Planar diagrammatics of self-adjoint functors and recognizable tree series

TL;DR

This paper explores how biadjoint functors generate central elements in categories via planar diagrams and investigates reconstructing functors and categories from diagram-based data, focusing on self-adjoint cases.

Contribution

It introduces a diagrammatic framework for understanding self-adjoint functors and addresses the inverse problem of recovering categories from diagrammatic invariants.

Findings

Established a correspondence between nested circle diagrams and central elements.

Developed methods to reconstruct categories and functors from diagrammatic data.

Analyzed the structure of self-adjoint functors using planar diagrammatics.

Abstract

A pair of biadjoint functors between two categories produces a collection of elements in the centers of these categories, one for each isotopy class of nested circles in the plane. If the centers are equipped with a trace map into the ground field, then one assigns an element of that field to a diagram of nested circles. We focus on the self-adjoint functor case of this construction and study the reverse problem of recovering such a functor and a category given values associated to diagrams of nested circles.