Pseudomonads and Descent, PhD Thesis (Chapter 1)

TL;DR

This thesis introduces advanced concepts in 2-dimensional category theory, focusing on doctrinal adjunctions, biadjoint lifting, and pseudocoalgebras, connecting multiple research papers to explore descent and exponentiability.

Contribution

It synthesizes and applies recent theoretical results in 2-category theory to deepen understanding of pseudomonads, descent, and biadjoint structures within a unified framework.

Findings

Exposition of doctrinal adjunction and Beck-Chevalley condition

Application of biadjoint triangle theorem to pseudocoalgebras

Generalized lifting of biadjoints in monad theory

Abstract

This is the introductory chapter of my PhD Thesis. This thesis consists of one introductory chapter and four single-authored papers written during my PhD studies at the University of Coimbra under supervision of Maria Manuel Clementino. In this first chapter, we give a glance of the scope of our work and briefly describe elements of the original contributions of each paper, including some connections between them. We also give a brief exposition of our main setting, which is $2$-dimensional category theory. In this direction: (1) we give an exposition on the doctrinal adjunction, focusing on the Beck-Chevalley condition as used in Chapter "Pseudo-Kan Extensions and Descent" (arXiv: 1606.04999), (2) we apply the results of "On lifting of biadjoints and lax algebras" (arXiv: 1607.03087) in a generalized setting of the formal theory of monads and (3) we apply the biadjoint triangle theorem of "On biadjoint triangles" (aXiv: 1606.05009) to study (pseudo)exponentiable pseudocoalgebras.