Presentations of strict monoidal categories and strict monoidal categories of welded tangle-oids

In this thesis, we address combinatorial descriptions of welded knotoids from the point of view of strict monoidal categories.

To this end, we address combinatorial presentations of strict monoidal categories, by generators and relations. We do this by addressing presentations of a closely related type of categorical object, which we call 1/2 -monoidal categories (essentially sesqui-categories on a single object). A key part of the construction relies on the construction of the free 1/2 -monoidal category over what we call a monoidal graph ( a graph with monoidal structure on the set of vertices). We prove that the category of what we called slideable 1/2 - monoidal categories is equivalent to the category of strict monoidal categories. We prove that there exists a slidealisation functor, sending a 1/2 -monoidal category to a slideable 1/2- monoidal category. We use this to obtain combinatorial presentations of strict monoidal categories from combinatorial presentations of 1/2 -monoidal categories.

We use this formalism to define presentations of strict monoidal categories of welded tangle-oids, generalising work of Lambropoulou, Turaev, Kaufmann and others on knotoids.

Given a finite group G, more generally a finite group acting on a finite abelian group, we construct a functor from the monoidal category of welded tangle-oids to a strictified version of the monoidal category of vector spaces.