Manifold diagrams for higher categories

Thesis

We develop a graphical calculus of manifold diagrams which generalises string and surface dia- grams to arbitrary dimensions. Manifold diagrams are pasting diagrams for (∞,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 geomet- rically 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 (∞,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 (∞,n)-category.

Authors: Heidemann, LV

• DOI: 10.5287/ora-erxqapyn4
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: homotopical algebra; framed combinatorial topology; string diagrams; stratified spaces; manifold diagrams; graphical calculi; higher category theory
• Subjects: Homotopy theory; Geometry, Algebraic