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

Technical Summary: Approaching the Continuous from the Discrete via Infinite Tensor Products

Problem Statement
Recent advancements in categorical probability have successfully axiomatized discrete probabilistic processes (e.g., finite sets and stochastic matrices) and Gaussian processes using string diagrammatic calculi. However, a significant gap remains in the axiomatization of general continuous probability, specifically categories like $\mathbf{BorelStoch}$ (standard Borel spaces and Markov kernels). The primary difficulty lies in encoding infinite behavior algebraically within the existing framework of string diagrams, which typically rely on finite generators and equations. While the Kolmogorov extension theorem suggests that continuous measures can be viewed as limits of discrete ones, a rigorous categorical construction that bridges discrete settings to continuous ones via infinite tensor products has been lacking.

Methodology
The authors propose a universal construction that adjoins infinite tensor products to a given semicartesian category $\mathcal{C}$. The methodology proceeds through several stages:

1. Universal Construction: The paper defines a category $\mathcal{C}^{\otimes\infty}$ obtained by adjoining infinite tensor products to a semicartesian category $\mathcal{C}$. This is achieved via a universal property: $\mathcal{C}^{\otimes\infty}$ is the initial semicartesian category with infinite tensor products receiving a symmetric monoidal functor from $\mathcal{C}$.
2. Concrete Representation (Finite Approximation Families): To make morphisms in $\mathcal{C}^{\otimes\infty}$ tractable, the authors restrict attention to categories with cancellative deletions (where $f \otimes \text{del} = g \otimes \text{del} \implies f=g$). In this setting, morphisms between infinite tensor products are characterized not as abstract limits, but as finite approximation families. A morphism is an equivalence class of families of finite morphisms $(f_{F,G})$ that satisfy naturality, covering, and hereditariness conditions.
3. Diagrammatic Syntax: The authors introduce "plate notation" to represent these finite approximation families as string diagrams. This allows for the lifting of axiomatic presentations: if $\mathcal{C}$ is presented by a symmetric monoidal theory $(\Sigma, E)$, then $\mathcal{C}^{\otimes\infty}$ is presented by a theory $\text{Free}_\infty(\Sigma, E)$ that includes the original generators and equations, plus rules for handling infinite tensor products (plates).
4. Application to $\mathbf{FinStoch}$: The construction is applied to $\mathbf{FinStoch}$ (finite sets and stochastic matrices) and its subcategory $\mathbf{BinStoch}$ (powers of $\{0,1\}$).

Key Results

• Characterization of $\mathbf{FinStoch}^{\otimes\infty}$: The authors prove that the category $\mathbf{FinStoch}^{\otimes\infty}$ is equivalent to a category of Stone spaces equipped with locally constant Markov kernels ($\mathbf{StoneStoch}_{lc}$). In this category, objects are infinite products of finite sets (which, by Brouwer's theorem, are either finite sets or the Cantor space $2^\mathbb{N}$), and morphisms are kernels $f: X \to Y$ where the probability of a clopen set depends only on finitely many coordinates of the input.
• Axiomatic Presentation of Continuous Probability: By restricting to $\mathbf{BinStoch}$, the authors derive an axiomatic presentation for $\mathbf{CantorStoch}_{lc}$ (the subcategory of $\mathbf{StoneStoch}_{lc}$ involving powers of 2 and the Cantor space). They show that $\mathbf{CantorStoch}_{lc}$ is isomorphic to $\text{Free}_\infty(\Sigma, E)$, where $(\Sigma, E)$ is the theory of $\mathbf{CausCirc}$ (the axiomatization of $\mathbf{BinStoch}$).
• Expressivity: The paper demonstrates that $\mathbf{CantorStoch}_{lc}$ is expressive enough to encompass all probability measures on the real line $\mathbb{R}$. This is because $\mathbb{R}$ is isomorphic to an infinite tensor product of 2 in $\mathbf{BorelStoch}$, and the construction establishes a bijective correspondence between measures on $\mathbb{R}$ and measures on the Cantor space $2^\mathbb{N}$.
• Limitations: The authors clarify that $\mathbf{FinStoch}^{\otimes\infty}$ does not recover the entirety of $\mathbf{BorelStoch}$. While it captures all probability measures, it cannot fully capture the behavior of arbitrary measurable functions (which may not be locally constant), thus forming a specific, expressive subcategory rather than the full continuous category.

Significance and Claims
The paper claims to provide the first tools for axiomatizing Markov categories with infinite tensor products using string diagrams. Its primary contribution is establishing a rigorous bridge between discrete and continuous probabilistic reasoning. By showing that continuous probability can be approached as limits of discrete settings via infinite tensor products, the work enables the use of diagrammatic calculi (specifically plate notation) to reason about continuous systems.

The authors emphasize that this approach allows for the derivation of semantic equalities in continuous probability directly from discrete axioms. They note that while the construction does not capture the full generality of $\mathbf{BorelStoch}$ (specifically regarding non-locally constant kernels), it provides a rich framework for studying probability measures on $\mathbb{R}$ and other continuous spaces through the lens of the Cantor space. The work is presented as a foundational step toward connecting semantic approaches with string diagrammatic methods for continuous probability, with future work suggested for extending these results to general Markov kernels and other probabilistic models like Gaussian mixtures.