Abelian 2-group (original) (raw)
In mathematics, an Abelian 2-group is a higher dimensional analogue of an Abelian group, in the sense of higher algebra, which were originally introduced by Alexander Grothendieck while studying abstract structures surrounding Abelian varieties and Picard groups. More concretely, they are given by groupoids which have a bifunctor which acts formally like the addition an Abelian group. Namely, the bifunctor has a notion of commutativity, associativity, and an identity structure. Although this seems like a rather lofty and abstract structure, there are several (very concrete) examples of Abelian 2-groups. In fact, some of which provide prototypes for more complex examples of higher algebraic structures, such as Abelian n-groups.
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| dbo:abstract | In mathematics, an Abelian 2-group is a higher dimensional analogue of an Abelian group, in the sense of higher algebra, which were originally introduced by Alexander Grothendieck while studying abstract structures surrounding Abelian varieties and Picard groups. More concretely, they are given by groupoids which have a bifunctor which acts formally like the addition an Abelian group. Namely, the bifunctor has a notion of commutativity, associativity, and an identity structure. Although this seems like a rather lofty and abstract structure, there are several (very concrete) examples of Abelian 2-groups. In fact, some of which provide prototypes for more complex examples of higher algebraic structures, such as Abelian n-groups. (en) |
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| dbo:wikiPageWikiLink | dbr:N-group_(category_theory) dbr:Bifunctor dbr:Mathematics dbr:Picard_group dbr:Spectrum_(topology) dbr:Alexander_Grothendieck dbr:Algebraic_variety dbc:Algebraic_topology dbr:Exalcomm dbr:Abelian_varieties dbr:Gerbe dbr:Postnikov_system dbr:Cotangent_complex dbc:Homological_algebra dbr:Abelian_group dbc:Abelian_group_theory dbr:Eilenberg–MacLane_space dbr:Sphere_spectrum dbr:Grothendieck_topology dbr:Scheme_(mathematics) dbr:∞-groupoid dbr:Arxiv:1007.4138 dbr:Arxiv:1101.2918 dbr:Arxiv:1201.2686 dbr:Arxiv:1606.07032v2 dbr:Arxiv:1702.02128 |
| dct:subject | dbc:Algebraic_topology dbc:Homological_algebra dbc:Abelian_group_theory |
| rdfs:comment | In mathematics, an Abelian 2-group is a higher dimensional analogue of an Abelian group, in the sense of higher algebra, which were originally introduced by Alexander Grothendieck while studying abstract structures surrounding Abelian varieties and Picard groups. More concretely, they are given by groupoids which have a bifunctor which acts formally like the addition an Abelian group. Namely, the bifunctor has a notion of commutativity, associativity, and an identity structure. Although this seems like a rather lofty and abstract structure, there are several (very concrete) examples of Abelian 2-groups. In fact, some of which provide prototypes for more complex examples of higher algebraic structures, such as Abelian n-groups. (en) |
| rdfs:label | Abelian 2-group (en) |
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