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Structures in two dimensional category theory and applications to polynomial functors

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posted on 28.03.2022, 23:41 authored by Charles Robert Walker
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.

History

Table of Contents

1. Introduction -- 2. Yoneda structures and KZ doctrines -- 3. Distributive laws via admissibility -- 4. Generic bicategories -- 5. Universal properties of bicategories of polynomials -- 6. An elementary view of familial pseudofunctors -- 7. Conclusion and future directions -- References.

Notes

"Centre of Australian Category Theory Department" -- title page. Bibliography: pages 249-253 Theoretical thesis.

Awarding Institution

Macquarie University

Degree Type

Thesis PhD

Degree

PhD, Macquarie University, Faculty of Science and Engineering, Department of Mathematics and Statistics

Department, Centre or School

Department of Mathematics and Statistics

Year of Award

2019

Principal Supervisor

Richard Garner

Rights

Copyright Charles Robert Walker 2019. Copyright disclaimer: http://mq.edu.au/library/copyright

Language

English

Extent

1 online resource (xii, 253 pages)

Former Identifiers

mq:71037 http://hdl.handle.net/1959.14/1270211