TallCat Seminar Talk: From zero to manifold diagrams (and beyond)
Abstract
String diagrams have become a ubiquitous computational tool, as they allow us to efficiently represent and transform certain compositional information. Higher category theory provides a general framework in which to understand higher compositional information. Based on the higher-categorical perspective on string diagrams it has long been conjectured that the “idea” of string diagrams should generalize to higher dimensions as well, namely, to a notion of manifold diagrams. In this talk we present (equivalent!) purely geometric and purely combinatorial definitions of manifold diagrams. Importantly, historically, the combinatorial definition of manifold diagrams preceded their geometric definition: indeed, distilling the “idea” of manifold diagrams from a geometric perspective isn’t at all obvious (mainly because higher-dimensional geometry isn’t at all obvious). So in this talk we will focus on the (rather beautiful) mutual interplay of the geometry and the combinatorics of manifold diagrams. If time permits, we will discuss models of higher categories that arise from working with manifold-diagrammatic notions of composition.
Talk notes
References
Paper references
- Dorn and Douglas. Framed combinatorial topology, Math. Surveys and Monographs, to appear (2025).
- arXiv 2208.13758 (combinatorial and geometric definition of manifold diagrams, latest versions)
Online resources
- $n$Lab article on $n$-trusses (overview of underlying combinatorial ideas)
- $n$Lab article on $n$-manifold diagrams (idea of manifold diagrams + geometric definition)
- $n$Lab article on mannifold-diagrammatic higher categories