monad, every monad is dened by its T -alebras, and eventually, Beck's Theorem. 2. ateories In this section, we introduce the idea o a cateory, and a ew relevant concepts, includin equalizers and adjoints. Intuitively a cateory is a collection o mathematical objects with a set o maps between the objects o the cateory. The objects o a cateory can be roups, topoloical spaces, etc. Thus one useul way a cateory can be understood as a type o mathematical objects. O course this is not the most inclusive way o thinkin about cateories, as a cateory can consist o only one object; but such cases are not particularly useul in explorin the relations between types o mathematical objects, such as that between monads and alebras, or eld extensions and roups, as those who are amiliar with Galois Theory may recall.

3 BEK'S THEOREM HARATERIZING ALGEBRAS 3 enition 2.1. A cateory satises the ollowin axioms: It contains a amily o objects, which are denoted in this paper with italic capital letters A, B,... The collection o all objects is denoted 0. It contains a amily o arrows, which are denoted with small italic letters,, h... The collection o all arrows is denoted 1. These arrows should be understood as maps between object o the cateory. Thereore, or each arrow, there are two objects A and B in the cateory that exist as the domain and codomain o, denoted as dom( ), cod( ), such that : A! B: For any two arrows : A! B and : B!, the composite o and, : A! is also an arrow in the cateory. For any object A, there exists an identity arrow 1 A such that 1 A : A! A. omposition is associative, meanin or any three composable arrows ; and h in the cateory: h ( ) = (h ) : For a map : A! B, 1 A = = 1 B. Immediately, we can see a lot o amiliar mathematical objects satisyin the axioms o cateories. Here is a short list o examples o objects and arrows o cateories, respectively: Grp: Groups and homomorphisms Vect: Vector spaces and linear maps Top: Topoloical spaces and continuous maps Pos: Partially ordered sets and monotone unctions One natural question arises whether there can be a cateory whose objects are cateories. The answer is yes, thouh the objects o the cateory at need to be restricted to small cateories. We will touch upon a brie explanation or this later, as we need to rst dene the maps between cateories, or at to be dened at all. enition 2.2. A unctor F :! between cateories and is a mappin rom objects to objects, and arrows to arrows, such that domains, codomains, identity and composition are preserved, or F ( : A! B) = F () : F (A)! F (B), F (1 A ) = 1 F (A), F ( ) = F () F (). It is simple to check that a unctor satises the axioms o arrows in cateories. Thereore at is a well-dened cateory. And eventually, let us distinuish amon three types o cateories: lare, small, and locally small. We impose a restriction on the cateories we talk about in this paper, namely, all cateories concerned are locally small, as when a cateory is \too lare", we encounter diculty with conventional set theory, which is the oundation o the approach to cateory theory in this paper. enition 2.3. A cateory is called small i both the collection 0 and 1 are sets. Otherwise, is lare.

4 4 SOFI GJING JOVANOVSKA enition 2.4. A cateory is called locally small i or all objects X, Y in the collection Hom (X; Y ) = 2 1 j : X! Y is a set (called a hom-set). All nite cateories are small. On the other hand, common cateories Pos, Top, and Grp are all lare and locally small. 3. Adjoints In this section, we aim to dene adjunct pair o unctors. In a way, an adjunct pair can be juxatposed with an inverse pair o maps, except it is a weaker notion. In this section we will introduce a ew relevant concepts that enable us to dene adjoints Isomorphisms and Natural Transormations. enition 3.1. For any cateory, an arrow : A! B is an isomorphism i there is an arrow : B! A in such that = 1 A and = 1 B. We say A is isomorphic to B, written as A = B. We are amiliar with various notions o isomorphisms or dierent structured sets, such as isomorphisms o roups, or topoloical spaces (called homeomorphisms), and the way they are dened in each area o mathematics. It is easy to check that this denition is equivalent to all the detions o isomorphism we have seen so ar, and is, in act, universally applicable to all cateories o mathematical objects. enition 3.2. For cateories, and unctors F; G :! ; a natural transormation # : F! G is a amily o arrows in (# : F! G) 2 0 ; such that or any :! 0 in, one has # 0 F () = G() #, that is, the ollowin diaram commutes: F # G F F 0 # 0 G G 0 : And i a natural transormation is an isomorphism in Fun(; ) (the cateory o unctors rom to ), we call it a natural isomorphism. Example 3.3. Let M(X) be the ree monoid with the eneratin set X. We can dene a natural transormation : 1 Sets! UM, where 1 Sets is the identity unctor in the cateory o sets, and U : Mon! Sets is the oretul unctor on the monoid (meanin it orets the structure, or operation, o the monoid), such that each X : X! UM(X) is iven by the \insertion o enerators" { takin every element x to itsel as a word. Then we can construct the ollowin diaram, which demonstrates its naturality. X X UM(X) UM() Y y UM(Y )

