A Generalised Exactness Structure for Sets

TL;DR

This paper extends the concept of adjoint functors from functions to relations within set theory, demonstrating that a key exactness structure in group theory also applies to sets in this broader context.

Contribution

It introduces a generalized exactness structure for sets based on relations, expanding the applicability of adjunctions beyond functions.

Findings

The exactness structure in self-dual group theory applies to the category of sets.

Relations can be used to generalize adjoint functors.

The framework accommodates the empty set, as noted in the paper.

Abstract

Two adjoint functors can be seen as generalisations of the two functions within a Galois connection. If instead the adjoints are not generalised from functions, but from relations, then analogously the object of study becomes a more general notion of an adjunction. A suitable method to express such functor-level relations is to consider functors into categories of families. This structure is then used to show that the central exactness structure in self-dual group theory, consisting of a chain of adjunctions, holds also for the category of sets when seen in this general form. EDIT: Please see the note about the empty set on page 4!