arxiv: 2605.01370 · v2 · submitted 2026-05-02 · 💱 q-fin.MF · math.CT

Recognition: no theorem link

Martingale Cohomology, Holonomy, and Homological Arbitrage

Takanori Adachi

Pith reviewed 2026-05-12 04:53 UTC · model grok-4.3

classification 💱 q-fin.MF math.CT

keywords martingale cohomologyholonomyhomological arbitragecategorical filtrationsprobabilistic transportsimplicial nerveconditional expectation

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The pith

Closed loop transport in categorical filtrations produces persistent probabilistic distortions despite matching endpoints.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a cohomological approach to transport in categorical filtrations by treating conditional expectations as operators between probabilistic states on the simplicial nerve of the category. It demonstrates that closed simplicial histories produce nontrivial distortions in these states even when the initial and terminal objects are the same. This matters because it frames arbitrage as a global homological effect arising from loop transport rather than from pointwise inconsistencies in the filtration.

Core claim

Given a contravariant filtration F from the opposite of a small category T to probability spaces, conditional expectation defines transport operators between local states. The simplicial nerve N of T is used to build simplex-local cochain complexes from these operators. Their transport cohomology yields holonomy operators that capture global effects of transport around closed simplicial histories. These operators detect obstructions and introduce homological arbitrage as the probabilistic distortion generated by such loop transport, providing an analogy to parallel transport and holonomy from differential geometry.

What carries the argument

The holonomy operators extracted from the cohomology of transport cochain complexes constructed via conditional expectations on the simplicial nerve of the category.

If this is right

Nontrivial probabilistic distortions arise from transport around closed simplicial histories even when initial and terminal objects coincide.
The associated holonomy operators encode global transport effects between probabilistic states.
Obstructions generated by loop transport are detected through the cohomology.
Homological arbitrage is defined as the global transport phenomenon emerging from probabilistic distortion along loops.
The framework supplies a geometric viewpoint on categorical filtrations and probabilistic transport structures.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This viewpoint could yield new invariants for checking no-arbitrage in filtered models by examining whether cohomology classes vanish.
Computation of these classes in concrete discrete-time examples might reveal hidden arbitrage opportunities.
The construction suggests possible links to other topological methods in stochastic analysis.

Load-bearing premise

The simplicial nerve of the category together with conditional-expectation transport maps naturally yields cochain complexes whose cohomology classes correspond to economically meaningful probabilistic distortions rather than artifacts of the chosen categorical encoding.

What would settle it

A concrete calculation on a small finite category with an explicit filtration in which every closed simplicial history produces a trivial cohomology class with no resulting probabilistic distortion.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper sketches a transport cohomology on categorical filtrations but the claimed nontrivial holonomy from loop transport is likely trivial by the tower property of conditional expectations.

read the letter

The core move here is to treat a contravariant filtration as a functor from a small category to probability spaces, let conditional expectations serve as transport maps along the nerve, and then build simplex-local cochain complexes whose cohomology is supposed to capture holonomy and homological arbitrage. That combination is new in the literature on mathematical finance, and the geometric analogy to parallel transport is drawn cleanly enough to be readable. The setup organizes loop-dependent inconsistencies in a single language, which could be useful if it actually produces something beyond standard martingale theory. The paper does a reasonable job laying out the formal scaffolding without overclaiming immediate applications. The central weakness is exactly the one flagged in the stress test. Conditional expectations obey the tower property, so any closed path in a filtration should compose to the identity. The abstract asserts that transport around closed simplicial histories can still produce nontrivial distortions, yet it gives no concrete category, no explicit cochain, and no calculation showing a holonomy operator different from the identity. Without that, the cohomology classes risk being formal artifacts rather than economically meaningful obstructions. If the full text contains a worked example where the operator is genuinely non-id, that would fix the issue; otherwise the framework stays at the level of rephrasing. This is aimed at readers already comfortable with categorical methods in stochastic processes who want to see whether the construction can be made nontrivial. It is coherent on its own terms and engages the relevant ideas honestly, so it deserves a serious referee to check the missing calculations and decide whether the cohomology is empty or not.

Referee Report

1 major / 2 minor

Summary. The manuscript introduces a transport cohomological framework for contravariant filtrations F: T^op → Prob on small categories T. Conditional expectations induce transport operators between local probabilistic states; the simplicial nerve N_•(T) is used to construct simplex-local cochain complexes whose transport cohomology yields holonomy operators. These operators are claimed to encode nontrivial probabilistic distortions along closed simplicial histories (even when initial and terminal objects coincide), detecting obstructions from loop transport and giving rise to the notion of homological arbitrage as a global transport phenomenon.

