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Hom $ω$-categories of a computad are free

Thibaut Benjamin, Ioannis Markakis

Abstract

We provide a new description of the hom functor on weak $ω$-categories, and we show that it admits a left adjoint that we call the suspension functor. We then show that the hom functor preserves the property of being free on a computad, in contrast to the hom functor for strict $ω$-categories. Using the same technique, we define the opposite of an $ω$-category with respect to a set of dimensions, and we show that this construction also preserves the property of being free on a computad. Finally, we show that the constructions of opposites and homs commute.

Hom $ω$-categories of a computad are free

Abstract

We provide a new description of the hom functor on weak -categories, and we show that it admits a left adjoint that we call the suspension functor. We then show that the hom functor preserves the property of being free on a computad, in contrast to the hom functor for strict -categories. Using the same technique, we define the opposite of an -category with respect to a set of dimensions, and we show that this construction also preserves the property of being free on a computad. Finally, we show that the constructions of opposites and homs commute.
Paper Structure (31 sections, 11 theorems, 174 equations)

This paper contains 31 sections, 11 theorems, 174 equations.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Related work.
  4. Overview.
  5. Globular pasting diagrams
  6. Computads and ω-categories
  7. Binary compositions in ω-categories
  8. The suspension and hom functors
  9. The suspension of a computad
  10. Computads.
  11. Cells and morphisms.
  12. Spheres.
  13. Fullness.
  14. Infinite-dimensional computads.
  15. Hom ω-categories
  16. ...and 16 more sections

Key Result

Proposition 10

For every bipointed computad $C$, the $\omega$-category $\Omega K^{T^{\star\star}}C$ is free on a computad.

Theorems & Definitions (38)

  • Definition 1
  • Definition 2
  • Definition 3
  • Example 4
  • Example 5
  • Example 6
  • Definition 7
  • Example 8
  • Definition 9
  • Proposition 10
  • ...and 28 more