arXiv subjects

Miloslav Štěpán

Publications and source records attributed to Miloslav Štěpán.

3 recordsLinked to original sources

arXiv2025

Lax structures in 2-category theory

This thesis focuses on topics in 2-category theory: in particular on double categories, pseudomonads and codescent objects. In Chapter 2 we recall all the necessary notions. In Chapter 3 we show that factorization systems can be equivalently described as double categories satisfying certain properties. In Chapter 4 we focus on turning weak (colax) structures into strict ones in a universal way - this covers for instance colax monoidal categories or lax functors. In Chapter 5 we study the Kleisli 2-category for a lax-idempotent pseudomonad, the application of which is establishing weak cocompleteness of 2-categories such as the one of monoidal categories and lax monoidal functors. Finally, Chapter 6 focuses on the process of turning any 2-monad into a lax-idempotent one.

math.CT↗

arXiv2024

Colax adjunctions and lax-idempotent pseudomonads

We prove a generalization of a theorem of Bunge and Gray about forming colax adjunctions out of relative Kan extensions and apply it to the study of the Kleisli 2-category for a lax-idempotent pseudomonad. For instance, we establish the weak completeness of the Kleisli 2-category and describe colax change-of-base adjunctions between Kleisli 2-categories. Our approach covers such examples as the bicategory of small profunctors and the 2-category of lax triangles in a 2-category. The duals of our results provide lax analogues of classical results in two-dimensional monad theory: for instance, establishing the weak cocompleteness of the 2-category of strict algebras and lax morphisms and the existence of colax change-of-base adjunctions.

math.CT↗

arXiv2023

Factorization systems and double categories

We show that factorization systems, both strict and orthogonal, can be equivalently described as double categories satisfying certain properties. This provides conceptual reasons for why the category of sets and partial maps or the category of small categories and cofunctors admit orthogonal factorization systems. The theory also gives an explicit description of various lax morphism classifiers and explains why they admit strict factorization systems.

math.CT↗