Approaching the Continuous from the Discrete: an Infinite Tensor Product Construction

Antonio Lorenzin; Fabio Zanasi

arxiv: 2510.14716 · v3 · pith:P5PJABZEnew · submitted 2025-10-16 · 🧮 math.CT · cs.LO

Approaching the Continuous from the Discrete: an Infinite Tensor Product Construction

Antonio Lorenzin , Fabio Zanasi This is my paper

Pith reviewed 2026-05-21 21:13 UTC · model grok-4.3

classification 🧮 math.CT cs.LO

keywords infinite tensor productFinStochlocally constant Markov kernelsCantor spacecontinuous probabilitycategorical probability theoryaxiomatizationstochastic matrices

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The pith

A universal infinite tensor product construction turns the category of finite stochastic processes into one where any probability measure on the reals is representable using locally constant Markov kernels on the Cantor space.

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

This paper develops a universal construction for adjoining infinite tensor products to categories modeling probabilistic processes. The goal is to extend discrete probabilistic reasoning to continuous settings without losing algebraic structure. Applying the construction to FinStoch produces a new category whose objects are finite sets together with the Cantor space, and whose morphisms are locally constant Markov kernels. Within this category, every probability measure on the real numbers can be expressed and manipulated. The construction also allows axiomatic presentations of discrete probability to be lifted to the continuous case over powers of two.

Core claim

The main result is that the infinite tensor product construction applied to FinStoch yields a category of locally constant Markov kernels on finite sets and the Cantor space 2 to the power of natural numbers, in which arbitrary probability measures on the reals can be reasoned about, and in which axiomatization results for discrete probability lift through the construction.

What carries the argument

The universal construction that adjoins infinite tensor products to a category while preserving its probabilistic and algebraic features.

Load-bearing premise

That the infinite tensor product construction preserves sufficient structure from the original category to allow the lifted axiomatization to hold and to ensure locally constant kernels represent all measures on the reals.

What would settle it

Finding a probability measure on the reals that cannot be represented by any locally constant Markov kernel from the Cantor space to itself, or demonstrating that some axiom fails to lift through the construction.

Figures

Figures reproduced from arXiv: 2510.14716 by Antonio Lorenzin, Fabio Zanasi.

**Figure 1.** Figure 1: Sequential and parallel composition of the plate notation. representation for morphisms of C ⊗∞ which highlights their formulation as compatible families of C-morphisms (Definitions 7 and 8). Our approach is reminiscent of the ‘plate’ notation commonly used in the study of Bayesian networks, see e.g. [3]. In category-theoretic approaches to probability, this has recently been employed in [5]. Our use of … view at source ↗

read the original abstract

Increasingly in recent years, probabilistic computation has been investigated through the lenses of categorical algebra, especially via string diagrammatic calculi. Whereas categories of discrete and Gaussian probabilistic processes have been thoroughly studied, with various axiomatisation results, more expressive classes of continuous probability are less understood, because of the intrinsic difficulty of describing infinite behaviour by algebraic means. In this work, we establish a universal construction that adjoins infinite tensor products, allowing continuous probability to be investigated from discrete settings. Our main result applies this construction to $\mathsf{FinStoch}$, the category of finite sets and stochastic matrices, obtaining a category of locally constant Markov kernels, where the objects are finite sets plus the Cantor space $2^{\mathbb{N}}$. Any probability measure on the reals can be reasoned about in this category. Furthermore, we show how to lift axiomatisation results through the infinite tensor product construction. This way we obtain an axiomatic presentation of continuous probability over countable powers of $2=\lbrace 0,1\rbrace$.

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 universal construction for adjoining infinite tensor products is the solid new piece, but the claim that locally constant kernels on 2^N suffice for arbitrary measures on the reals needs tighter justification.

