Seminário de Álgebra e Topologia 

Categorical algebra is a fundamental branch of mathematics that lies at the intersection of category theory and algebra. On the one hand, it captures the fruitful properties and structures studied in algebra via category theory. On the other hand, it investigates the global categorical properties that algebraic objects enjoy when collected together. Both these endeavors are essential to extend and transport the fundamental concepts and theorems of algebra to different and broader settings. In this talk, we present an innovative theory that generalizes categorical algebra to the framework of 2-dimensional category theory. This has the notable advantage that the second dimension can be used both to weaken conditions that are too strict in nature and to refine algebraic invariants, obtaining a richer theory which encompasses a broader range of examples. Furthermore, 2-dimensional categorical algebra is essential to effectively compare different algebraic categories with each other.

This talk is based on a joint work with Elena Caviglia and Zurab Janelidze.

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.