BorelStoch: Markov Category Framework

• BorelStoch is a categorical framework that organizes standard Borel spaces and probability kernels via the Giry monad, providing a unified measure-theoretic probability structure.
• It employs symmetric monoidal and comonoidal structures to model tensor products, copying, and conditional independence, making it pivotal for Bayesian inference and ergodic theory.
• The framework supports practical applications in stochastic processes, hidden Markov models, and statistical experiments, while also highlighting limitations for non-atomic measures.

The Markov category BorelStoch constitutes the categorical framework in which standard Borel spaces and probability kernels (Markov kernels) are organized to capture the essential structure of measure-theoretic probability in a symmetric monoidal and comonoidal setting. It provides a foundational basis for categorical probability, Bayesian inference, stochastic processes, and the structural study of conditional independence, ergodic theory, and statistical experiments.

1. Objects, Morphisms, and Composition

BorelStoch is defined as the Kleisli category of the Giry monad $G$ on the category of standard Borel spaces, denoted $\mathbf{SB}$. Its objects are standard Borel spaces $(X,\Sigma_X)$, i.e., measurable spaces isomorphic to complete separable metric spaces with their Borel σ-algebra (Culbertson et al., 2013, Fritz et al., 2020, Rischel, 17 Dec 2025).

Morphisms in BorelStoch, called Markov kernels or regular conditional probabilities, are functions

$k\colon X \times \Sigma_Y \rightarrow [0,1]$

such that:

• For each $x\in X$, $k(x,-)$ is a probability measure on $(Y,\Sigma_Y)$.
• For each measurable set $A \subseteq Y$, $x\mapsto k(x,A)$ is measurable.

A deterministic measurable function $f: X \to Y$ embeds as the Dirac kernel: $$\delta_f(x,A) = \begin{cases} 1, & f(x)\in A \ 0, & f(x)\notin A \end{cases}$$ Composition of kernels $k:X\to Y$ and $h:Y\to Z$ is given by Chapman–Kolmogorov (i.e., Kleisli) composition: $$(h \circ k)(x,C) = \int_{Y} h(y,C) \, k(x,dy),\quad C\in \Sigma_Z$$ This structure ensures that BorelStoch models conditional probability and pushforward under noisy channels (Culbertson et al., 2013, Fritz et al., 2020).

2. Symmetric Monoidal Structure and Comonoid Operations

BorelStoch is a symmetric monoidal category:

• The monoidal product is the standard measurable-space product: $$(X,\Sigma_X) \otimes (Y,\Sigma_Y) = (X\times Y,\Sigma_X\otimes\Sigma_Y)$$
• The unit is the one-point measurable space.

On morphisms, the tensor product is “independent product of kernels”: $(f\otimes g)((x,y),A\times B) = f(x,A) g(y,B)$ BorelStoch is equipped with a commutative comonoid structure:

• Copying: $\mathsf{copy}_X:X\to X\otimes X$, realized as the diagonal Dirac kernel.
• Deleting: $\mathsf{del}_X:X\to I$, the unique kernel to the terminal object.

These satisfy all coherence laws for a semicartesian or “Markov” category, such as coassociativity, counitality, and cocommutativity, and their naturality or compatibility with the tensor structure (Fritz et al., 2020, Rischel, 17 Dec 2025, Moss et al., 2022).

3. Representability, The Giry Monad, and Disintegration

BorelStoch is the paradigmatic representable Markov category. The Giry monad $G$ sends each standard Borel space to the space of probability measures on it. The Kleisli category $\text{Kl}(G)$ is precisely BorelStoch, and the “sampling kernel” provides the adjunction between deterministic and general kernels (Fritz et al., 2020): $\mathrm{samp}_X: G(X) \to X$ This structure allows the existence of Bayesian inverses (disintegrations) and conditional probability as an internal operation. For every morphism $f:A \to X\otimes Y$, there exists a regular conditional $f_{|X}:A\otimes X\to Y$ satisfying the joint law factorization, which is central to Bayesian inference and stochastic process composition (Culbertson et al., 2013, Fritz et al., 26 Jan 2024, Stein, 4 Mar 2025).

