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 $(\infty, n)$-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.