5 o 3.2. Adjoints. BEK'S THEOREM HARATERIZING ALGEBRAS 5 enition 3.4. Given two cateories and and two unctors F :! and U :!, F is the let adjoint o U (written F a U) i there exists a natural isomorphism between hom-sets : Hom (F ; ) = Hom (; U) : 1 The map determines two maps (1 F ) = :! UF, and 1 (1 U ) =, called unit and counit o the adjunction, respectively. We denote the adjoint pair (F; G; ; ; ), where F a U, : Hom (F ; ) = Hom (; U), 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; ; ). enition 3.5. Given two adjoint pairs o unctors (F; G; ; ; ) :!, and (F 0 ; G 0 ; 0 ; 0 ; 0 ) : 0! 0, a map o adjunctions rom the rst to the second pair is a pair o unctors K :! 0 and L :! 0 such that the ollowin diram o unctors (3.6) K G L F 0 G 0 0 F 0 0 commutes, and such that the diaram o hom-sets and adjunctions K Hom(F ; ) Hom(; G) (3.7) K Hom(KF ; K) L Hom(L; LG) = Hom(F 0 L; K) commutes or all objects 2 and 2. restricted F! and! G, respectively. 0 = Hom(L; G 0 K) 3.3. Trianle Identities. Given a pair o adjoint unctors with unit and counit F : : U : 1! U F : F U! 1 ; Here the unctors K and L are we have : Hom (F ; ) = Hom (; U) : 1. Thus, or any : F!, () = U() ives a map rom to U. And similarly, 1 () = F () ives a map rom F to. These properties ives rise to the trianle identities. First, look at 1 U : U! U. Recall rom the denition that = 1 (1 U ). Thereore (3.8) 1 U = ( ) = U( ) U :

6 6 SOFI GJING JOVANOVSKA And under the same loic, (3.9) 1 F = 1 ( ) = F F ( ): We see that the trianle identities are equivalent to that the ollowin diarams commute. U U UF U 1 U $ U U F F F UF 1 F $ F Theorem Given two cateories and and two unctors F :! and U :!, and natural transormations: F : 1! U F : F U! 1 ; then F a U i and only i the trianle identities (3.8) and (3.9) hold. Proo. The proo or one direction is done throuh the construction o the trianle identities above. Thereore we only need to prove that i the trianle 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., : Hom (F ; ) = Hom (; U) : ; or equivalently, or any 2 Hom (F ; ) and 2 Hom (; U), such that ( ()) = and (()) = : We can dene elements rom Hom (F ; ) and Hom (; U) by and Thereore () = U() 2 Hom (; U); () = F () 2 Hom (F ; ): ( ()) = ( F ()) = U( F () = U( ) UF () = U( ) U = (()) = (U() ) = F U( ) = F U() F = F F =

7 o ~ o BEK'S THEOREM HARATERIZING ALGEBRAS 7 And this isomorphism is natural, as the equations above hold or all compatible objects and arrows in the cateories. Remark Given the trianle identities, we see that not only determines, but also is determined by the unit and counit and. orollary Given two pairs o adjuntions (F; G; ; ; ) :!, and (F 0 ; G 0 ; 0 ; 0 ; 0 ) : 0! 0, and the maps between them, K and L, diaram (3.7) commutes i and only i L = 0 L and 0 K = K. 4. Monads and Alebras 4.1. Monads and Adjoints. Given the diaram o a pair o adjoint unctors: F : : U one observes that the composite o the adjoint pair is an endounctor. So one natural thin to think about is whether the converse holds. Given any endounctor T = U F :!, is T always a composite o an adjoint pair F a U to and rom another cateory? And how can we nd the cateory and the unit and counit? To answer this, we rst need to introduce monads. Then we will see that monads are special endounctors that ive rise to adjoint pairs. enition 4.1. A monad on a cateory consists o an endounctor T :! and two natural transormations : 1! T : T 2! T denoted as T = (T; ; ), such that T = T T = 1 T = T Or, equivalently, that the ollowin diarams commute T 3 T T 2 T T 2 T T T T 2 T T = T Theorem 4.2. Every adjoint pair o unctors F a U such that U :!, with unit : UF! 1 and counit : 1! F U ives rise to a monad (T; ; ) on with (4.3) T = U F :! (4.4) : 1! T (4.5) = U T : T 2! T =

