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(9)
Abelian Group
Axiom of Choice
Category Theory
Chain Complex
Enriched Categories
Natural Transformation
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Abelian categories in dimension 2
Abelian categories in dimension 2,Mathieu Dupont
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Abelian categories in dimension 2
(
Citations: 7
)
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Mathieu Dupont
The goal of this thesis is to define a 2dimensional version of abelian categories, where symmetric 2groups play the role that abelian groups played in 1dimensional algebra. Abelian and 2abelian 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 2groups. The examples include, in addition to symmetric 2groups, the 2modules on a 2ring, which form a 2abelian groupoid enriched category. Moreover, internal groupoids, functors and natural transformations in an abelian category C (in particular, BaezCrans 2vector spaces) form a 2abelian groupoid enriched category if and only if the
axiom of choice
holds in C.
Published in 2008.
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Citation Context
(1)
...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...
...2categories. � 3. The 2category 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 2subcategory of the 2category A [1] consisting of objects a : A1 → A0...
...We let A [1] c be the full 2subcategory of the 2category A [
1
] consisting of objects a : A1 → A0...
...If A is an abelian category with enough projective objects, then A [
1
] c is a 2abelian...
...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 2kernels and 2cokernels in A [
1
] c . Let (f0, f1) : a → b be a morphism in A [1] c ...
...Now we discuss 2kernels and 2cokernels in A [1] c . Let (f0, f1) : a → b be a morphism in A [
1
] c ...
...According to [
1
] the 2cokernel of (f0, f1) in A[1] is (q′ : Q → B0,(id, q), ξ);...
...Thanks to [
1
] the 2kernel 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 2kernel of...
...Indeed, we have to show that for any object x in A[1]c the groupoids HomA[1](x, c) and the 2kernel 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 2kernel 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 2subcategory of the 2category A [1] consisting of objects a : A1 →...
...Namely, we let A [1] f be the full 2subcategory of the 2category A [
1
] consisting of objects a : A1 →...
...A0 such that A1 is an injective object of A. Then A [
1
] f is a 2abelian Gpdcategory and the category...
...of discrete and codiscrete objects of A [
1
] f are equivalent to A...
Teimuraz Pirashvili
.
Abelian categories versus abelian 2categories
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Higher Dimensional Homology Algebra II:Projectivity
(
Citations: 3
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Fang Huang
,
ShaoHan Chen
,
Wei Chen
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ZhuJun Zheng
Published in 2010.
Higher Dimensional Homology Algebra III:Projective Resolutions and Derived 2Functors in (2SGp)
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Published in 2010.
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Published in 2010.
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