Units of skew monoidal categories and skew monoidales in Span

This thesis is about skew monoidal categories and consists of two relatively independent chapters, the first of which shows that the units of a skew monoidal category are unique up to a unique isomorphism, and internalises this fact to skew monoidales. Some benefits of certain extra structure on the unit maps are also discussed. We include some remarks on the unit conditions for a monoidal functor between skew monoidal categories that generalises the earlier uniqueness result. In the second, an interesting characterisation of a skew monoidale in the monoidal bicategory Span is given, generalising the case where the unit of the skew monoidale is of a certain restricted form, along with an example. Finally in an appendix, we show that the five axioms of a skew monoidal category are independent.