Programme

Notions as category, functor, groupoid, 2-category and double category can be internalized in any category with finite limits. These internal structures are well-behaved in algebraic categories such as the categories of abelian groups, of groups, of rings, and of algebras. For example, it becomes a property for a reflexive graph to be an internal category or not, and any internal category is automatically an internal groupoid. If the category C is abelian, then the category Grpd(C) of internal groupoids in C is equivalent to the arrow category of C. If C is a semi-abelian category, then Grpd(C) is equivalent to the category of internal crossed modules in C. In this talk, we want to study the relationship between the categories of internal n-fold groupoids and of internal n-groupoids in the semi-abelian and abelian setting.

In this talk, I will present the construction of a skeletal diagram category, which we call the square with bands (SWB) category. This category extends the Temperley-Lieb (TL) category, so that morphisms now include embedded curves in (possibly) non-orientable bounded surfaces and involves three parameters associated to simple closed curves. We define a tensor product on this category and give a full set of monoidal generators, which includes the TL generators, a family of orientable genus one diagrams, and a family of non-orientable diagrams. These latter monoidal generators have interpretations as natural transformations between structural functors present in pivotal categories, suggesting a universal characterisation of the SWB category. A presentation of this category is not yet known, but generators satisfy analogues of well-known equations such as the Yang-Baxter and reflection equations.

Skew braces provide an algebraic framework for bijective non-degenerate set-theoretic solutions of the Yang–Baxter equation. In this talk I discuss recent work on free skew braces in certain classes and their relation to such solutions. I present a construction of free right nilpotent skew braces of class n. This yields a concrete realization of the free object and allows one to derive new structural properties. As an illustration, I discuss the one-generated case and its relation to free solutions in this setting.

While braided monoidal categories formalize the theory of R-matrices and solutions to the Yang-Baxter equation, braided module categories do the same for K-matrices and the reflection equation. As such, they also provide representations of Artin groups of type B, and additionally they have found interesting applications in the theory of nondegenerate extensions of fusion categories and of centres for monoidal 2-categories. I will present a reconstruction result for braided module categories in terms of braided algebra objects, in the sense of Johnson-Freyd and Reutter. This is based on work in progress joint with Tyler Franke and Tony Zorman.