8 ! 8 SOFI GJING JOVANOVSKA Proo. It is easy to check that (4.3), (4.4), and (4.5) hold rom the trianle identities Alebras or a monad. As we have introduced a monad, which is an endouctor, one miht think that a monad is precisely the type o endounctor that rises rom an adjoint pair. And the answer is yes, and we can actually construct two dierent pairs o adjunctions. In the ollowin part o this paper, we will ocus on the pair o adjunct to and rom a cateory T called the Eilenber-Moore ateory o T -alebras. enition 4.6. Given a monad (T; ; ), a T alebra, (; h) 2 T, is a pair consistin o an object 2 (the underlyin object o the alebra) and an arrow h : T! o (called the structure o the alebra), such that the ollowin diarams commute, with the rst bein the associative law, the second the unit law. T 2 T T T h h T 1 A morphism : (; h)! (; ) o T-alebras is an arrow :! o such that the diaram below commuttes. T T h h T Theorem 4.7. Every monad is dened by its T -alebra. That is to say, i (T; ; ) is a monad on, then the collection o all T -alebras and their morphisms orm a cateory T. And there is an adjunction (F T ; G T ; T ; T ) :! T in which the unctors G T and F T are iven by the respective assiments (; h) (; ) ; (T ; ) ; T (T ; ) where T = and T (; h) = h or each T -alebra (; h). The monad dened on by this is the iven monad (T; ; ).

9 o BEK'S THEOREM HARATERIZING ALGEBRAS 9 Proo. We need to check rst that the T -alebras orm a cateory, and secondly, the adjunction is well-dened. Given the denition o T -alebras ives us associativity and unit laws or ree, we only need to check the composition axiom or it to satisy all requirements or a cateory. Suppose : (; h)! (; k) and : (; k)! (B; l) are two arrows in T, then their composite : (; h)! (B; l) is sel-evidently an arrow in T. Now we show that the adjunction is well-dened. From the construction o G T :! T, it is clear that G T is the oretul unctor that simply \orets" the structure imposed on the set. While (T ; ) is a T -alebra (recall that by denition, : T (T )! T ), called the ree T -alebra on. Hence by mappin each 7! (T ; ), we see that there is actually a uctor F T :! T. Then or any 2 0, the composition o ucntors G T F T = G T (T ; ) = T, so, G T F T = T and T = : 1! T is a natural transormation. As or the other composition F T G T (; h) = (T ; ), while by denition, the map h : T! is a morphism (T ; )! (; h) o T -alebras, thereore resultin transormation T (;h) = h : F T G T (; h)! (; h) is a natural transormation. And so we have the trianle identities: T T T T 1 T # U T ; T 1! h : And by Theorem 4.3, F T a G T. Finally, a monad rises rom this adjunction, let T = G T F T, : 1! G T F T, and T = G T T F T implies T = G T T (T ; ) = G T =, which are the requirements or a monad The omparison with Alebras. In this section we prove the omparison Theorem. I we have an adjunction (F; G; ; ) such that F :!, and the monad T dened by it, and T the cateory o T -alebras denin the monad, we would like to know how the cateory T is realted to. The omparison Theorem shows that there is a unctor that preserves the adjunction. Theorem 4.8. Let with unit and counit: F : : U : 1! U F : F U! 1 be a pair o adjoint unctors, and T = (GF; ; GF ) be the monad dened on. Then there is a unique unctor K :! T with G T K = G and KF = F T, such

