Manifold Diagrams for Higher Categories

TL;DR

This paper introduces a graphical calculus called manifold diagrams for higher categories, enabling geometric and combinatorial analysis of complex categorical structures in arbitrary dimensions.

Contribution

It develops a new diagrammatic framework for $( , n)$-categories that generalizes existing string and surface diagrams, with a focus on semi-strict composition and isotopy-based interchange laws.

Findings

Manifold diagrams provide a geometric representation of higher categories.

A classification of critical points in isotopies aids in understanding diagram transformations.

The framework allows generating free $( , n)$-categories from labeled singularities.

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.