Significance. If the central claims are substantiated, the work supplies a geometric and homological viewpoint on categorical filtrations and martingale transport that is structurally parallel to parallel transport and holonomy in differential geometry. This could furnish new algebraic invariants for detecting global inconsistencies in probabilistic models that are invisible to local martingale conditions, with potential relevance to arbitrage theory in mathematical finance.

major comments (1)

[Framework construction and holonomy operators (as described in the abstract and § on transport cohomology)] The central claim that transport around closed simplicial histories generates nontrivial probabilistic distortions (non-identity holonomy operators) appears to conflict with the tower property of conditional expectations. When morphisms in T correspond to nested σ-algebras, the composition of conditional expectations along any closed path must return the identity; the manuscript must therefore specify (with explicit cochain-level definitions and at least one concrete example) how the simplicial nerve construction or the choice of transport maps produces a nontrivial holonomy class rather than the zero class in cohomology.

minor comments (2)

[Cochain complex construction] The definition of the coboundary operators in the simplex-local cochain complexes should be written out explicitly (including the precise role of the contravariant functor F) to allow direct verification of the claimed cohomology.
[Examples / Applications] A short illustrative example (e.g., a small poset filtration with explicit conditional expectations) would greatly clarify whether the holonomy is indeed nontrivial and how homological arbitrage manifests numerically.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments, which help clarify the scope of our framework. We address the major comment below and will incorporate the requested clarifications in a revised version of the manuscript.

read point-by-point responses

Referee: The central claim that transport around closed simplicial histories generates nontrivial probabilistic distortions (non-identity holonomy operators) appears to conflict with the tower property of conditional expectations. When morphisms in T correspond to nested σ-algebras, the composition of conditional expectations along any closed path must return the identity; the manuscript must therefore specify (with explicit cochain-level definitions and at least one concrete example) how the simplicial nerve construction or the choice of transport maps produces a nontrivial holonomy class rather than the zero class in cohomology.

Authors: We appreciate this observation. Our construction applies to general small categories T equipped with a contravariant functor F: T^op → Prob, rather than being restricted to posets whose morphisms correspond to nested σ-algebras. In this broader setting, morphisms need not represent inclusions, so the induced transport operators (defined via conditional expectations only where the structure of Prob permits) do not obey a tower property that would force every closed loop composition to act as the identity. The simplicial nerve N_•(T) is used to build simplex-local cochain complexes in which the coboundary maps are twisted by these transport operators; the resulting cohomology therefore admits nontrivial classes that encode holonomy even when the underlying objects coincide. We will revise the relevant sections to supply explicit cochain-level definitions of the transport cohomology and at least one concrete example of a categorical filtration on a small category containing a closed simplicial loop that yields a non-trivial holonomy operator. revision: yes

Circularity Check

0 steps flagged

No circularity: definitional construction of a new cohomological framework

full rationale

The paper introduces a transport cohomological framework by defining contravariant filtrations F: T^op → Prob, inducing transport operators via conditional expectations, and constructing simplex-local cochain complexes on the nerve N_•(T). Holonomy operators and homological arbitrage are defined as emergent structures within this construction. No equations reduce a claimed prediction or result to a fitted input by construction, no self-citations are load-bearing for the central claims, and no uniqueness theorems or ansatzes are smuggled from prior author work. The framework is self-contained as a categorical encoding; any nontriviality of loop transport follows from the chosen simplicial and probabilistic data rather than tautological redefinition of inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 3 invented entities

The framework rests on standard category theory and simplicial sets plus the assumption that conditional expectation supplies well-behaved transport maps; it introduces several new named concepts without independent empirical or formal verification.

axioms (3)

domain assumption A contravariant filtration F: T^op → Prob exists on a small category T
Invoked in the first sentence of the abstract as the starting object for the construction.
domain assumption Conditional expectation induces transport operators between local probabilistic states
Stated as the mechanism that produces the transport operators used to build the cochain complexes.
standard math The nerve N_•(T) supplies a simplicial structure on which simplex-local cochain complexes can be defined
Standard fact from category theory used to construct the cohomology.

invented entities (3)

transport cohomology no independent evidence
purpose: Cohomology groups that capture obstructions to consistent probabilistic transport along simplices
New named object constructed in the paper; no independent evidence supplied in abstract.
homological arbitrage no independent evidence
purpose: Global probabilistic distortion arising from loop transport
New concept defined as the economic interpretation of nontrivial holonomy; no independent evidence or examples given.
holonomy operators no independent evidence
purpose: Operators that encode global transport effects and detect loop obstructions
Introduced as the output of the cohomology construction; no concrete realization provided.

pith-pipeline@v0.9.0 · 5484 in / 1725 out tokens · 58920 ms · 2026-05-12T04:53:26.658393+00:00 · methodology

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Reference graph

Works this paper leans on

4 extracted references · 4 canonical work pages · 1 internal anchor

[1]

Adachi, T. (2025). A geometric model of synthetic filtrations via context-dependent time. arXiv:2509.12919 [math.PR]

work page arXiv 2025
[2]

Adachi, T. (2026). A haronov– B ohm type arbitrage and homological obstructions in financial markets. arXiv:2604.10492 [q-fin.MF]

work page internal anchor Pith review Pith/arXiv arXiv 2026
[3]

Adachi, T., Nakajima, K., and Ryu, Y. (2020). Generalized filtrations and its application to binomial asset pricing models. arXiv:2011.08531 [q-fin.MF]

work page arXiv 2020
[4]

and Ryu, Y

Adachi, T. and Ryu, Y. (2019). A category of probability spaces. J. Math. Sci. Univ. Tokyo , 26(2):201--221

work page 2019