  The study of Markov chains on discrete spaces, such as digraphs, has
captivated mathematicians in recent decades due to its interconnectedness with
topology, geometry, dynamics, spectral theory, and differential equations.
Furthermore, extensive exploration of these multifaceted relationships has been
pursued for their practical utility in diverse fields, including machine
learning and image segmentation. In recent times, these interrelations have
been generalized to higher dimensions within the framework of
finite-dimensional simplicial complexes.
  In this paper, we embark on a further extension of these concepts. Initially,
we introduce a cohomology of infinite (though locally finite) digraphs in
arbitrary dimensions. Subsequently, in the latter portion of this manuscript,
we define a fresh family of Laplace operators and conduct an examination of
their spectrum, culminating in the proof of the Hodge Decomposition Theorem
within this framework. Finally, we conclude by presenting a Markov chain, the
$p$-Lazy Random Walk, whose asymptotic behavior is intrinsically linked to
these cohomologies, while its mixing time is related to the the spectrum of our
Laplace operators.
  This development opens doors to numerous unexplored questions, particularly
regarding potential generalizations of the Ollivier-Ricci curvature to this
topology and these Laplacians.
