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String diagrams for $4$-categories and fibrations of mapping $4$-groupoids

Manuel Araújo

Abstract

We introduce a string diagram calculus for strict $4$-categories and use it to prove that given a cofinite inclusion of $4$-categorical presentations, the induced restriction functor on mapping spaces to a fixed target strict $4$-category is a fibration of strict $4$-groupoids.

String diagrams for $4$-categories and fibrations of mapping $4$-groupoids

Abstract

We introduce a string diagram calculus for strict -categories and use it to prove that given a cofinite inclusion of -categorical presentations, the induced restriction functor on mapping spaces to a fixed target strict -category is a fibration of strict -groupoids.

Paper Structure

This paper contains 28 sections, 26 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Results
  3. Related work
  4. Future work
  5. Background
  6. Globular pasting diagrams and strict $n$-categories
  7. Simple string diagrams and n-sesquicategories
  8. Computads and string diagrams for $n$-sesquicategories
  9. Equivalences
  10. String diagrams for strict $4$-categories
  11. Homotopy generators
  12. String diagrams in strict $4$-categories
  13. Functor $4$-categories
  14. Natural transformations
  15. Modifications
  16. ...and 13 more sections

Key Result

Theorem 1

Let $\mathcal{C}$ be a strict $4$-category, $\mathcal{P}$ a presentation and $\mathcal{Q}$ another presentation, obtained by adding a finite number of cells to $\mathcal{P}$. Then the restriction map is a fibration of strict $4$-groupoids.

Theorems & Definitions (76)

  • Theorem 1
  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • remark 1
  • Definition 8
  • ...and 66 more