  In spacetime dimensions of 4D (i.e., 3+1D) and higher, topological orders
exhibit spatially extended excitations like loops and membranes, which support
diverse topological data characterizing braiding, fusion, and shrinking
processes, despite the absence of anyons. Our understanding of these
topological data remains less mature compared to 3D, where anyons have been
extensively studied and can be fully described through diagrammatic
representations. Inspired by recent advancements in field theory descriptions
of higher-dimensional topological orders, this paper systematically constructs
diagrammatic representations for 4D and 5D topological orders, generalizable to
higher dimensions. We introduce elementary diagrams for fusion and shrinking
processes, treating them as vectors in fusion and shrinking spaces,
respectively, and build complex diagrams by combining these elementary
diagrams. Within these vector spaces, we design unitary operations represented
by \(F\)-, \(\Delta\)-, and \(\Delta^2\)-symbols to transform between different
bases. We uncover \textit{pentagon equations} and \textit{(hierarchical)
shrinking-fusion hexagon equations} that impose constraints on the legitimate
forms of these unitary operations. We conjecture that all anomaly-free
higher-dimensional topological orders must satisfy these conditions, with any
violations indicating a quantum anomaly. This work opens promising avenues for
future research, including the exploration of diagrammatic representations
involving braiding and implications for noninvertible symmetries and Symmetry
Topological Field Theory.
