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Beck's Theorem Characterizing Algebras BECK'S THEOREM CHARACTERIZING ALGEBRAS 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 functors F and G between categories C and D. It determines a monad T on C. We can associate a T -algebra to the monad, and Beck's Theorem demonstrates when the catego- ry of T -algebras is equivalent to the category 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. Coequalizers 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 functor 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 image 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 category theory 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 Saunders Mac Lane'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 arrow 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 natural transformation # : 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.
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