In this thesis, we explore several topics in the theory of monoidal and skew monoidal categories.
In Chapter 3, we give definitions of dual pairs in monoidal categories, skew monoidal categories, closed skew monoidal categories and closed monoidal categories. In the case of monoidal and closed monoidal categories, there are multiple well-known definitions of a dual pair. We generalise these definitions to skew monoidal and closed skew monoidal categories.
In Chapter 4, we introduce semidirect products of skew monoidal categories. Semidirect products of groups are a well-known and well-studied algebraic construction. Semidirect products of monoids can be defined analogously. We provide a categorification of this construction, for semidirect products of skew monoidal categories. We then discuss semidirect products of monoidal, closed skew monoidal and closed monoidal categories, in each case providing sufficient conditions for the semidirect product of two skew monoidal categories with the given structure to inherit the structure itself.
In Chapter 5, we prove a coherence theorem for monoidal adjunctions between closed monoidal categories, a fragment of Grothendieck's 'six operations' formalism.
