Partializable Markov Category

• Partializable Markov categories are symmetric monoidal frameworks that generalize Markov categories and categorical models of partial maps by integrating quasi-totality and domain restriction.
• They offer a robust foundation for handling partial probabilistic computations, conditional operations, and domain-theoretic constructs while preserving key probabilistic structures.
• The partialization construction forms span categories that inherit monoidal and restriction properties, enabling practical applications in stochastic analysis and probabilistic programming.

A partializable Markov category is a monoidal categorical framework that generalizes both Markov categories and categorical models of partial maps—such as p-categories, dominical categories, and restriction categories—to encompass nondeterministic and noncartesian settings. These categories provide a robust theoretical basis for handling partial probabilistic computation, partial kernels, and conditionals with quasi-totality, while preserving key structures of categorical probability theory. The canonical construction, called partialization, freely generates a category of partial kernels from a suitable Markov category, ensuring compatibility with probabilistic and domain-theoretic constructs (Mohammed, 5 Sep 2025).

1. Structure of Partializable Markov Categories

A partializable Markov category is a symmetric monoidal category $(C,\otimes,I)$ equipped with a cocommutative comonoid structure on every object:

$$\delta_X: X \to X\otimes X, \qquad \epsilon_X: X \to I$$

with $\delta$ and $\epsilon$ monoidally natural in $X$. A Markov category is a CD (copy-discard) category in which all morphisms are total, enforcing $\epsilon_Y\circ f = \epsilon_X$ for any $f: X \to Y$.

To generalize to the partial setting, the concepts of quasi-totality and restriction arise:

• The domain of $f: X \to Y$ is

$$\mathsf{dom}\,f := X \xrightarrow{\delta_X} X\otimes X \xrightarrow{f\otimes\epsilon_X} Y\otimes I \cong Y$$

• A morphism is quasi-total if it absorbs its domain:

$$\delta_X\circ\mathsf{dom}\,f = (\mathsf{dom}\,f)\otimes(\mathsf{dom}\,f) \circ \delta_X$$

• Each hom-set admits a natural partial order given by $$\delta_X: X \to X\otimes X, \qquad \epsilon_X: X \to I$$0 iff $$\delta_X: X \to X\otimes X, \qquad \epsilon_X: X \to I$$1.

A partializable Markov category is a Markov category $$\delta_X: X \to X\otimes X, \qquad \epsilon_X: X \to I$$2 that is:

1. Positive (copyable maps compose as in deterministic case).
2. The class of deterministic monomorphisms is stable under pullback.
3. Closed under tensoring with identities.

These requirements ensure that partialization preserves key categorical and probabilistic structures, such as the existence of conditionals, representability, and positivity (Mohammed, 5 Sep 2025).

2. The Partialization Construction

Given a partializable Markov category $$\delta_X: X \to X\otimes X, \qquad \epsilon_X: X \to I$$3, the partialization $$\delta_X: X \to X\otimes X, \qquad \epsilon_X: X \to I$$4 is constructed as a span category:

• Objects: Same as $$\delta_X: X \to X\otimes X, \qquad \epsilon_X: X \to I$$5.
• Morphisms: Equivalence classes of spans $$\delta_X: X \to X\otimes X, \qquad \epsilon_X: X \to I$$6 where $$\delta_X: X \to X\otimes X, \qquad \epsilon_X: X \to I$$7 is a deterministic mono.
• Composition: By pullback in $$\delta_X: X \to X\otimes X, \qquad \epsilon_X: X \to I$$8.
• Monoidal structure: Inherited componentwise, i.e.,

$$\delta_X: X \to X\otimes X, \qquad \epsilon_X: X \to I$$9

$\delta$0 confers a monoidal restriction category structure:

• Restriction idempotents: $\delta$1.
• The extension-order is preserved: $\delta$2 iff $\delta$3.
• Total spans are those with $\delta$4 an isomorphism (hence from $\delta$5 itself).
• Every morphism is quasi-total, so $\delta$6 is a quasi–Markov category.
• Positivity transfers to $\delta$7.

$\delta$8 satisfies a universal property: It is the free way to adjoin partiality to $\delta$9 in the context of positive quasi–Markov categories, serving as a left adjoint to the forgetful functor back to Markov categories (Mohammed, 5 Sep 2025).

