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Doubly weak double categories

Aaron David Fairbanks, Michael Shulman

Abstract

We propose a definition of double categories whose composition of 1-cells is weak in both directions. Namely, a doubly weak double category is a double computad -- a structure with 2-cells of all possible double-categorical shapes -- equipped with all possible composition operations, coherently. We also characterize them using "implicit" double categories, which are double computads having all possible compositions of 2-cells, but no compositions of 1-cells; doubly weak double categories are then obtained by a simple representability criterion. Finally, they can also be defined by adding a "tidiness" condition to the double bicategories of Verity, or to the cubical bicategories of Garner.

Doubly weak double categories

Abstract

We propose a definition of double categories whose composition of 1-cells is weak in both directions. Namely, a doubly weak double category is a double computad -- a structure with 2-cells of all possible double-categorical shapes -- equipped with all possible composition operations, coherently. We also characterize them using "implicit" double categories, which are double computads having all possible compositions of 2-cells, but no compositions of 1-cells; doubly weak double categories are then obtained by a simple representability criterion. Finally, they can also be defined by adding a "tidiness" condition to the double bicategories of Verity, or to the cubical bicategories of Garner.

Paper Structure

This paper contains 15 sections, 50 theorems, 84 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The problem of doubly weak double categories
  3. Implicit structures
  4. Other definitions
  5. Outline
  6. Acknowledgments
  7. Bicategories
  8. Doubly weak double categories
  9. Double computads
  10. Algebraic definitions
  11. Icons and 2-monads
  12. Double bicategories
  13. Cubical bicategories
  14. A finite axiomatization
  15. Transformations and modifications

Key Result

Proposition 1

Given a bicategory $\mathcal{C}$, the following data amount to a represented implicit 2-category:

Figures (3)

  • Figure 1: $\lC_2$ consists of the "shapes of cell" in a 2-computad.
  • Figure 2: A generic 2-cell $(\alpha\xspace,\beta\xspace)$ in $\mathbf{C} \otimes \mathbf{D}$.
  • Figure 3: The colax transformation naturality axiom

Theorems & Definitions (144)

  • Remark 1
  • Definition 1
  • Remark 2
  • Definition 2
  • Remark 3
  • Proposition 1
  • proof
  • Proposition 2
  • Proposition 3
  • proof
  • ...and 134 more