Martingale Cohomology, Holonomy, and Homological Arbitrage

Abstract

We introduce a transport cohomological framework for categorical filtrations. Given a contravariant filtration $F:\mathcal T^{op}\to\mathbf{Prob}$ on a small category \(\mathcal T\), conditional expectation induces transport operators between local probabilistic states. Using the simplicial structure of the nerve \(N_\bullet(\mathcal T)\), we construct simplex-local cochain complexes associated with parametrized simplices and study their transport cohomology. The resulting framework naturally produces loop effects and holonomy structures. In particular, transport around closed simplicial histories may generate nontrivial probabilistic distortions, even when the initial and terminal objects coincide. The associated holonomy operators encode global transport effects between probabilistic states and detect obstructions generated by loop transport. This leads to the notion of homological arbitrage, understood as a global transport phenomenon emerging from probabilistic distortion along loops. From this viewpoint, the essential source of loop effects is the probabilistic distortion generated by transport around closed simplicial histories. The present framework is structurally analogous to parallel transport and holonomy in differential geometry, providing a geometric viewpoint on categorical filtrations and probabilistic transport structures.