2008/09/10 by Mathieu Dupont, M. Dupont, Dupont, Mathieu
Mathematics · #18A20 #18A22 #18A32 #18B40 #18D05 #18D10 #18E05 #18E10 #18G35 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #math.CT #msc:18A20 #msc:18A22 #msc:18A32 #msc:18B40 #msc:18D05 #msc:18D10 #msc:18E05 #msc:18E10 #msc:18G35
paper · pdf · doi:10.48550/arxiv.0809.1760
x + 268 pages. This is the English version of my PhD thesis; the original French version is available at http://edoc.bib.ucl.ac.be:81/ETD-db/collection/available/BelnUcetd-06112008-231800/
arxiv created 2008/09/10 · openalex publication_date 2008/09/10 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.