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Planar diagrammatics of self-adjoint functors and recognizable tree series

Mikhail Khovanov, Robert Laugwitz

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.

Planar diagrammatics of self-adjoint functors and recognizable tree series

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.

Paper Structure

This paper contains 10 sections, 3 theorems, 41 equations, 24 figures.

Table of Contents

  1. Introduction
  2. Self-adjoint functors and circle diagrams
  3. Diagrammatics for biadjoint pairs
  4. Diagrammatics for a self-adjoint functor and the category U
  5. Circular triples obtained from wrapping actions on centers of categories
  6. The skein category SUZ of a circular triple
  7. Universality of RU and w-evaluation
  8. An adjunction of circular triples and monoidal categories with a self-adjoint endofunctor
  9. Pairings and monoidal envelopes
  10. Pairings, negligible morphisms, and the gligible quotient category Uomega of SUomega

Key Result

Proposition 1

Horizontal refection (and hence also vertical reflection) is the identity on the set $\mathbb{U}^0_0$ of (isomorphism classes) of circle diagrams.

Figures (24)

  • Figure 1: Diagrams of $1_F$ and $1_G$. In this diagrammatics, the planar region between two parallel horizontal dashed lines describes a natural transformation from the composition of functors read off the bottom dashed line to the composition given by boundary points at the top dashed line. Regions of the diagram correspond to categories.
  • Figure 2: Diagrams of biadjointness natural transformations. For instance, the leftmost diagram is the transformation $\delta_1$ from the identity functor $1_B$ to $FG$. When no arcs end on a dashed line, then we assign the identity functor, on the category which labels the region, to it.
  • Figure 3: Biadjointness relations are the isotopy relations on strands.
  • Figure 4: A diagram built out of the biadjointness transformations.
  • Figure 5: Examples of nested circles giving elements in $Z({\mathcal{A}})$ and $Z({\mathcal{B}})$.
  • ...and 19 more figures

Theorems & Definitions (9)

  • Remark
  • Proposition 1
  • proof
  • Definition 2
  • Example 3
  • Example 4
  • Proposition 5
  • Proposition 6
  • proof