Geometric constructions preserve fibrations

TL;DR

This paper demonstrates that certain geometric constructions in 2-category theory inherently preserve fibrations and opfibrations, extending Street's characterization through a new 2-monad framework.

Contribution

It introduces a 2-monad on the arrow 2-category that accounts for change of base, showing geometric constructions preserve fibrations and opfibrations.

Findings

Preservation of fibrations and opfibrations by specific 2-endofunctors.

Development of a 2-monad framework for change of base in 2-categories.

Extension of Street's characterization to a broader class of geometric constructions.

Abstract

Let $\mathcal{C}$ be a representable 2-category, and $\mathfrak{T}_\bullet$ a 2-endofunctor of the arrow 2-category $\mathcal{C}^\downarrow$ such that (i) $\mathsf{cod} \mathfrak{T}_\bullet = \mathsf{cod}$ and (ii) $\mathfrak{T}_\bullet$ preserves proneness of morphisms in $\mathcal{C}^\downarrow$. Then $\mathfrak{T}_\bullet$ preserves fibrations and opfibrations in $\mathcal{C}$. The proof takes Street's characterization of (e.g.) opfibrations as pseudoalgebras for 2-monads $\mathfrak{L}_B$ on slice categories $\mathcal{C}/B$ and develops it by defining a 2-monad $\mathfrak{L}_\bullet$ on $\mathcal{C}^\downarrow$ that takes change of base into account, and uses known results on the lifting of 2-functors to pseudoalgebras.