Abelian categories in dimension 2
Abstract
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.
 Publication:

Ph.D. Thesis
 Pub Date:
 September 2008
 arXiv:
 arXiv:0809.1760
 Bibcode:
 2008PhDT.......197D
 Keywords:

 Mathematics  Category Theory;
 18E10;
 18D05;
 18A20;
 18A22;
 18A32;
 18B40;
 18D10;
 18E05;
 18G35;
 Mathematics  Category Theory;
 18E10;
 18D05;
 18A20;
 18A22;
 18A32;
 18B40;
 18D10;
 18E05;
 18G35
 EPrint:
 x + 268 pages. This is the English version of my PhD thesis