Category theory seminar

Coalgebraic flavour of metric compact Hausdorff spaces - Marco Abbadini

In regular categories, morphisms have a well-behaved (regular epi)-mono factorization, allowing for a nice calculus of relations. Regularity is a pleasant and common feature of categories with algebraic traits, such as any class of algebras defined by equations. In contrast, coregularity—the dual concept—is common among categories with a topological character, the category of topological spaces being a prime example.

This talk focuses on the topologically-flavoured category of metric compact Hausdorff spaces. These structures generalize classical compact metric spaces (which form a poorly behaved category) and consist of a metric space equipped with a compatible compact Hausdorff topology, which need not be the induced topology. Our main result is that the category of metric compact Hausdorff spaces is coregular and that every equivalence corelation is effective, making it Barr-coexact. 

The proof techniques, which had already been used in joint work with Luca Reggio on Nachbin's compact ordered spaces, show promise for adaptation to other concrete categories with a topological flavour.

This talk is based on the preprint "Barr-coexactness for metric compact Hausdorff spaces", joint with Dirk Hofmann: https://arxiv.org/abs/2408.07039

Cartesian and additive opindexed categories - Alan Cigoli

We give a characterization of cartesian objects in the cartesian 2-category OpICat of opindexed categories. They are given by pseudofunctors F: B --> Cat, where B has finite products and the canonical oplax monoidal structure L on F admits a right adjoint R (in a suitable sense), which makes F a lax monoidal pseudofunctor. As a special case, if we restrict our attention to functors F: B --> Set, the cartesian ones are just finite-product preserving functors. When moreover B is additive, such F factorizes through the category Ab of abelian groups, and the corestriction is an additive functor.

Then we consider opindexed groupoids, i.e. pseudofunctors F: B --> Gpd. The cartesian objects here are pseudofunctors preserving finite products up to equivalences. When moreover B is additive, we find that such F factorizes through the 2-category Sym2Gp of symmetric 2-groups. In fact, we characterize the latter as 2-additive pseudofunctors (in the sense of Dupont).

This is joint work with S. Mantovani and G. Metere

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.