On Adjoint and Brain Functors

TL;DR

This paper offers a novel heterodox approach to adjoint functors using heteromorphisms, leading to the introduction of brain functors that model perception and action in cognitive systems.

Contribution

It introduces a new perspective on adjoint functors through heteromorphisms and defines brain functors to model cognitive functions, expanding the applications of category theory.

Findings

Heteromorphisms provide a natural way to interpret adjoint functors.

Brain functors can model perception and action in cognitive systems.

The approach simplifies the understanding of adjunctions in category theory.

Abstract

There is some consensus among orthodox category theorists that the concept of adjoint functors is the most important concept contributed to mathematics by category theory. We give a heterodox treatment of adjoints using heteromorphisms (object-to-object morphisms between objects of different categories) that parses an adjunction into two separate parts (left and right representations of heteromorphisms). Then these separate parts can be recombined in a new way to define a cognate concept, the brain functor, to abstractly model the functions of perception and action of a brain. The treatment uses relatively simple category theory and is focused on the interpretation and application of the mathematical concepts. The Mathematical Appendix is of general interest to category theorists as it is a defense of the use of heteromorphisms as a natural and necessary part of category theory.