Structures in two dimensional category theory and applications to polynomial functors

This thesis seeks to further develop two-dimensional category theory, with a focus on Yoneda structures, (lax-idempotent) pseudomonads, pseudo-distributive laws, and familial representability, in order to gain new insights and tools in the study of polynomial functors. The first contribution of this thesis concerns Yoneda structures, which give a formalization of the presheaf construction. Our main result shows that any fully faithful lax-idempotent pseudomonad almost gives rise to a Yoneda structure, with all of the axioms holding except for one condition. The second contribution of this thesis concerns pseudo-distributive laws of a pseudomonad and a lax-idempotent pseudomonad. We show that such distributive laws have a simple algebraic description which only requires three out the usual eight coherence conditions, and another simple description in terms of the data of the near-Yoneda structurere covered from the lax-idempotent pseudomonad. Our third contribution is to introduce a class of bicategories, which we term generic bicategories. These are the bicategories for which horizontal composition admits generic factorisations, and have the interesting property that oplax functors out of them have a reduced description, similar to the axioms of a comonad. The fourth contribution of this thesis is to establish the universal properties of the bicategory of polynomials, with general and cartesian 2-cells, using the properties of generic bicategories to avoid the majority of the coherence conditions. In addition, we give a new proof of the universal properties of the bicategory of spans and establish the universal properties of the bicategory of spans with invertible 2-cells. The fifth contribution of this thesis is to give an appropriate notion of familial representability for pseudofunctors L : A - B of bicategories, and to describe an equivalence with an analogue of generic factorisations. This improves on work of Weber, who did not provide such an equivalence, and required A to have a terminal object.