10 O O 10 SOFI GJING JOVANOVSKA that the ollowin diaram commutes: (4.9) F G K T F T G T = Proo. By the trianle identities or the counit, we have the ollowin diarams: G G GF G 1 G $ G G GF GF G GF G GF G G GF G G G Thereore, or any :! 0 in, we dene K by K = (G; G ) G K = G : (G; G )! (G 0 ; G 0): Since is a natural transormation, K commutes with G and so is a morphism o T -alebras. To check that KF = F T and G T K = G, let 2, compute (4.10) KF () = (GF (); G F () ) (4.11) F T () = (GF (); G F () ); and let 2, (4.12) G T K() = G T (G; G ) = G = G(): Thus the equalities hold. Finally, we show K is unique. Since K = (G; G ) is a T-alebra, we need to check that its underlyin set and the structure morphism are unique. Its underlyin set is G, which is clearly uniquely determined. Thereore, we only need to check the structure morphism h o the T-alebra, which is G. To do this, we dene a map between the two adjunct pairs (F; G; ; ) and (F T ; G T ; T ; T ). Since diaram (4.9) commutes, usin equations (4.10) and (4.11), we et that or any 2, T = GT F T = G T KF = GF =, so = T, and urthermore, K :! T and I :! dene a map between the two adjunct pairs. By proposition (3.9), K = T K. And since we have dened or any :! 0 in, K = G, K = G, and K = G or any 2. On the other hand, Theorem 4.7 shows that T (;h) = h; so in this case, T K = T (G; h) = h. But since K is dened as K = (G; G ), T K = T (G; G ) = h. Thus G = h. The structure map is unique.

11 o BEK'S THEOREM HARATERIZING ALGEBRAS oequalizers. enition For any cateory, iven two parallel arrows A B ; a coequalizer o and is an object Q with an arrow q : B! Q, such that q = q. Further, iven the ollowin diaram such that z = z, there exists a unique u such that the ollowin diaram commutes. A B We need to introduce a certain type o coequalizers to prove our nal result. enition A ork in a cateory is a diaram such that q = q : z q Q Z 9u A B q Q enition A split ork in is a ork with two more arrows such that the ollowin equations are satised: A t B s q = q ; q s = 1 Q ; t = 1 B ; t = s q; or equivalently, such that the ollowin diaram commutes: = Q : B t A B q Q s B q Q; = in which case s and t split the ork, and q is a split morphism with s as its riht inverse. Lemma In every split ork, q is the coequalizer o and. Proo. From the denition o a split ork already requires that q = q. We just need to check that or any z : B! Z, there is a unique u such that z = q u; equivalently, the diaram in enition 3.2 commutes. So let u = z s, usin the equations in the denition o a split ork, we have: u q = z s q = z t = z t = z q

12 o 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. enition An absolute coequalizer is a coequalizer that is preserved under all unctors. That is, iven a cateory and the ollowin A B where q is the coequalizer or and, or any F :!, F (q) is the coequalizer o F () and F (). orollary In every split ork, q is an absolute coequalier o and. Proo. Since a unctor preserves composites and identities, it preserves all equations in the denition o a split ork, and thus the coequalizer is uniquely determined Beck's Threorem. Now we prove Beck's Theorem. This theorem proves three equivalent statements under the assumption o the existence o an adjunction and a monad dened by it. The rst statement as a result is the most interestin one, as it demonstrates when we can know that the cateory o T -alebra denin the monad is isomorphic to the cateory the let adjoint unctor maps to. Theorem Given a pair o adjoint unctors with unit and counit: F : : U : 1! U F : F U! 1 and the monad dened by it (T; ; ), and T the cateory o T -alebras denin the monad, the ollowin conditions are equivalent: (i) The (unique) comparison unctor K :! T is an isomorphism; (ii) For a parallel pair, in such that G, G has an absolute coequalizer in, the unctor G :! creates coequalizers or the pair, ; (iii) For a parallel pair, in such that G, G has a split coequalizer in, the unctor G :! creates coequalizers or the pair,. Proo. We will prove the theorem in the order o (i) ) (ii), (ii) ) (iii), and (iii) ) (i). To show (i) ) (ii), take ; 2 0, ; 2 1, such that and have an absolute coequalizer q and consider two maps o T -alebras q Q ; (; h) (; k) : We want to extend the coequalizer q to the maps or T -alebras. To do so, we need to nd a unique T -alebra structure m : T q! q on q such that the ollowin

13 b {!! = " BEK'S THEOREM HARATERIZING ALGEBRAS 13 diaram commutes: (4.20) T h T T T k T q q and aterwards, we need to show that the map m is the structure map or (Q; m) as a T -alebra, and urther, this construction o m implies that q is a coequalizer in T. Now, both squares o the let side o the diaram commute by the denition o T -alebras. And since q is an absolute coequalizer o and, T q is a coequalizer o T and T as maps o T -alebras. This leaves only to check that the arrow m closes up the diaram. Notice q k (T ) = q k (T ). Thus the map q k serves as the map needed in the denition o a coequalizer illustrated below: T T T T And so such a unique m exists by the denition o a coequalizer. Next, we need to show that m is the unique structure map or Q. The associative law or m (outer square) can be deduced rom the associative law o the structure map k (inner square), i we can show that the ollowin diaram commutes: qk T q T Q Q; T Q Q m 9m T 2 Q T 2 q T m T q T Q T 2 T k T q T q T k k q m T Q m Since is natural, the let trapezoid commutes, and the other three commute by the denition o m in the earlier diaram. Thereore m T m T 2 q = m Q T 2 q Since q is an absolute coequalizer, T 2 q is also an coequalizer, which has a riht inverse. So we can cancel T 2 q, and that leaves m T m = m Q, which is the associative law or m. Similarly, take the trianle diaram Q 1 T h ;

