Category theory seminar

Logic from the enriched categorical point of view

In logic, regular theories are those whose axioms are built using only equations, relation symbols, conjunctions, and existential quantification. The categories of models of such theories have been widely studied and characterised in purely category theoretical terms through the notions of exact and abielian category, and of injectivity class; I will recall these during the talk.

When moving to the context of categories enriched over a base V, corresponding notions of "exact V-category" and "V-injectivity class" have been studied by several authors, but no enriched notion of regular logic was considered in the literature before. The aim of this talk, which is based on joint work with Rosicky, is to fill this gap by introducing a notion of "enriched regular logic" that interacts well with the category theoretical counterparts mentioned above. Among others, we'll see examples from the additive, differentially graded, and 2-categorical setting.

Do torsion theories form a 2-torsion theory? - Zurab Janelidze

The category of 1-categories forms a 2-category. Recently, I showed that a suitable category of Puppe exact categories forms a 2-category that satisfies 2-dimensional counterparts of the axioms of a Puppe exact category (joint work in progress with Ülo Reimaa).

1-cells in this 2-category are to be called Serre functors, as they are closely linked with Serre subcategory inclusions and Serre quotients of abelian categories, and include those as special cases. This is a promising example of a potentially general phenomenon: a 2-category of 1-categories defined by an algebraic exactness property exhibits a 2-dimensional counterpart of the same exactness property.

In this talk we discuss work in progress on showing that a similar phenomenon could be exhibited for algebraically structured categories: we make first steps in showing that the 2-category of categories equipped with a torsion theory itself has a 2-dimensional torsion theory. What is common to both 2-dimensional situations is the use of the same and the usual notions of 2-zero object, 2-kernels and 2-cokernels.