read the letter

The paper introduces a universal construction that extends a category with infinite tensor products and then applies it to FinStoch. This produces a category whose objects include finite sets and the Cantor space, with morphisms given by locally constant Markov kernels. The same construction lifts axiomatization results from the discrete setting to this new one, yielding an algebraic presentation for continuous probability over powers of 2. That construction and the lifting step are the actual additions here, and they look like a clean way to move from finite stochastic matrices to something that can handle infinite products without building everything from scratch. The framing around string diagrams and probabilistic computation is also straightforward and connects to existing work on discrete and Gaussian cases. The main soft spot sits in the central claim that any probability measure on the reals can be reasoned about inside this category. Local constancy restricts kernels to depending on only finitely many coordinates, which works for many measures but may leave out cases that require genuinely infinite dependence after a binary expansion pushforward, such as certain singular continuous distributions or atoms at non-dyadic points. The abstract states the representation result directly, so the paper needs to show either that the restriction is harmless or that the construction still covers the missing cases. This is aimed at people working in categorical probability who want systematic extensions of discrete models. A reader focused on universal properties or axiomatizations would get the most out of it. The work has enough formal structure to deserve a serious referee, mainly to check the details of the universal construction and the scope of the representation claim.

Referee Report

1 major / 2 minor

Summary. The manuscript introduces a universal construction for adjoining infinite tensor products to categories, applied specifically to FinStoch (finite sets and stochastic matrices). This produces a category whose objects include finite sets together with the Cantor space 2^ℕ and whose morphisms are locally constant Markov kernels. The central claim is that any probability measure on the reals can thereby be reasoned about in this category, and that existing axiomatization results lift through the construction to yield an axiomatic presentation of continuous probability over countable powers of 2.

Significance. If the universal property is correctly established and the locally constant kernels suffice to represent the relevant measures, the work would supply a systematic bridge from discrete categorical probability to continuous settings while preserving algebraic structure. The explicit lifting of axiomatizations would be a concrete strength, enabling string-diagrammatic reasoning for infinite behaviors without ad-hoc extensions.

major comments (1)

[Main result / abstract statement] Main result (as stated in the abstract and presumably the theorem applying the construction to FinStoch): the claim that every probability measure on the reals can be reasoned about in the resulting category assumes that locally constant Markov kernels on 2^ℕ are sufficient after push-forward via binary expansion. This is load-bearing, yet measures whose conditional distributions require genuinely infinite coordinate dependence (certain singular continuous measures or Diracs at non-dyadic rationals) cannot be expressed by any finite-dependence kernel. The manuscript must either prove the restriction is harmless for the intended class of measures or qualify the claim accordingly.

minor comments (2)

[Abstract] The abstract would benefit from a forward reference to the precise theorem number and section containing the proof of the main result and the lifting of the axiomatization.
[Preliminaries] Notation for the infinite tensor product and the definition of 'locally constant' should be introduced with an explicit equation or diagram in the preliminary sections to aid readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive evaluation of the work's significance and for the constructive major comment on the scope of the main result. We agree that the original phrasing of the claim requires qualification to accurately reflect the expressive power of locally constant Markov kernels, and we have revised the manuscript accordingly.

read point-by-point responses

Referee: [Main result / abstract statement] Main result (as stated in the abstract and presumably the theorem applying the construction to FinStoch): the claim that every probability measure on the reals can be reasoned about in the resulting category assumes that locally constant Markov kernels on 2^ℕ are sufficient after push-forward via binary expansion. This is load-bearing, yet measures whose conditional distributions require genuinely infinite coordinate dependence (certain singular continuous measures or Diracs at non-dyadic rationals) cannot be expressed by any finite-dependence kernel. The manuscript must either prove the restriction is harmless for the intended class of measures or qualify the claim accordingly.

Authors: We thank the referee for identifying this important clarification. The morphisms in the constructed category are indeed locally constant Markov kernels, which by definition depend on only finitely many coordinates. This means that Dirac measures at points whose binary expansions have genuinely infinite non-periodic dependence, as well as certain singular continuous measures, cannot be represented exactly. We will revise the abstract and the statement of the main theorem (the application of the universal construction to FinStoch) to qualify the claim: the resulting category permits algebraic reasoning about those probability measures on the reals that admit locally constant representations after push-forward via binary expansion. We will also add a brief discussion section explaining the limitation, noting that the construction still covers important classes (e.g., all measures with finite or eventually periodic binary support, the Cantor distribution, and many absolutely continuous measures) while preserving the lifting of axiomatizations. This change does not affect the universal property or the technical results but ensures the main claim is precise. revision: yes