4. Categorical Probability Theory and Absolute Notions

Within BorelStoch, categorical analogues of classical probabilistic concepts are defined:

• Absolute continuity ($\ll$): A kernel $p:A\to X$ is absolutely continuous with respect to $q:B\to X$ precisely if for every Borel set $S$, $q(S|b)=0 \Rightarrow p(S|a)=0$, for all $a\in A, b\in B$ (Fritz et al., 2023).
• Supports: The categorical support of a kernel coincides with the set of atoms in measure theory; a kernel has a support if and only if it is atomic.
• Idempotents and splitting: Every idempotent kernel in BorelStoch splits through another standard Borel space. Concretely, for $e:X\to X$, there exist $r:X\to Y$, $s:Y\to X$ such that $e=s\circ r$ and $r\circ s=\mathrm{id}_Y$. This facilitates ergodic decompositions and measure-theoretic quotients (Fritz et al., 2023).

5. Universal and Structural Properties

BorelStoch is characterized by a universal property: it is initial among Markov categories that are countably extensive, Boolean, admit countable Kolmogorov products, and possess a unique “coinflip” morphism—i.e., the canonical unbiased binary choice (Rischel, 17 Dec 2025). All countable diagrams required for probability theory (products, coproducts, Kolmogorov extensions) are present, and deterministic inclusions correspond to coproduct injections.

These universal properties establish BorelStoch as the canonical model of classical measure-theoretic probability, encompassing discrete, countable, and continuous settings in a unified framework.

6. Conditionals, Hidden Markov Models, and Filtering

BorelStoch admits conditional probabilities and supports categorical generalizations of Markov chains, hidden Markov models (HMMs), Bayesian filtering, and smoothing (Fritz et al., 26 Jan 2024). For hidden Markov models structured in BorelStoch, all essential operations (predict, update, forward-backward smoothing) and their diagrammatic representations are encapsulated categorically:

• Bayesian filtering is realized via sequential composition and conditionalization, with state and observation kernels corresponding to morphisms.
• Conditional independence is structurally characterized, with string diagrams directly encoding Markov properties and conditional-independence constraints.

The filter process (sequence of posterior measures) itself forms a Markov chain under categorical evolution.

7. Limitations, Atomicity, and Nonexistence of Traces

Not all categorical structures extend seamlessly to BorelStoch. It is not an atomic Markov category: for non-atomic measures (e.g., Lebesgue measure), certain contraction identities fail, and BorelStoch cannot embed into a traced monoidal category (Stein et al., 2 Apr 2024). This precludes the existence of a categorical trace in the general (non-atomic) case. However, in the subclass of atomic Markov categories (kernels supported only on atoms), contraction identities and a partial trace for non-signalling morphisms can be restored, enabling intrinsic comb calculus and causal trace structures.

A key example is the failure of contraction identity for the diagonal kernel and Lebesgue measure, demonstrating categorical limitations in modeling quantum-like structures (Stein et al., 2 Apr 2024).

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References

• (Culbertson et al., 2013) Bayesian machine learning via category theory
• (Fritz et al., 2020) Representable Markov Categories and Comparison of Statistical Experiments in Categorical Probability
• (Moss et al., 2022) A category-theoretic proof of the ergodic decomposition theorem
• (Fritz et al., 2023) Absolute continuity, supports and idempotent splitting in categorical probability
• (Fritz et al., 26 Jan 2024) Hidden Markov Models and the Bayes Filter in Categorical Probability
• (Stein et al., 2 Apr 2024) Combs, Causality and Contractions in Atomic Markov Categories
• (Stein, 4 Mar 2025) Random Variables, Conditional Independence and Categories of Abstract Sample Spaces
• (Rischel, 17 Dec 2025) The Universal Property of Measure-Theoretic Probability