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Combinatorial skeletons of 2-cobordism and annular categories with applications to equational logic

Karl Auinger, Mikhail Volkov

Abstract

We introduce a complete set of combinatorial data that encode the category $2\mathfrak{Cob}$ of all $2$-cobordisms. As an application, we show that the local monoids of $2\mathfrak{Cob}$ do not have finitely axiomatizable equational theories. As yet another application, we construct a von-Neumann-regular extension of this category. Similar results are provided for the topological annular category and various quotients of the latter, like the affine Temperley--Lieb category.

Combinatorial skeletons of 2-cobordism and annular categories with applications to equational logic

Abstract

We introduce a complete set of combinatorial data that encode the category of all -cobordisms. As an application, we show that the local monoids of do not have finitely axiomatizable equational theories. As yet another application, we construct a von-Neumann-regular extension of this category. Similar results are provided for the topological annular category and various quotients of the latter, like the affine Temperley--Lieb category.

Paper Structure

This paper contains 39 sections, 59 theorems, 197 equations, 13 figures, 1 table.

Table of Contents

  1. Introduction
  2. Partition categories and their regular extensions
  3. Undeformed and deformed partition categories
  4. The category $2\mathfrak{Cob}$ of $2$-cobordisms and its discrete skeleton
  5. Types of dead blocks in partition categories
  6. Regular versions of $\mathfrak{P}^{\mathrm{d}}$ and $2\mathfrak{Cob}$
  7. Categories of annular type
  8. Topological annular category
  9. Quotients and a subcategory of $\mathfrak{tAnn}$
  10. Combinatorial description of affine Temperley--Lieb categories and the morphism $\mathfrak{aLT}^{\mathrm e}\twoheadrightarrow \mathfrak{Ann}$
  11. Regular versions of the affine Temperley--Lieb categories
  12. Combinatorial description of $\mathfrak{tAnn}$ and $\mathfrak{L}$, and their regular versions
  13. Basics of the combinatorial description
  14. Composition of arrows
  15. Formation of increments
  16. ...and 24 more sections

Key Result

Proposition 2.1

Figures (13)

  • Figure 1: Two t$A$-diagrams $\bm\alpha$ and $\bm\beta$
  • Figure 2: The t$A$-diagram $\bm{\alpha\beta}$
  • Figure 3: Graphical representation of the element $c_0+\bigcirc+c_1+\cdots+c_{k-1}+\bigcirc+c_k$.
  • Figure 4: $\bm\alpha\circ\bm\beta\circ\bm\gamma$
  • Figure 5: $\bm\alpha\bm\beta\bm\gamma$
  • ...and 8 more figures

Theorems & Definitions (115)

  • Remark 1
  • Proposition 2.1
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.2
  • Remark 2
  • Remark 3
  • Proposition 2.3
  • proof
  • Remark 4
  • ...and 105 more