Morphisms and Elements of Group-like Type in Weak Multiplier Bialgebras

With the purpose of providing a categorical treatment of weak multiplier bialgebras (introduced by Bohm, Gomez-Torrecillas and Lopez-Centella in 2015), an appropriate notion of morphism for these algebraic objects is proposed. This allows us to define a category wmb of (regular) weak multiplier bialgebras (with a right full comultiplication), containing as a full subcategory the category wba of weak bialgebras defined by Bohm, Gomez-Torrecillas and Lopez-Centella in 2014. We present a great source of examples of these morphisms proving that, under some assumption, a functor between small categories induces a morphism of this kind between the natural weak multiplier bialgebra structures carried by the linear spans of the arrow sets of the categories. We explore the notion of elements of group-like type in a weak multiplier bialgebra, proposing a definition in the line of the one by the aforementioned authors for weak bialgebras. We show a big number of its properties and provide more general versions of many results known in the context of weak bialgebras. In particular, in analogy with the classical bialgebra setting (where the set of group-like elements is a monoid), we prove that the set of these elements possesses a structure of category.