  We define the topological Dirac equation describing the evolution of a
topological wave function on networks or on simplicial complexes. On networks,
the topological wave function describes the dynamics of topological signals or
cochains, i.e. dynamical signals defined both on nodes and on links. On
simplicial complexes the wave function is also defined on higher-dimensional
simplices. Therefore the topological wave function satisfies a relaxed
condition of locality as it acquires the same value along simplices of
dimension larger than zero. The topological Dirac equation defines eigenstates
whose dispersion relation is determined by the spectral properties of the Dirac
(or chiral) operator defined on networks and generalized network structures
including simplicial complexes and multiplex networks. On simplicial complexes
the Dirac equation leads to multiple energy bands. On multiplex networks the
topological Dirac equation can be generalized to distinguish between different
mutlilinks leading to a natural definition of rotations of the topological
spinor. The topological Dirac equation is here initially formulated on a
spatial network or simplicial complex for describing the evolution of the
topological wave function in continuous time. This framework is also extended
to treat the topological Dirac equation on $1+d$ spaces describing a discrete
space-time with one temporal dimension and $d$ spatial dimensions with $d\in
\{1,2,3\}$. This work includes also the discussion of numerical results
obtained by implementing the topological Dirac equation on simplicial complex
models and on real simple and multiplex network data.
