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Abelian Categories Abelian Categories Simon Ertl 10. September 2020 Simon Ertl Abelian Categories 10. September 2020 1 / 21 1 Introduction Null object and null morphism Kernels and Cokernels 2 Additive Categories Factorization of morphisms Additive functors 3 Abelian Categories Basic properties Five lemma The Freyd-Mitchell embedding theorem Simon Ertl Abelian Categories 10. September 2020 2 / 21 Definition Let C be a category with a zero object. The morphism f : a 7! b is called null morphism if it factors through 0. For each pair a; b 2 obj(C), the zero morphism 0c;d : c 7! 0 7! d is unique as 0 is both initial and terminal. Example In Gr the zero object is the trivial group and the zero morphisms are exactly the morphisms, that map all elements to the neutral element. Null object and null morphism Definition In a category C an object is called null object or zero object 0 if it is both initial and terminal. A category with zero objects is called pointed category. Simon Ertl Abelian Categories 10. September 2020 3 / 21 For each pair a; b 2 obj(C), the zero morphism 0c;d : c 7! 0 7! d is unique as 0 is both initial and terminal. Example In Gr the zero object is the trivial group and the zero morphisms are exactly the morphisms, that map all elements to the neutral element. Null object and null morphism Definition In a category C an object is called null object or zero object 0 if it is both initial and terminal. A category with zero objects is called pointed category. Definition Let C be a category with a zero object. The morphism f : a 7! b is called null morphism if it factors through 0. Simon Ertl Abelian Categories 10. September 2020 3 / 21 Example In Gr the zero object is the trivial group and the zero morphisms are exactly the morphisms, that map all elements to the neutral element. Null object and null morphism Definition In a category C an object is called null object or zero object 0 if it is both initial and terminal. A category with zero objects is called pointed category. Definition Let C be a category with a zero object. The morphism f : a 7! b is called null morphism if it factors through 0. For each pair a; b 2 obj(C), the zero morphism 0c;d : c 7! 0 7! d is unique as 0 is both initial and terminal. Simon Ertl Abelian Categories 10. September 2020 3 / 21 Null object and null morphism Definition In a category C an object is called null object or zero object 0 if it is both initial and terminal. A category with zero objects is called pointed category. Definition Let C be a category with a zero object. The morphism f : a 7! b is called null morphism if it factors through 0. For each pair a; b 2 obj(C), the zero morphism 0c;d : c 7! 0 7! d is unique as 0 is both initial and terminal. Example In Gr the zero object is the trivial group and the zero morphisms are exactly the morphisms, that map all elements to the neutral element. Simon Ertl Abelian Categories 10. September 2020 3 / 21 Kernels and cokernels Definition For a pointed category C the kernel of a morphism f : A 7! B is the equalizer of f and 0a;b. The cokernel is the dual concept. 0K;B K k A f B k0 9!u K 0 So (i) f ◦ k = 0 ◦ k = 0 (ii) for any other k0 with f ◦ k0 = 0, k0 factors uniquely to k0 = k ◦ u. Simon Ertl Abelian Categories 10. September 2020 4 / 21 Proposition Let z be an object of an Ab-category A. The following conditions are equivalent: (i) z is initial. (ii) z is terminal. (iii) A has a zero object. pre-additive categories Definition A category is called pre-additive or Ab-category if every hom-set C(a; b) is an additive abelian group and composition is bilinear in a way that the it distributes over the addition. h ◦ (f + g) = h ◦ f + h ◦ g and (f + g) ◦ h = f ◦ h + g ◦ h Simon Ertl Abelian Categories 10. September 2020 5 / 21 pre-additive categories Definition A category is called pre-additive or Ab-category if every hom-set C(a; b) is an additive abelian group and composition is bilinear in a way that the it distributes over the addition. h ◦ (f + g) = h ◦ f + h ◦ g and (f + g) ◦ h = f ◦ h + g ◦ h Proposition Let z be an object of an Ab-category