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Manifold Diagrams for Higher Categories

Lukas Heidemann

Abstract

We develop a graphical calculus of manifold diagrams which generalises string and surface diagrams to arbitrary dimensions. Manifold diagrams are pasting diagrams for $(\infty, n)$-categories that admit a semi-strict composition operation for which associativity and unitality is strict. The weak interchange law satisfied by composition of manifold diagrams is determined geometrically through isotopies of diagrams. By building upon framed combinatorial topology, we can classify critical points in isotopies at which the arrangement of cells changes. This allows us to represent manifold diagrams combinatorially and use them as shapes with which to probe $(\infty, n)$-categories, presented as $n$-fold Segal spaces. Moreover, for any system of labels for the singularities in a manifold diagram, we show how to generate a free $(\infty, n)$-category.

Manifold Diagrams for Higher Categories

Abstract

We develop a graphical calculus of manifold diagrams which generalises string and surface diagrams to arbitrary dimensions. Manifold diagrams are pasting diagrams for -categories that admit a semi-strict composition operation for which associativity and unitality is strict. The weak interchange law satisfied by composition of manifold diagrams is determined geometrically through isotopies of diagrams. By building upon framed combinatorial topology, we can classify critical points in isotopies at which the arrangement of cells changes. This allows us to represent manifold diagrams combinatorially and use them as shapes with which to probe -categories, presented as -fold Segal spaces. Moreover, for any system of labels for the singularities in a manifold diagram, we show how to generate a free -category.

Paper Structure

This paper contains 65 sections, 138 theorems, 179 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Acknowledgements
  4. Background
  5. Infinity Categories
  6. Quasicategories and Kan Complexes
  7. Localisation and Model Categories
  8. Fibrations
  9. Segal Spaces
  10. Polynomial Functors
  11. Maps of Polynomials
  12. Composition of Polynomials
  13. Polynomial Functors of Quasicategories
  14. Geometry
  15. Topology and Simplices
  16. ...and 50 more sections

Key Result

Lemma 2.1.2.7

Suppose that $f : \mathcal{C} \to \mathcal{D}$ is a map of simplicial sets such that $\mathcal{C}$ is a quasicategory and $\mathcal{D}$ is the nerve of a $1$-category. Then $f$ is an inner fibration.

Figures (3)

  • Figure 1: The $3$-diagram which braids a blue and an orange string, and its corresponding open $3$-mesh which detects the critical moment at which the two strata pass over each other.
  • Figure 2: Starting with a string diagram on the left, we can find the coarsest open $2$-mesh that refines it. The open $2$-mesh can then be described as a combinatorial object via the open $2$-truss on the right.
  • Figure 3: The stratified standard simplices $\|\Delta\lbrack 1 \rbrack\|$ and $\|\Delta\lbrack 2 \rbrack\|$.

Theorems & Definitions (408)

  • Definition 2.1.1.1
  • Lemma 2.1.2.7
  • proof
  • Lemma 2.1.3.19
  • proof
  • Example 2.2.0.2
  • Example 2.2.0.3
  • Definition 2.2.0.4
  • Lemma 2.2.0.6
  • proof
  • ...and 398 more