Abelian Categories and the Freyd-Mitchell Embedding Theorem Geillan Aly May 15, 2008 1 Categories 1.1 Introduction to Category Theory The field of category theory may be the closest a mathematician can get to a Meta- Theory. By understanding the foundation of mathematical structures, similarities be- tween seemingly disparate fields of study can be observed. While abstracting the ab- stract may seem counter-productive, studying the underlying structures and relation- ships between mathematical objects has opened up new and fascinating fields in math- ematics. Definition 1. A category C consists of the following: • a class Ob j(C) of objects • a set MorC(A,B) of morphisms for every ordered pair (A, B) of objects in C. When the category in question is clear, we will say Mor(A,B). • a composition function of morphisms for every ordered triple (A, B, C) of objects in C i.e. a function Mor(A,C) such that Mor(B; C) × Mor(A; B) ! Mor(A; C) Denote f : A ! B to indicate that f is a morphism in Mor(A, B), and g f or g ◦ f for the composition of f 2 Mor(A; B) with g 2 Mor(B; C): A category must satisfy the following properties: • For each object A in a category C, there exists an identity morphism idA 2 Mor(A, A) with the property that idB ◦ f = f = f ◦idA for f : A ! B. • The Associativity Axiom: (hg) f = h(g f ) for f : A ! B, g : B ! C, h : C ! D. We begin by discussing examples of categories and their morphisms. The collection of all sets forms the category Sets, where the set of morphisms in Sets are set maps. The collection of all groups forms the category Groups, where the morphisms are group 1 homomorphisms. Of special interest will be our final two examples. The category G of Abelian groups where, again, morphisms are group homomorphisms. Our final example will be R-mod (mod-R), the category of left (right) R-modules, where R is a ring with unity and the morphisms are left (right) module homomorphisms. Definition 2. A subcategory C of a category C is a subclass of objects and morphisms in C such that the collections of morphisms MorC(A; B) for A; B in C are closed under composition, and include the identity morphism idA for every object A 2 C. 0 0 0 Definition 3. A subcategory C in which MorC(A; A ) = MorC(A; A ) for all A and A in C is a full subcategory. Clearly the category G is a subcategory of Groups. Definition 4. Let fCigi2I be a set of objects in C.A product of fCigi2I is an object C 0 together with a set of morphisms fpi : C ! Cigi2I such that for any set fp˜i : C ! Cigi2I 0 there exists a unique morphism f˜ : C ! C such that pi f˜ = p˜i. 0 0 If A is a product of fAig then, for all A 2 C the set of morphisms Mor(A ; A) is in Q 0 one-to-one correspondence with the set i2I Mor(A ; Ai): Definition 5. Let fCigi2I be a set of objects in C.A coproduct of fCigi2I is an object C 0 together with a set of morphisms fpi : Ci ! Cgi2I such that for any set fp˜i : Ci ! C gi2I 0 there is a unique morphism f˜ : C ! C such that f˜ pi = p˜i. 1.2 Morphisms Between Categories Definition 6. A covariant functor (or just functor)F : C!D from a category C to a category D is a relation such that every object A 2 Ob j(C) is associated to an object F(A) in D and every morphism f : A ! B in C is associated to a morphism F( f ): F(A) ! F(B) in D. A functor F must satisfy the following properties: • preserve identity morphisms, F(idA) = idF(A) • preserve composition, F(g f ) = F(g)F( f ) for gf a composition of morphisms in C where g : B ! C in C. Observe that F induces set maps MorC(A; B) ! MorD F(A); F(B) for every A; B in C. Lastly, if F : C!D and G : D!E, the composite GF : C!E is defined in the following manner: (GF)(A) = G(F(A)) and (GF)( f ) = G(F( f )) for all objects A in C. Similarly, a contravariant functor is a relation F : C!D defined such that ev- ery morphism f : A ! B is associated to F( f ): F(B) ! F(A). Furthermore, F(g f ) = F( f )F(g) in D for gf a composition of morphisms in C.1 1Every category C induces a naturally defined opposite category Copp. Both categories have the same opp objects, while the morphisms in C are