A geometric model of synthetic filtrations via context-dependent time
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915605726101504 |
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| author | Adachi, Takanori |
| author_facet | Adachi, Takanori |
| contents | 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 $Σ$, extending the simplex category $Δ$ so that each moment of time carries contextual structure. Synthetic filtrations are realized as contravariant functors $Σ^{op} \to Prob$, where $Prob$ is the category of probability spaces with null-preserving maps.
We then develop a homological analysis of $Σ$-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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_12919 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A geometric model of synthetic filtrations via context-dependent time Adachi, Takanori Probability 60A99, 16B50, 60G20, 91B82 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 $Σ$, extending the simplex category $Δ$ so that each moment of time carries contextual structure. Synthetic filtrations are realized as contravariant functors $Σ^{op} \to Prob$, where $Prob$ is the category of probability spaces with null-preserving maps. We then develop a homological analysis of $Σ$-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. |
| title | A geometric model of synthetic filtrations via context-dependent time |
| topic | Probability 60A99, 16B50, 60G20, 91B82 |
| url | https://arxiv.org/abs/2509.12919 |