14 o 14 SOFI GJING JOVANOVSKA and construct a larer trianle or Q, and it is routine to show that it commutes usin the similar reasonin, which ives unit law or m. Thereore m is the unique structure map or (Q; m). Thirdly, we need to show that q is a coequalizer in T. onsider a map d : (; k)! (W; n) o T -alebras with d = d. Then :! W is an arrow in such that d = d, and q is a coequalizer o and. Thereore there is a map d 0 : Q! W such that d = d 0 q. Followin the same loic o the previous proos,, d 0 is a map o T -alebras, and it is unique with d = d 0 e. Thereore q is a coequalizer in T. (ii) ) (iii) is straihtorward. Every split coequalizer is an absolute coequalizer by orollary So i a pair G, G has an absolute coequalizer in ives certain property or,, then it is necessarily true or when G, G has a split coequalizer. So (ii) implies (iii). We will prove (iii) ) (i) thouh two claims. First, suppose is a ork split by T 2 T h T h T 2 T T o ; then h : T! is a structure map or the T -alebra (; h): Proo o claim: The ork reads h = h T h, which is also the associative law rom h, and the composite h = 1 is the unit law or (; h), and T = 1 and T h T = h hold because T is a monad. This proves the claim. Let 2, the adjunction (F; G; ; :! ) ives a ork: F GF G F G F G F G in which we call the \canonical presentation" o. Apply G to the ork, we et a split ork in. Now let (F 0 ; G 0 ; 0 ; 0 ) :! be another adjunction which denes the same monad in. And let M :! such that MF 0 = F and GM = G 0 be a comparision unctor. Theorem 4.8 shows the comparison unctor is a morphism o adjuntions and thereore satises M 0 = M. The scond claim we will prove is that M is unique. And this will complete the proo rom (iii) to (i) Proo o claim: The comparison map satises F GM = MF 0 G 0 and M 0 = M, we have just shown in the earlier part o the proo that M must carry the canonical presentation o 0 to the canonical presentation o M 0, or equivalently, that there exists a unique k such that the ollowin holds On the other hand, k = M 0 0 F GF G 0 0 = F G 0 F 0 G 0 0 F G 0 0 F G F G 0 0 k M 0 : = M0. Apply G to the ork, we et that GF G 0 F 0 G 0 0 GF G 0 0 GF G 0 0 T G G 0 0 G 0 0 is a split ork in, as = G 0 0. Thereore the choices o k and M 0 are unique as above. Thus M is unique.

15 BEK'S THEOREM HARATERIZING ALGEBRAS 15 To show that M is a unctor, let 2 0 be such that : 0! E 0. Then we obtain the diaram below: F G 0 F 0 G 0 0 F G 0 0 k M 0 F G 0 F 0 G 0 F G 0 M F G 0 F 0 G 0 E 0 F G 0 E 0 k E ME 0 : Usin same loic as the earlier part o the proo (look at diaram (4.19)), since k is a coequalizer, M : M 0! ME 0 is a unique map, and thus M : 0! is a unctor. Thus we proved the claim. Now that we have constructed two comparison unctors K :! T, and M : T!. The composites MK :! and KM : T! T are then comparisons that are the identities, accoridn to the lemma. We have M K = KM = 1. Thereore K is an isomorphism, and we proved (iii) ) (i). Acknowledments. It is a pleasure to thank my mentors, Henry han and Rol Hoyer, or meetin with me twice per week durin the proram, suestin this wonderul topic, reviewin the material I read and explain the parts that I ailed to understand, and eventually, readin and editin this paper. I would also like to thank Pater May or oranizin this proram and providin comments. Reerences [1] Saunders Mac Lane. ateories or the Workin Mathematician. Spriner-Verla Publishin, [2] Steve Awodey. ateory Theory. Oxord University Press