Martingale Cohomology, Holonomy, and Homological Arbitrage

We develop a cohomological framework for martingale theory based on categorical filtrations, where time is modeled by a small category and a filtration is defined as a contravariant functor to the category of probability spaces. By constructing a simplicial cochain complex associated with such filtrations, we show that martingales arise naturally as $0$-cocycles. A key feature of the construction is the presence of a multiplicative distortion encoded by a density operator, which prevents the naive coboundary from forming a cochain complex. We introduce a normalization procedure, called the $\beta$-gauge, which removes this obstruction and yields a well-defined cochain complex. Within this framework, $1$-cochains represent gain systems, and the first cohomology group captures consistent gains that cannot be generated by any price process. This leads to the notion of homological arbitrage, interpreted as a global cohomological obstruction. We further introduce an additive holonomy along loops in the time category, defined by transporting gains via conditional expectation. This provides an observable quantity measuring total gain accumulation along loops. By factoring out transport effects arising from price systems, we define a cohomological holonomy that depends only on the cohomology class and isolates the intrinsic loop-level arbitrage component. These results suggest a geometric perspective on financial markets in which arbitrage arises from global structures associated with nonlinear time.