# Hilsum-Skandalis maps as Frobenius adjunctions with application to geometric morphisms

## TL;DR

This paper demonstrates that Hilsum-Skandalis maps can be represented as stably Frobenius adjunctions within a cartesian category, providing a unified framework for understanding geometric morphisms and inverting essential equivalences.

## Contribution

It introduces a new representation of Hilsum-Skandalis maps as stably Frobenius adjunctions and characterizes the connected components adjunction of internal groupoids.

## Key findings

- Hilsum-Skandalis maps are represented as stably Frobenius adjunctions.
- A new characterization of the connected components adjunction is provided.
- Properties of geometric morphisms are recovered through this framework.

## Abstract

Hilsum-Skandalis maps, from differential geometry, are studied in the context of a cartesian category. It is shown that Hilsum-Skandalis maps can be represented as stably Frobenius adjunctions. This leads to a new and more general proof that Hilsum-Skandalis maps represent a universal way of inverting essential equivalences between internal groupoids. To prove the representation theorem, a new characterisation of the con- nected components adjunction of any internal groupoid is given. The charaterisation is that the adjunction is covered by a stable Frobenius adjunction that is a slice and whose right adjoint is monadic. Geometric morphisms can be represented as stably Frobenius adjunctions. As applications of the study we show how it is easy to recover properties of geometric morphisms, seeing them as aspects of properties of stably Frobenius adjunctions.

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Source: https://tomesphere.com/paper/1703.10636