A. The following conditions are equivalent: (i) z is initial. (ii) z is terminal. (iii) A has a zero object. Simon Ertl Abelian Categories 10. September 2020 5 / 21 Definition Let A be an Ab-category. For objects a; b 2 A a biproduct is diagram i1 p2 a c b that fulfils p1i1 = 1a; p2i2 = 1b; i1p1 + i2p2 = 1c . p1 i2 Example In the abelian groups the biproduct is exactly the direct sum. pre-additive categories Proof. (iii))(i) and (iii))(ii) by definition. By duality it suffices to proof (i) ) (ii). As z is initial there exists a unique map 1z : z 7! z, thus MorA(z; z) has only one element, the zero element. For any other arrow f : b 7! z the composition 1z ◦ f has to be zero as well. And therefore z is terminal. Simon Ertl Abelian Categories 10. September 2020 6 / 21 Example In the abelian groups the biproduct is exactly the direct sum. pre-additive categories Proof. (iii))(i) and (iii))(ii) by definition. By duality it suffices to proof (i) ) (ii). As z is initial there exists a unique map 1z : z 7! z, thus MorA(z; z) has only one element, the zero element. For any other arrow f : b 7! z the composition 1z ◦ f has to be zero as well. And therefore z is terminal. Definition Let A be an Ab-category. For objects a; b 2 A a biproduct is diagram i1 p2 a c b that fulfils p1i1 = 1a; p2i2 = 1b; i1p1 + i2p2 = 1c . p1 i2 Simon Ertl Abelian Categories 10. September 2020 6 / 21 pre-additive categories Proof. (iii))(i) and (iii))(ii) by definition. By duality it suffices to proof (i) ) (ii). As z is initial there exists a unique map 1z : z 7! z, thus MorA(z; z) has only one element, the zero element. For any other arrow f : b 7! z the composition 1z ◦ f has to be zero as well. And therefore z is terminal. Definition Let A be an Ab-category. For objects a; b 2 A a biproduct is diagram i1 p2 a c b that fulfils p1i1 = 1a; p2i2 = 1b; i1p1 + i2p2 = 1c . p1 i2 Example In the abelian groups the biproduct is exactly the direct sum. Simon Ertl Abelian Categories 10. September 2020 6 / 21 pre-additive categories Proposition In an Ab-category the following statements are equivalent: (i) The product of a; b exists. (ii) The coproduct of a; b exists. (iii) The biproduct of a; b exists. Simon Ertl Abelian Categories 10. September 2020 7 / 21 a 1a 0a;b sa p a c b b pa sb 0b;a 1b b Proof. By duality we only have to proof the equivalence of (i) and (iii). Let us p p assume we have a product a a c b b: By the definition of the product there exists a unique morphism sa : a 7! c, such that pa ◦ sa = 1a; pb ◦ sb = 0 and vice versa for sb. Then pa ◦ (sa ◦ pa + sb ◦ pb) = pa + 0 = pa pb ◦ (sa ◦ pa + sb ◦ pb) = 0 + pb = pb and sa ◦ pa + sb ◦ pb = 1c . Simon Ertl Abelian Categories 10. September 2020 8 / 21 Proof. We now have a biproduct as in our definition. d f g h i1 i2 a c b p1 p2 Define h : d 7! c as h = i1 ◦ f + i2 ◦ g. Then p1 ◦ h = f and p2 ◦ h = g. 0 0 0 For h d 7! c with p1 ◦ h = f and p2 ◦ h = g we deduce 0 0 0 0 0 h = 1c h = (i1p1 + i2p2)h = i1p1h + i2p2h = i1f + i2g = h p p and therefore a 1 c 2 b is indeed a product. Simon Ertl Abelian Categories 10. September 2020 9 / 21 Definition If A; B are Ab-categories, a functor F : A 7! B is additive when for all parallel arrows f ; f 0 : b 7! c in A F (f + f 0) = F (f ) + F (f 0) holds, so F is a group homomorphism. The composite of of additive functors is additive. Proposition F is additive if and only if F preserves biproducts. Additive category Definition An additive category is defined as a pre-additive category with a zero object and a biproduct for each pair of objects. Simon Ertl Abelian Categories 10. September 2020 10 / 21 Proposition F is additive if and only if F preserves biproducts. Additive category Definition An additive category is defined as a pre-additive category with a zero object and a biproduct for each pair of objects. Definition If A; B are Ab-categories, a functor F : A 7! B is additive when for all parallel arrows f ; f 0 : b 7! c in A F (f + f 0) = F (f ) + F (f 0) holds, so F is a group homomorphism. The composite of of additive functors is additive. Simon Ertl Abelian Categories 10.
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