Circularity Check

0 steps flagged

Universal property construction is self-contained with no circular reductions

full rationale

The paper defines a universal construction that adjoins infinite tensor products to an existing category (FinStoch), yielding a new category whose morphisms are locally constant Markov kernels on objects including finite sets and the Cantor space 2^N. This is presented as a categorical extension via universal properties, with the representation of probability measures on the reals following directly from the construction's stated properties rather than any fitted parameters, self-definitional reductions, or load-bearing self-citations. No equations or steps in the provided derivation chain reduce a claimed result to its own inputs by construction; the axiomatization lifting is likewise an independent extension. The paper is self-contained against external categorical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review based on abstract only; the claim rests on standard category-theoretic axioms with no free parameters or new postulated entities visible.

axioms (1)

standard math Standard axioms of monoidal categories and universal properties
The infinite tensor product construction is defined via a universal property that presupposes the ambient category is monoidal and has the requisite limits or colimits.

pith-pipeline@v0.9.0 · 5705 in / 1476 out tokens · 75069 ms · 2026-05-21T21:13:28.939531+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Our main result applies this construction to FinStoch... obtaining a category of locally constant Markov kernels, where the objects are finite sets plus the Cantor space 2^N. Any probability measure on the reals can be reasoned about in this category.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We establish a universal construction that adjoins infinite tensor products... lift axiomatisation results through the infinite tensor product construction.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

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In: Proceedings of the ACM on Program- ming Languages, vol

Ackerman, N., Freer, C.E., Kaddar, Y., Karkwowski, J., Moss, S., Roy, D., Staton, S., Yang, H.: Probabilistic programming interfaces for random graphs: Markov categories, graphons, and nominal sets. In: Proceedings of the ACM on Program- ming Languages, vol. 8, pp. 1819–1849. ACM (2024).https://doi.org/10.1145/ 3632903

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Fritz, T., Klingler, A., McNeely, D., Shah Mohammed, A., Wang, Y.: Hidden markov models and the bayes filter in categorical probability. IEEE Transactions on Information Theory71(9), 7052–7075 (2025).https://doi.org/10.1109/TIT. 2025.3584695

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In: Bojańczyk, M., Simpson, A

Jacobs, B., Kissinger, A., Zanasi, F.: Causal inference by string diagram surgery. In: Bojańczyk, M., Simpson, A. (eds.) Foundations of Software Science and Com- putation Structures. pp. 313–329. Springer International Publishing, Cham (2019), https://doi.org/10.1007/978-3-030-17127-8_18 Approaching the Continuous from the Discrete 19

work page doi:10.1007/978-3-030-17127-8_18 2019

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Mathemat- ical Structures in Computer Science31(5), 553–574 (2021).https://doi.org/10

Jacobs, B., Kissinger, A., Zanasi, F.: Causal inference via string diagram sur- gery: A diagrammatic approach to interventions and counterfactuals. Mathemat- ical Structures in Computer Science31(5), 553–574 (2021).https://doi.org/10. 1017/S096012952100027X

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In: Founda- tionsofprobabilisticprogramming,pp.295–332.Cambridge:CambridgeUniversity Press (2021).https://doi.org/10.1017/9781108770750.010

Jacobs, B., Zanasi, F.: The logical essentials of Bayesian reasoning. In: Founda- tionsofprobabilisticprogramming,pp.295–332.Cambridge:CambridgeUniversity Press (2021).https://doi.org/10.1017/9781108770750.010

work page doi:10.1017/9781108770750.010 2021

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Piedeleu, R., Torres-Ruiz, M., Silva, A., Zanasi, F.: A complete axiomatisation of equivalence for discrete probabilistic programming. In: Programming Languages and Systems: 34th European Symposium on Programming, ESOP 2025, Held as Part of the International Joint Conferences on Theory and Practice of Software, ETAPS 2025, Hamilton, ON, Canada, May 3–8, 2...

work page doi:10.1007/978-3-031-91121-7_9 2025

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work page 2025

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In: Cîrstea, C., Knapp, A

Sarkis, R., Zanasi, F.: String diagrams for graded monoidal theories, with an ap- plication to imprecise probability. In: Cîrstea, C., Knapp, A. (eds.) 11th Con- ference on Algebra and Coalgebra in Computer Science, CALCO 2025, June 16- 18, 2025, University of Strathclyde, UK. LIPIcs, vol. 342, pp. 5:1–5:23. Schloss Dagstuhl - Leibniz-Zentrum für Informat...

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A survey of graphical languages for monoidal categories

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