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Abelian categories in dimension 2

Abelian categories in dimension 2,Mathieu Dupont

Abelian categories in dimension 2   (Citations: 7)
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The goal of this thesis is to define a 2-dimensional version of abelian categories, where symmetric 2-groups play the role that abelian groups played in 1-dimensional algebra. Abelian and 2-abelian groupoid enriched categories are defined and it is proved that homology can be developed in them, including the existence of a long exact sequence of homology corresponding to an extension of chain complexes. This generalises known results for symmetric 2-groups. The examples include, in addition to symmetric 2-groups, the 2-modules on a 2-ring, which form a 2-abelian groupoid enriched category. Moreover, internal groupoids, functors and natural transformations in an abelian category C (in particular, Baez-Crans 2-vector spaces) form a 2-abelian groupoid enriched category if and only if the axiom of choice holds in C.
Published in 2008.
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    • ...In the recent preprint Mathieu Dupont [1] rediscovered this notion...
    • ...Our original axioms were equivalent but not the same as one given in [1]...
    • ...The paper [1] contains several interesting results unknown to us, however some of the results of...
    • ...Dupont were known to us including Corollary 192 [1], which claims that the category of discrete and codiscrete (or connected) objects are equivalent abelian categories...
    • ...In this note we follow [1] with few exceptions...
    • ...For objects a : A1 → A0 and b : B1 → B0 we let HomA[1](a, b) be the corresponding homgroupoid...
    • ...It is clear that π1(HomA[1](a, b)) = HomA(Coker(a), Ker(b))...
    • ...in A, then one has an exact sequence 0 → Ext1A(Coker(a), Ker(b)) → π0(HomA[1](a, b)) → HomE(Ch(a), Ch(b)) → 0...
    • ...induced morphism of groupoids HomA[1](a, b) → HomA[1](a, b ′ )...
    • ...induced morphism of groupoids HomA[1](a, b) → HomA[1](a, b ′ )...
    • ...Proof. i) In this case we have a nice description for πi(HomA[1](a, −)), which shows that the functor HomA[1](a, b) → HomA[1](a, b′) yields an isomorphism on π0 and π1 and hence is an equivalence of categories...
    • ...Proof. i) In this case we have a nice description for πi(HomA[1](a, −)), which shows that the functor HomA[1](a, b) → HomA[1](a, b′) yields an isomorphism on π0 and π1 and hence is an equivalence of categories...
    • ...Proof. i) In this case we have a nice description for πi(HomA[1](a, −)), which shows that the functor HomA[1](a, b) → HomA[1](a, b′) yields an isomorphism on π0 and π1 and hence is an equivalence of categories...
    • ...ii) By the same reason the induced functor HomA[1](x, a) → HomA[1](x, b) is an equivalence of categories for all x ∈ A [1] c and hence we can use the Yoneda lemma for...
    • ...ii) By the same reason the induced functor HomA[1](x, a) → HomA[1](x, b) is an equivalence of categories for all x ∈ A [1] c and hence we can use the Yoneda lemma for...
    • ...ii) By the same reason the induced functor HomA[1](x, a) → HomA[1](x, b) is an equivalence of categories for all x ∈ A [1] c and hence we can use the Yoneda lemma for...
    • ...2-categories. � 3. The 2-category A [1] c In this section we will assume that A is an abelian category with enough projective objects...
    • ...We let A [1] c be the full 2-subcategory of the 2-category A [1] consisting of objects a : A1 → A0...
    • ...We let A [1] c be the full 2-subcategory of the 2-category A [1] consisting of objects a : A1 → A0...
    • ...If A is an abelian category with enough projective objects, then A [1] c is a 2-abelian...
    • ...Lemma 3.2. (i) A morphism (f0, f1) : a → b is faithful in A [1] c iff the morphism � −a...
    • ...is a monomorphism in A. (ii) Let (f0, f1) : a → b be a morphism in A [1] c and let...
