Manifold $n$-diagrams generalize string diagrams to higher dimensions—they recover string diagrams in dimension $n = 2$. For $n = 3$, they specialize to the notion of surface diagrams.
The idea of manifold diagrams has its roots in the relation between stratified manifolds and higher categories. One way to think about this relation is by noting that stratified manifolds arise after dualizing (in the sense of Poincaré duality) pasting diagrams of directed cells in higher categories.
A well-known instance of this relation materializes in the generalized tangle hypothesis: indeed, tangles can (in a canonical way) be regarded as examples of manifold diagrams.
From the perspective of the tangle hypothesis, manifold diagrams can be used to express the structure of coherently dualizable objects. More precisely, this translates between “manifold singularities” and relations satisfied by the higher morphisms associated to a coherently dualizable objects. For instance, the manifold diagram in Figure 2 expresses the “swallowtail identity” satisfied by coherent biadjunctions (this is a manifold 4-diagram: think about the two surfaces as embedded in 3-dimensional space, and in the 4th spatial dimension we deform the top surface into the bottom one). The terminology derives from Thom’s classification of classical singularities (cf. this picture).
Many other interesting categorical and higher-algebraic identities can be expressed in manifold diagrams in a similar way. As an example, the Yang–Baxter equation (or, relatedly, the Reidemeister moves) can be expressed in manifold 4-diagrams as well, see Figure 3.
In higher dimensions the types of identites become more and more complicated: Figure 4 shows a manifold 5-diagram representing the Zamolodchikov tetrahedron equation). Much diagrammatic algebra of surfaces along these lines as been developed by Carter and collaborators.
Manifold diagrams are meant to provide a definitional framework for higher algebraic laws such as those above in all dimensions. Moreover, while the above examples only illustrate such laws for tangles (i.e. for embedded ordinary manifolds), in full generality manifold diagrams are diagrams of stratified manifolds.
Several definitional variants exist in dimensions $\leq 4$ (see also at surface diagrams). In general dimensions, manifold diagrams can be defined as the local models of conical cellulable stratifications on directed spaces.
A manifold $n$-diagram is a conical cellulable stratification of standard $n$-dimensional directed space.
In the definition, note:
A more detailed description of the situation can be found in Dorn and Douglas 22 and Dorn and Douglas 22B. Note that, “cellulable” is also called “meshable”, or “tame”.
John Baez and James Dolan, Higher-dimensional Algebra and Topological Quantum Field Theory, 1995 (arXiv)
Scott Carter, An excursion in diagrammatic algebra, 2012, World Scientific (doi:10.1142/8315)
Christoph Dorn and Christopher Douglas, Manifold diagrams and tame tangles, 2022 (pdf)
Christoph Dorn and Christopher Douglas, A brief introduction to framed combinatorial topology, 2022 (pdf)
Last revised on November 17, 2022 at 15:25:33. See the history of this page for a list of all contributions to it.