15/05/2026, 12:00 — 13:00 —
Room P3.10, Mathematics Building
Elena Caviglia, Stellenbosch University
Abelian categories, triangulated categories and 2-dimensional exactness
Abelian categories and triangulated categories provide fundamental frameworks to study homological and cohomological problems across algebraic geometry, topology and representation theory.
In this talk we will explain how we can study the 2-category AbCat of abelian categories and the 2-category Triang of triangulated categories through the lenses of 2-dimensional categorical algebra. Surprisingly, through these lenses, AbCat and Triang look extremely similar.
We will show that the important notions of Serre subcategories and Serre quotients of abelian categories precisely correspond respectively with the 2-dimensional kernels and cokernels in AbCat. In a similar way, thick triangulated subcategories and Verdier localizations of triangulated categories are exactly the 2-kernels and the 2-cokernels inTriang.
Furthermore, even more striking similarities between the two contexts arise when characterizing these 2-kernels and 2-cokernels in terms of categorical properties satisfied by their underlying functors.
These results will allow us to show that both the 2-categories AbCat and Triang are exact, in appropriate 2-dimensional senses. In particular, AbCat is 2-Puppe exact in a 2-dimensional sense, while Triang satisfies the weaker exactness property of a 2-homological category.
This talk is based on a joint work in progress with Zurab Janelidze, Luca Mesiti and Ulo Reimaa.
