The core of adjoint functors

TL;DR

This paper refines the definition of adjoint functors by identifying their essential core, simplifying the concept across various categorical contexts and providing a unified, minimal framework.

Contribution

It introduces a core definition of adjoint functors, reducing redundancy and extending the concept to hom-enriched, bicategorical, and doctrinal settings.

Findings

Core of adjoint functors identified and formalized

Unified framework applicable to multiple categorical contexts

Simplified definitions facilitate broader applications

Abstract

There is a lot of redundancy in the usual definition of adjoint functors. We define and prove the core of what is required. First we do this in the hom-enriched context. Then we do it in the cocompletion of a bicategory with respect to Kleisli objects, which we then apply to internal categories. Finally, we describe a doctrinal setting.