Abelian categories in dimension 2

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.

💡 Research Summary

The thesis introduces a genuine two‑dimensional analogue of abelian categories, called 2‑abelian categories, in which symmetric 2‑groups replace ordinary abelian groups as the basic additive objects. The author proceeds in four main stages.

First, the necessary background on symmetric 2‑groups (also known as Picard groupoids) is reviewed. A symmetric 2‑group is a groupoid equipped with a monoidal structure that is strictly associative, unital, and symmetric, and whose objects and morphisms both carry a compatible addition. This structure is the natural “categorified” version of an abelian group and serves as the prototypical example of an additive object in a higher‑categorical setting.

Second, the notion of a 2‑abelian groupoid‑enriched category is defined. Instead of enriching over the category of sets, the hom‑objects are taken to be groupoids (the “2‑cells” between 1‑morphisms). The enrichment is required to satisfy a list of axioms that categorify the classical abelian axioms: (i) every hom‑groupoid is equipped with a binary operation that is associative, commutative, and has a zero object; (ii) the 2‑cells themselves form an abelian groupoid under a compatible addition; (iii) kernels, cokernels, images and coimages exist and behave as expected; (iv) every monomorphism and epimorphism is normal, and the usual exactness conditions hold for sequences of 1‑morphisms and 2‑cells. These axioms guarantee that the familiar homological algebra of abelian categories can be lifted to the 2‑dimensional context.

Third, the author develops a homology theory for chain complexes internal to a 2‑abelian category. A chain complex consists of objects (C_n) together with differential 1‑morphisms (d_n:C_n\to C_{n-1}) and 2‑cells witnessing the relation (d_{n-1}\circ d_n\cong 0). Using the additive structure on 2‑cells, the author defines a boundary operator satisfying (\partial^2=0) up to coherent 2‑isomorphism, and then defines homology objects (H_n(C)=\ker d_n / \operatorname{im} d_{n+1}) in the usual way, but now each quotient is taken in the enriched sense. The central result is a long exact sequence of homology associated to any short exact sequence of chain complexes \