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• W3011712018 abstract "In 2-dimensional algebra, symmetric 2-groups (symmetric monoidal groupoids in which every object has an inverse up to isomorphism) play a similar role to that of abelian groups in 1-dimensional algebra. Since abelian categories are defined in the image of the category of abelian groups, a 2-dimensional version of the notion of abelian category should be a solution to the equation abelian categories abelian groups = ?? symmetric 2-groups . I give two solutions to this equation. The first — abelian groupoid enriched categories — is a generalisation of ordinary abelian categories. In such a context, we can develop the theory of exact sequences and homology in a way close to homology in an abelian category: we prove several classical diagram lemmas as well as the existence of the long exact sequence of homology corresponding to an extension of chain complexes. This generalises known results for symmetric 2-groups [18, 26]. The other solution — 2-abelian groupoid enriched categories, which are also abelian in the sense of the previous paragraph — mimics the specifically 2dimensional properties of symmetric 2-groups, in particular the existence of two factorisation systems: surjective/full and faithful, and full and surjective/faithful [52]. Moreover, in a 2-abelian groupoid enriched category, the category of discrete objects is equivalent to that of connected objects, and these categories are abelian. This is close to the project of Marco Grandis of “developing homotopical algebra as an enriched version of homological algebra” [40]. 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, internal functors and internal natural transformations in an abelian category C (which includes as a special case Baez-Crans 2-vector spaces on a field K [4]) form a 2-abelian groupoid-enriched category if and only if the axiom of choice holds in C." @default.
• W3011712018 created "2020-03-23" @default.
• W3011712018 creator A5088799859 @default.
• W3011712018 date "2008-09-01" @default.
• W3011712018 modified "2023-09-26" @default.
• W3011712018 title "Abelian categories in dimension 2" @default.
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