  Higher-order topological insulators (HOTIs) or multipole insulators, hosting
peculiar corner states, were discovered [arXiv:1611.07987, arXiv:1708.03636].
It was independently discovered [arXiv:1702.00624] that continuum 5-dimensional
(5D) Weyl semimetals generically host the corner states, and so do 4D class A
and 3D class AIII topological insulators. In this paper we further confirm that
the 5D Weyl semimetals, upon dimensional reduction, lead to universal
higher-order topology. First we explain a discrete symmetry protecting the 5D
Weyl semimetals, and describe dimensional reductions of the 5D Weyl semimetals
to the popular HOTIs in the continuum limit. We calculate the topological
charge carried by edge states of the 5D Weyl semimetal, for the most generic
boundary condition. The topological charge is a Dirac monopole, which can also
be seen from that edge Hamiltonians are always of the form of a 3D Weyl
semimetal. This edge topology leads to the edge-of-edge states, or the corner
states, generically, suggesting that the 5D Weyl semimetal is thought of as a
physical structural origin of corner states in HOTIs. In addition, we
explicitly calculate a nested Wilson loop of the 5D Weyl semimetal and find
that the topological structure is identical to that of a Wilson loop of a Dirac
monopole.
