  Contextuality has long been associated with topological properties. In this
work, such a relationship is elevated to identification in the broader
framework of generalized contextuality. We employ the usual identification of
states, effects, and transformations as vectors in a vector space and encode
them into a tangent space, rendering the noncontextual conditions the generic
condition that discrete closed paths imply null phases in valuations, which are
immediately extended to the continuous case. Contextual behavior admits two
equivalent interpretations in this formalism. In the geometric or
intrinsic-realistic view, termed "Schr\"odinger", flat space is imposed,
leading to contextual behavior being expressed as non-trivial holonomy of
probabilistic functions, analogous to the electromagnetic tensor. As a
modification of the valuation function, we use the equivalent curvature to
connect contextuality with interference, noncommutativity, and signed measures.
In the topological or participatory-realistic view, termed "Heisenberg",
valuation functions must satisfy classical measure axioms, resulting in
contextual behavior needing to be expressed in topological defects in events,
resulting in non-trivial monodromy. We utilize such defects to connect
contextuality with non-embeddability and to construct a generalized Vorob'ev
theorem, a result regarding the inevitability of noncontextuality. We identify
in this formalism the contextual fraction for models with outcome-determinism,
and a pathway to address disturbance in ontological models as non-trivial
transition maps. We also discuss how the two views for encoding contextuality
relate to interpretations of quantum theory.