reversed i.e. MorCopp (A; B) = MorC(B; A) with composition in Copp similarly reversed. In this context, a covariant functor F : C!D constructs a contravariant functor G : Copp !D. 2 Definition 7. Consider the elements A; B; C; D and the maps α : A ! B, ' : A ! C, : B ! D and β : C ! D. If in the following diagram α A / B ' β C / D ψα = βϕ : A ! D then the diagram commutes. Note that A; B; C; D may either be objects or categories so long as the maps and elements in the diagram are well defined. Definition 8. Given the following commutative diagram i.e. ψγ = ', the morphism ' factors through B. B ? ? ? γ ?? ?? ' ? A / C Definition 9. A natural transformation between two functors F; G : C!D is a relation ρ : Ob j(C) ! Ob j(D) such that • for A 2 Ob j(C), ρ(A) 2 MorF(A); G(A) • for any f 2 MorC(A; B) the following diagram commutes: F( f ) F(A) / F(B) ρ(A) ρ(B) G(A) / G(B) G( f ) If each ρ is an isomorphism then ρ is a natural isomorphism and the functors F and G are isomorphic. Definition 10. A category C is small if Obj(C), the set of objects in C and Morph(C), the collection of morphisms in C is a set, not just a class. The motivation for defining a small category lies in the foundation of set theory. One may be tempted to define the set of all functors F : C!D as the objects of a new category, with their natural transformations as the morphisms. Yet, the natural trans- formations of functors may not form a set. This difficulty can be circumvented by requiring that C be a small category. An example of a small category is the category of isomorphic groups IG, which is created from the category Groups as follows. Define an equivalence class on objects in Groups where two objects A and B are equivalent if and only if they are isomor- phic groups. Each equivalence class therefore is an object in IG. The morphisms 3 between the objects A, B in IG are likewise a set of equivalent classes of morphisms between any two representatives of A and B. Choose a; a0; b; b0 2 Obj(Groups) and let ' : a ! a0 and : b ! b0 be two isomorphisms so that A = [a] [a0] and B = [b] [b0] for A; B 2 Ob j(IG). Then f; g 2 Mor(A; B) are equivalent if and only if the following diagram commutes. ' a o / a0 f g b o / b0 Definition 11. A functor F : C!D is an equivalence of categories if there exists a functor G : D!C such that natural isomorphisms GF idC and FG idD exist. Two categories are therefore equivalent if there exist such functors that satisfy these relations. The categories R-mod and mod-R are naturally equivalent when R is a commutative ring using a quasi-identity functor F : R-mod ! mod-R such that F(A) = A and F( f ) = f . Definition 12. A functor F : C!D is called a faithful functor if the set maps 0 0 0 MorC(A; A ) ! MorD F(A); F(A ) are injections for all A; A 2 C. That is, if f1 and 0 f2 are distinct maps from A to A in C, then F( f1) , F( f2). A subcategory C of a category C gives rise to a natural inclusion functor F : C!C. The inclusion functor is clearly faithful. 0 0 Definition 13. A functor F : C!D is full if the maps MorC(A; A ) ! MorD F(A); F(A ) are all surjections. That is, every g : F(A) ! F(A0) in D is of the form g = F( f ) for some f : A ! A0. Definition 14. A functor F : C!D that is both full and faithful is fully faithful. 2 Abelian Categories In an Abelian category A the set of morphisms between two objects A; B form an Abelian group. The properties of Abelian categories make them very suitable for study. The final result of this paper, the Freyd-Mitchell Embedding Theorem allows for a concrete approach to understanding Abelian categories. Definition 15. A category A is an Ab-category if every set of morphisms MorA(C; D) in A is given the structure of an Abelian group in such a way that composition dis- tributes over addition. For example, given a diagram A of the form 0 f g h A / B g +3 C / D we have h(g + g0) f = hg f + hg0 f in Mor(A, D). 4 Letting A=B=C=D of objects A, B, C, D in C, we see that each Mor(A; A) is an as- sociative ring since by definition elements in Mor(A; A) possess the characteristics of a ring.