A. L. Agore’s research while affiliated with Vrije Universiteit Brussel and other places

We prove that the category of solutions of the set-theoretic Yang–Baxter equation of Frobenius-Separability (FS) type is equivalent to the category of pointed Kimura semigroups. As applications, all involutive, idempotent, non-degenerate, surjective, finite order, unitary or indecomposable solutions of FS type are classified. For instance, if $|X| = n$, then the number of isomorphism classes of all such solutions on X that are (a) left non-degenerate, (b) bijective, (c) unitary or (d) indecomposable and left-non-degenerate is: (a) the Davis number d(n), (b) $\sum _{m|n}\, p(m)$, where p(m) is the Euler partition number, (c) $\tau (n) + \sum _{d|n}\left\lfloor \frac{d}{2}\right\rfloor$, where $\tau (n)$ is the number of divisors of n, or (d) the Harary number $\mathfrak {c} (n)$. The automorphism groups of such solutions can also be recovered as automorphism groups $\textrm{Aut}(f)$ of sets X equipped with a single endo-function $f:X\rightarrow X$. We describe all groups of the form $\textrm{Aut}(f)$ as iterations of direct and (possibly infinite) wreath products of cyclic or full symmetric groups, characterize the abelian ones as products of cyclic groups, and produce examples of symmetry groups of FS solutions not of the form $\textrm{Aut}(f)$.

Hopf braces are the quantum analogues of skew braces and, as such, their cocommutative counterparts provide solutions to the quantum Yang-Baxter equation. We investigate various properties of categories related to Hopf braces. In particular, we prove that the category of Hopf braces is accessible while the category of cocommutative Hopf braces is even locally presentable. We also show that functors forgetting multiple antipodes and/or multiplications down to coalgebras are monadic. Colimits in the category of cocommutative Hopf braces are described explicitly and a free cocommutative Hopf brace on an arbitrary cocommutative Hopf algebra is constructed.

The first historical encounter with Poisson-type algebras is with Hamiltonian mechanics. With the abstraction of many notions in Physics, Hamiltonian systems were geometrized into manifolds that model the set of all possible configurations of the system, and the cotangent bundle of this manifold describes its phase space, which is endowed with a Poisson structure. Poisson brackets led to other algebraic structures, and the notion of Poisson-type algebra arose, including transposed Poisson algebras, Novikov–Poisson algebras, or commutative pre-Lie algebras, for example. These types of algebras have long gained popularity in the scientific world and are not only of their own interest to study, but are also an important tool for researching other mathematical and physical objects.

In general, universal (co)measuring (co)monoids and universal (co)acting bi/Hopf monoids, which prove to be a useful tool in the classification of quantum symmetries, do not always exist. In order to ensure their existence, the support of a given object was recently introduced in \cite{AGV3} and used to restrict the class of objects considered when defining universal (co)acting objects. It is well-known that, in contrast with the universal coacting Hopf algebra, for actions on algebras over a field it is usually difficult to describe the universal acting Hopf algebra explicitly and this turns the duality theorem into an important investigation tool. In the present paper we establish duality results for universal (co)measuring (co)monoids and universal (co)acting bi/Hopf monoids in pre-rigid braided monoidal categories $\mathcal C$. In addition, when the base category $\mathcal C$ is closed monoidal, we provide a convenient uniform approach to the aforementioned universal objects in terms of the cosupports, which in this case become subobjects of internal hom-objects. In order to explain our constructions, we use the language of locally initial objects. Known results from the literature are recovered when the base category is the category of vector spaces over a field. New cases where our results can be applied are explored, including categories of (co)modules over (co)quasitriangular Hopf algebras, Yetter-Drinfeld modules and dg-vector spaces.

The universal (co)acting bi/Hopf algebras introduced by Yu.\,I.~Manin, M.~Sweedler and D.~Tambara, the universal Hopf algebra of a given (co)module structure, as well as the universal group of a grading, introduced by J.~Patera and H.~Zassenhaus, find their applications in the classification of quantum symmetries. Typically, universal (co)acting objects are defined as initial or terminal in the corresponding categories and, as such, they do not always exist. In order to ensure their existence, we introduce the support of a given object, which generalizes the support of a grading and is used to restrict the class of objects under consideration. The existence problems for universal objects are formulated and studied in a purely categorical manner by seeing them as particular cases of the lifting problem for a locally initial object. We prove the existence of a lifting and, consequently, of the universal (co)acting objects under some assumptions on the base (braided or symmetric monoidal) category. In contrast to existing constructions, our approach is self-dual in the sense that we can use the same proof to obtain the existence of universal actions and coactions. In particular, when the base category is the category of vector spaces over a field, the category of sets or their duals, we recover known existence results for the aforementioned universal objects. The proposed approach allows us to apply our results not only to the classical categories of sets and vectors spaces and their duals but also to (co)modules over bi/Hopf algebras, differential graded vector spaces, G-sets and graded sets.

This chapter introduces the fundamental concepts needed in the sequel. Important notions such as (sub)categories, functors, natural transformations, representable functors which form the backbone of category theory are well illustrated by many familiar examples. A concise description of the duality principle, a crucial reasoning process in category theory, is also presented. The first important result we present is Yoneda’s lemma which allows us to embed any (locally small) category into a category of functors on that category.

Selected exercises are solved in full detail.

Treats the general theory of (co)limits. Both are very general concepts which arise in various forms in all fields of mathematics. We introduce them gradually starting with some special cases which might be familiar to the reader such as: (co)products, (co)equalizers, pullbacks and pushouts. A variety of detailed examples are included to illustrate the newly introduced concepts. (Co)products and (co)equalizers are not only important special cases of (co)limits but also generic in the sense that all (co)limits can be constructed out of these two special cases. Certain types of functors are considered in connection to the existence of (co)limits. The existence of (co)limits in several important categories such as functor categories or comma categories is investigated in detail as well.