The codual quotient vector bundle

TL;DR

This paper introduces the concept of the codual quotient vector bundle, exploring its properties and establishing a new adjunction between quotient vector bundles and certain locales, advancing the theoretical framework.

Contribution

It defines the codual bundle for quotient vector bundles and establishes a novel adjunction linking quotient bundles and locales with contravariant morphisms.

Findings

The codual bundle's fiber at each point is isomorphic to the dual of the kernel.

A new adjunction between quotient vector bundles and locale-based categories is developed.

The framework extends the categorical understanding of vector bundles and locales.

Abstract

Given a quotient vector bundle $\mathcal A$ over $X$ with kernel map $\kappa: X\to\mathrm{Max}\,A$ we study the codual bundle with fiber at each point $x\in X$ isomorphic to the dual of $\kappa(x)$. Applying the adjunction between quotient vector bundles and linearized locales to the codual bundle leads to a new adjunction between the category of all quotient vector bundles and a category whose objects are locales $L$ equipped with a contravariant morphism $L\to\mathrm{Max}\, A$.