Idea
Idea
Manifold $n$diagram generalize [[string diagram  string diagrams]] to higher dimensions—they recover string diagrams in dimension $n = 2$. For $n = 3$, they specialize to the notion of [[surface diagram  surface diagrams]]. 
The idea of manifold diagrams roots in the relation between [[stratified space  stratified]] [[manifold  manifolds]] and [[higher category theory  higher categories]]. One way to think about this relation is by the [[PontrjaginThom collapse map#ReferencesPTCellular  PontrjaginThom construction]] which links homotopy groups of [[cell complex  cell complexes]] and [[stratified space  stratified]] [[cobordism  cobordisms]]. Another (really, the same) way is to think of stratified manifolds as arising after dualizing (in the sense of [[Poincaré duality]]) [[pasting diagram  pasting diagrams of directed cells]] in higher categories. 
A wellknown instance of this relation materializes in the [[generalized tangle hypothesis]]: indeed, [[tangle  tangles]] can (in a canonical way) be regarded as an example of manifold diagrams. 
Figure 1: $m$Tangles in dimension $n$ can be regarded as manifold diagrams (picture reproduces part of Fig. 30 in BaezDolan95; note that $k = nm$ is the codimension of the tangle)
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 [[biadjunction  coherent biadjunctions]] (this is a manifold 4diagram: think about the two surfaces as embedded in 3dimensional 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). 
Figure 2: The swallowtail identity as a manifold 4diagram (reproducing title logo of MSRI program #323: higher categories and categorification)
Many other interesting categorical and [[higher algebra  higheralgebraic 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 as a manifold 4diagram, see Figure 3. 
Figure 3: The YangBaxterat identity as a manifold 4diagram, illustrated as movie of 1tangles in dimension 3 read from left to right.
In higher dimensions the types of identites become more and more complicated: Figure 4 shows a manifold 5diagram representing the Zamolodchikov tetrahedron equation). Much diagrammatic algebra of surfaces along these lines as been developed by Carter and collaborators.
Figure 4: The Zamolodchikov tetrahedron equation as a manifold 5diagram, illustrated as a movie from left to right (but the movie only has 2 frames).
Defining manifold diagrams is no easy task, as they are meant to provide a definitional framework for higher algebraic laws such as those above in all dimensions.
Definition
Several definitional variants exist in dimensions $\leq 4$ (see also at [[surface diagram  surface diagrams]]). In general dimensions, manifold diagrams can be defined as the local models of conical cellulable stratifications on [[directed space  directed spaces]]. 
\begin{defn} A manifold $n$diagram is a conical cellulable stratification of directed combinatorial standard $n$dimensional space. \end{defn}
Here, the term “standard” is another way of saying “local model” (concretely, we can take this to be [[directed space#framed_spaces  standard $\mathbb{R}^n$ with standard directions]]). The term “cellulable” simply says that the stratification has a subdivision into a complex of finitely many (open) directed topological cells—this condition is morally similar to requiring the existence of a finite triangulation, but cells can be much better adapted to the directed setting than simplices. The term “conical” is described at [[stratified space#conical_strat  stratified space]] in the setting of [[topological space  undirected spaces]]; in the setting of directed spaces one requires tubular neighborhoods to interact nicely with the directions of the underlying directed space. This is a sketch, a more detailed description of the situation can be found e.g. in DornDouglas and DornDouglas (Note 1: “cellulable” is also called “meshable” or “tame” to indicate a slight increase in generality: technically, one may also work with “cocells”, i.e. the Poincaré duals of cells.) (Note 2: The relation to tangles and manifold diagrams alluded to in the introduction can also be made precise.) 
References
For abelian groups in the context of [[homotopy type theory]]:

{#BaezDolan95} [[John Baez]] and [[James Dolan]], Higherdimensional Algebra and Topological Quantum Field Theory 1995 (arXiv)

{#Carter} Scott Carter, An excursion in diagrammatic algebra, 2012

{#DornDouglas22} Christoph Dorn and Christopher Douglas, Manifold diagrams and tame tangles, 2022 (pdfs)

{#DornDouglas22B} Christoph Dorn and Christopher Douglas, A brief introduction to framed combinatorial topology, 2022 (pdfs)