3. Preservation and Transfer of Categorical Probability Structures

Partialization preserves key probabilistic, categorical, and algebraic structures:

• Representability: If $\epsilon$0 is representable (admiting a right adjoint $\epsilon$1 to the inclusion of copyable maps), so is $\epsilon$2, and pushforwards and companions lift canonically.
• Conditionals: Conditional morphisms in $\epsilon$3 induce corresponding conditionals in $\epsilon$4. Thus, $\epsilon$5 meets the criteria for a partial Markov category as per (Lavore et al., 2023).
• Idempotents: Endospans are idempotent iff the corresponding map in $\epsilon$6 is idempotent, with splittings and balanced idempotents transferred as well.
• Kolmogorov products: Strict and lax Kolmogorov products (i.e., limits of product diagrams) in $\epsilon$7 are preserved in $\epsilon$8, and families of partial maps induce (strict and lax) Kolmogorov products in the span structure.

4. Alternative Kolmogorov Products and Structural Generalizations

In a general CD category, one can define lax cones over diagrams of finite marginals, supporting universal lax cones (lax infinite tensors) whose structure maps remain deterministic. In the quasi–Markov setting, this gives functoriality for $\epsilon$9-indexed tensors even without strict limits, which is crucial for infinite collections and continuous probabilistic phenomena. If $X$0 has a strict Kolmogorov product, lax and strict products coincide, and any family of partial maps forms a lax cone, ensuring a unique induced arrow by universality (Mohammed, 5 Sep 2025).

5. Partial Algebras and the Distribution Monad

Partial Markov categories naturally admit partial algebras for probability monads:

• If $X$1 is representable with monad $X$2, then $X$3 inherits this monad.
• A partial $X$4-algebra is a span $X$5 with $X$6 a deterministic mono, subject to span-level analogues of unit and multiplication axioms.
• This encompasses structures such as partial expectation integration, illustrated by the span collecting integrable measures and integrating them in $X$7, forming a partial algebra for the Giry monad.

This connection generalizes many traditional constructions in probabilistic programming and stochastic analysis into a categorical and partial framework (Mohammed, 5 Sep 2025).

6. Key Examples and Non-Examples

Several canonical examples and counterexamples elucidate the scope of this framework:

• Standard Borel spaces: Objects are standard Borel spaces, deterministic monos are inclusions of measurable subsets, and morphisms are partially defined Markov kernels—spans $X$8.
• Finite-support distributions: In the discrete case ("Dist" as finitely-supported $X$9-kernels), morphisms are partially defined finite-support distributions.
• Multivalued maps: For $\epsilon_Y\circ f = \epsilon_X$0 (Kleisli of the positive powerset monad), $\epsilon_Y\circ f = \epsilon_X$1 has as morphisms multivalued maps defined on a subset, with composition only defined where the domains overlap.

Key distinctions:

• The categories of sub-stochastic kernels (e.g., $\epsilon_Y\circ f = \epsilon_X$2) and relations ($\epsilon_Y\circ f = \epsilon_X$3) do not coincide with the partializations $\epsilon_Y\circ f = \epsilon_X$4 or $\epsilon_Y\circ f = \epsilon_X$5, as the latter require full domain compatibility rather than composition by the Chapman–Kolmogorov calculus or relational composition.

7. Restriction Structure and Order-Theoretic Enrichment

Partializable Markov categories and their partializations inherently carry a restriction structure:

• In a positive quasi–Markov category, $\epsilon_Y\circ f = \epsilon_X$6 satisfies the restriction axioms (R.1–R.4).
• In $\epsilon_Y\circ f = \epsilon_X$7, the domain operation coincides with the span-theoretic domain.
• Every partial Markov category is canonically enriched over the category of preorders and monotone maps (Lavore et al., 25 Jul 2025). The existence of codiagonal maps (comparators) is characterized order-theoretically as least conditionals, and comparators supply a (synthetic) Cauchy-Schwarz inequality supporting inequational reasoning.

Thus, the partializable Markov category framework unifies categorical probability and partiality, yielding a powerful, order-enriched categorical semantics for partial stochastic maps, conditionals, normalization, and Bayesian inference (Mohammed, 5 Sep 2025, Lavore et al., 25 Jul 2025, Lavore et al., 2023, Lavore et al., 24 Jan 2025).