    • ...Then (f0, f1) is fully faithful in A [1] c iff g is an isomorphism and...
    • ...π1(HomA[1](x, a)) → π1(HomA[1](x, b)) is a monomorphism for all x ∈ A [1] c . But the homomorphism in the question is the same as...
    • ...π1(HomA[1](x, a)) → π1(HomA[1](x, b)) is a monomorphism for all x ∈ A [1] c . But the homomorphism in the question is the same as...
    • ...π1(HomA[1](x, a)) → π1(HomA[1](x, b)) is a monomorphism for all x ∈ A [1] c . But the homomorphism in the question is the same as...
    • ...Since A has enough projective objects, any object in A is isomorphic to an object of the form Coker(x) for a suitable x ∈ A [1] c . Thus (f0, f1) : a → b is faithful iff the induced homomorphism...
    • ...ii) By definition (f0, f1) : a → b is fully faithful iff the induced functor (f0, f1)x : HomA[1](x, a)) → HomA[1](x, b))...
    • ...ii) By definition (f0, f1) : a → b is fully faithful iff the induced functor (f0, f1)x : HomA[1](x, a)) → HomA[1](x, b))...
    • ...This happens iff the functor (f0, f1)x yields an isomorphism on π1 and a monomorphism on π0. Thus for all x ∈ A [1] c the induced homomorphism...
    • ...is an isomorphism and the induced map π0(HomA[1](x, a)) → π1(HomA[1](x, b))...
    • ...is an isomorphism and the induced map π0(HomA[1](x, a)) → π1(HomA[1](x, b))...
    • ...Hence an object a : A1 → A0 in A [1] c is discrete iff...
    • ...The functor Dis(A [1] c ) → A given by a 7→ Coker(a) is an equivalence of categories...
    • ...is an epimorphism in A. (ii) Let (f0, f1) : a → b be a morphism in A [1] c and let...
    • ...Then (f0, f1) is fully cofaithful in A [1] c iff h is an isomorphism...
    • ...(f0, f1)x : HomA[1](b, x)) → HomA[1](a, x)) is faithful (resp...
    • ...(f0, f1)x : HomA[1](b, x)) → HomA[1](a, x)) is faithful (resp...
    • ...Since any object of A is of the form Ker(a) for a suitable a ∈ A [1] c ,...
    • ...it follows from the description of πi(HomA[1]) given in Section 2 (essentially by the same argument as in Lemma 3.2) that this happens iff the map h is a monomorphism (resp...
    • ...It is clear that the induced morphisms Ker(p) → Ker(x) and Coker(p) → Coker(x) are isomorphisms and p ∈ A [1] c . We call p a replacement of x. Sometimes it is denoted by x rep . The following easy...
    • ...If a ∈ A [1] c then f has the lifting to x rep , which is unique up to unique homotopy...
    • ...Now we discuss 2-kernels and 2-cokernels in A [1] c . Let (f0, f1) : a → b be a morphism in A [1] c ...
    • ...Now we discuss 2-kernels and 2-cokernels in A [1] c . Let (f0, f1) : a → b be a morphism in A [1] c ...
    • ...According to [1] the 2-cokernel of (f0, f1) in A[1] is (q′ : Q → B0,(id, q), ξ);...
    • ...Thanks to [1] the 2-kernel of (f0, f1) : a → b in A[1] is (k′ : A1 → K,(k, id), κ):...
    • ...Indeed, we have to show that for any object x in A[1]c the groupoids HomA[1](x, c) and the 2-kernel of...
    • ...Indeed, we have to show that for any object x in A[1]c the groupoids HomA[1](x, c) and the 2-kernel of HomA[1](x, a) → HomA[1](x, b)...
    • ...Indeed, we have to show that for any object x in A[1]c the groupoids HomA[1](x, c) and the 2-kernel of HomA[1](x, a) → HomA[1](x, b)...
    • ...are equivalent. But we know that the last groupoid is equivalent to HomA[1](x, k′)...
    • ...Since �(r) = (Ker(k′) → 0) we see that the morphism ωf : Coroot(π) → k′ defined in [1] is an equivalence...
    • ...Namely, we let A [1] f be the full 2-subcategory of the 2-category A [1] consisting of objects a : A1 →...
    • ...Namely, we let A [1] f be the full 2-subcategory of the 2-category A [1] consisting of objects a : A1 →...
    • ...A0 such that A1 is an injective object of A. Then A [1] f is a 2-abelian Gpd-category and the category...
    • ...of discrete and codiscrete objects of A [1] f are equivalent to A...

    Teimuraz Pirashvili. Abelian categories versus abelian 2-categories

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