A geometric model of synthetic filtrations via context-dependent time

Abstract

Classical filtrations in probability theory formalize the accumulation of information along a linear time axis: the past is unique and the present evolves into an uncertain future. In reality, however, this linearity may itself be an illusion - an artifact of human perception that collapses multiple possible histories into a single apparent path. In this paper, we propose a geometric and homological model of synthetic filtrations, where the present arises as a synthesis of many potential pasts. To achieve this, we introduce a new category $\Sigma$, extending the simplex category $\Delta$ so that each moment of time carries contextual structure. Synthetic filtrations are realized as contravariant functors $\Sigma^{op} \to Prob$, where $Prob$ is the category of probability spaces with null-preserving maps. We then develop a homological analysis of $\Sigma$-filtrations, constructing chain complexes whose boundaries are given by conditional expectations. Their homology groups measure informational "holes" - probabilistic obstructions arising from the incompatibility of contextual expectations. As a concrete realization, we define Dirichlet filtrations, in which measures on simplices arise from Dirichlet distributions, reflecting both parameter and contextual uncertainty. Bayesian updating is then interpreted as a categorical transformation of a Dirichlet functor, revealing learning as a reconstruction of coherence across contexts. This framework suggests that what appears as a linear temporal order is merely a projection of a higher contextual geometry. It unifies categorical probability, homological algebra, and Bayesian reasoning, offering a new language for uncertainty in mathematics, finance, and cognition.