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A geometric model of synthetic filtrations via context-dependent time

Takanori Adachi

TL;DR

This work reframes time as a context-dependent manifold by introducing the category $\mathbf{\Sigma}$, extending the simplex category $\mathbf{\Delta}$, and modeling information flow as a synthetic filtration $X:\mathbf{\Sigma}^{op}\to \mathbf{Prob}$. It develops a homological theory for these filtrations via chain complexes built from conditional expectations, producing invariants $H^c_t$ that quantify coherence gaps across contexts. A concrete instantiation, Dirichlet filtrations, assigns Dirichlet measures on simplices and recasts Bayesian updating as a categorical transformation of a Dirichlet functor, capturing both parameter and contextual uncertainty. The framework unifies categorical probability, simplicial geometry, and Bayesian reasoning with potential impact in finance, cognition, and uncertainty quantification.

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 $Σ$, 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.

A geometric model of synthetic filtrations via context-dependent time

TL;DR

This work reframes time as a context-dependent manifold by introducing the category , extending the simplex category , and modeling information flow as a synthetic filtration . It develops a homological theory for these filtrations via chain complexes built from conditional expectations, producing invariants that quantify coherence gaps across contexts. A concrete instantiation, Dirichlet filtrations, assigns Dirichlet measures on simplices and recasts Bayesian updating as a categorical transformation of a Dirichlet functor, capturing both parameter and contextual uncertainty. The framework unifies categorical probability, simplicial geometry, and Bayesian reasoning with potential impact in finance, cognition, and uncertainty quantification.

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 , extending the simplex category so that each moment of time carries contextual structure. Synthetic filtrations are realized as contravariant functors , where is the category of probability spaces with null-preserving maps. We then develop a homological analysis of -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.

Paper Structure

This paper contains 18 sections, 14 theorems, 68 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Review of $\mathcal{T}$-filtrations
  3. Category of probability spaces
  4. $\mathcal{T}$-filtration
  5. Synthetic filtration
  6. Simplex category
  7. Geometric realization
  8. Functor $\Delta$
  9. Geometric realizers of $\mathbf{\Delta}$
  10. $\mathbf{\Sigma}$-filtration
  11. Homological analysis of $\mathbf{\Sigma}$-filtration
  12. Chain complex derived from $\mathbf{\Sigma}$-filtration
  13. Homology of $\mathbf{\Sigma}$-filtration
  14. Dirichlet filtration
  15. Dirichlet distribution
  16. ...and 3 more sections

Key Result

Theorem 2.3

(AR_2019) Let $\varphi : X \to Y$ be a null-preserving map. For any random variable $f \in L^{1}X$, there exists a random variable $g$ on $Y$ such that for every $B \in \mathcal{F}_Y$, We write $E^{\varphi}[f]$ for the random variable $g$, and call it a conditional expectation of $f$ along $\varphi$.

Figures (7)

  • Figure 1.1: Synthetic model
  • Figure 2.1: $\mathcal{T}$-Filtration as a contravariant functor from $\mathcal{T}$ to $\mathbf{Prob}$
  • Figure 2.2: $\mathcal{T}$-filtration corresponding to a classical filtration
  • Figure 3.1: Standard $n$-simplexes
  • Figure 3.2: Time evolution in $\mathbf{\Sigma}$
  • ...and 2 more figures

Theorems & Definitions (44)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • proof
  • Proposition 2.4
  • proof
  • Definition 2.5
  • Definition 2.6
  • Example 2.7
  • Definition 3.1
  • ...and 34 more