CD Category of Partial Kernels

• CD Category of Partial Kernels is a symmetric monoidal framework that extends partial Markov kernels into the probabilistic and non-cartesian regime.
• It is constructed via the partialization of a Markov category, using spans with deterministic monomorphisms and pullback composition to capture partiality.
• This framework applies to measurable partial Markov kernels, finitely supported distributions, and multivalued maps, enhancing categorical probabilistic reasoning.

A CD category of partial kernels refers to the categorical framework arising from the partialization of a suitable Markov category, producing a symmetric monoidal category that generalizes various settings of partial morphisms into the probabilistic, non-deterministic, and non-cartesian regime. This construction, developed in the context of categorical probability theory, retains crucial structural properties, enabling the systematic study of partially defined kernels and probabilistic morphisms, as exemplified by measurable partial Markov kernels between standard Borel spaces (Mohammed, 5 Sep 2025).

1. Definitions and Structural Foundations

A CD-category is a symmetric monoidal category $(C, \otimes, I)$ in which every object $X$ is endowed uniformly with a commutative comonoid structure $\cop_X : X \to X \otimes X$ and $\discard_X : X \to I$, satisfying coassociativity, counitality, commutativity, and uniformity with respect to the monoidal product: $\cop_{X \otimes Y} = (\cop_X \otimes \cop_Y) \cong (X \otimes Y) \otimes (X \otimes Y), \qquad \discard_{X \otimes Y} = (\discard_X \otimes \discard_Y) \cong I$ A Markov category is a CD-category in which every morphism is total, i.e., $\discard_Y \circ f = \discard_X$ for all $f: X \to Y$.

Within a CD-category, morphisms are further classified as:

• Copyable: if the comultiplication is natural with respect to $f$,
• Deterministic: if $f$ is both total and copyable.

A partializable Markov category is a Markov category $C$ that satisfies positivity of copyable composites and stability of deterministic monomorphisms under pullbacks and tensor products. Each morphism in such a category also comes equipped with a poset-enrichment via a restriction operation given by a domain idempotent ($X$0), with the order $X$1 defined by $X$2 (Mohammed, 5 Sep 2025).

2. The Partialization Construction

Given a partializable Markov category $X$3, its partialization $X$4 is constructed as follows:

• Objects: identical to those of $X$5.
• Morphisms: equivalence classes of spans $X$6 where $X$7 is a deterministic monomorphism in $X$8. Two spans $X$9 and $\cop_X : X \to X \otimes X$0 are equivalent if there is an isomorphism $\cop_X : X \to X \otimes X$1 with $\cop_X : X \to X \otimes X$2 and $\cop_X : X \to X \otimes X$3.
• Composition: given by pullback on the "domain" legs; i.e., for $\cop_X : X \to X \otimes X$4 and $\cop_X : X \to X \otimes X$5, their composite uses the pullback of $\cop_X : X \to X \otimes X$6 over $\cop_X : X \to X \otimes X$7.

An identity morphism for $\cop_X : X \to X \otimes X$8 is the span $\cop_X : X \to X \otimes X$9 (Mohammed, 5 Sep 2025). The resulting category admits a strict symmetric monoidal structure, and the comonoid maps $\discard_X : X \to I$0, $\discard_X : X \to I$1 from $\discard_X : X \to I$2 lift to $\discard_X : X \to I$3 as spans.

3. Poset Enrichment and Restriction Structure

$\discard_X : X \to I$4 acquires a canonical poset-enrichment. Each hom-set is a poset under the order imposed by span-restriction: the restriction idempotent of a morphism $\discard_X : X \to I$5 is the span $\discard_X : X \to I$6. All morphisms are quasi-total: the restriction of a morphism to its domain is always defined. This makes $\discard_X : X \to I$7 a positive quasi-Markov category and a restriction category in the formal sense. The partialization construction preserves key properties such as positivity, representability, and the existence of conditionals:

• Positivity: copyability and positivity conditions for morphisms persist under partialization.
• Representability: if $\discard_X : X \to I$8 is representable (possesses a right-adjoint $\discard_X : X \to I$9 encoding probabilistic "sampling" structure), so is $\cop_{X \otimes Y} = (\cop_X \otimes \cop_Y) \cong (X \otimes Y) \otimes (X \otimes Y), \qquad \discard_{X \otimes Y} = (\discard_X \otimes \discard_Y) \cong I$0.
• Conditionals: if conditionals exist in $\cop_{X \otimes Y} = (\cop_X \otimes \cop_Y) \cong (X \otimes Y) \otimes (X \otimes Y), \qquad \discard_{X \otimes Y} = (\discard_X \otimes \discard_Y) \cong I$1, they transfer to $\cop_{X \otimes Y} = (\cop_X \otimes \cop_Y) \cong (X \otimes Y) \otimes (X \otimes Y), \qquad \discard_{X \otimes Y} = (\discard_X \otimes \discard_Y) \cong I$2 via explicit span-based constructions (Mohammed, 5 Sep 2025).

4. Universal Properties and Lax Kolmogorov Products

The partialization $\cop_{X \otimes Y} = (\cop_X \otimes \cop_Y) \cong (X \otimes Y) \otimes (X \otimes Y), \qquad \discard_{X \otimes Y} = (\discard_X \otimes \discard_Y) \cong I$3 features universal constructions analogous to those in Markov categories, but with necessary adaptations due to partiality. In particular:

• Lax Kolmogorov Product: for a family $\cop_{X \otimes Y} = (\cop_X \otimes \cop_Y) \cong (X \otimes Y) \otimes (X \otimes Y), \qquad \discard_{X \otimes Y} = (\discard_X \otimes \discard_Y) \cong I$4, the Kolmogorov product in $\cop_{X \otimes Y} = (\cop_X \otimes \cop_Y) \cong (X \otimes Y) \otimes (X \otimes Y), \qquad \discard_{X \otimes Y} = (\discard_X \otimes \discard_Y) \cong I$5 is defined as a lax limit of finite tensor diagrams. A lax cone consists of an object $\cop_{X \otimes Y} = (\cop_X \otimes \cop_Y) \cong (X \otimes Y) \otimes (X \otimes Y), \qquad \discard_{X \otimes Y} = (\discard_X \otimes \discard_Y) \cong I$6 and spans $\cop_{X \otimes Y} = (\cop_X \otimes \cop_Y) \cong (X \otimes Y) \otimes (X \otimes Y), \qquad \discard_{X \otimes Y} = (\discard_X \otimes \discard_Y) \cong I$7 for each finite $\cop_{X \otimes Y} = (\cop_X \otimes \cop_Y) \cong (X \otimes Y) \otimes (X \otimes Y), \qquad \discard_{X \otimes Y} = (\discard_X \otimes \discard_Y) \cong I$8, with commutativity up to the restriction order.
• Whenever $\cop_{X \otimes Y} = (\cop_X \otimes \cop_Y) \cong (X \otimes Y) \otimes (X \otimes Y), \qquad \discard_{X \otimes Y} = (\discard_X \otimes \discard_Y) \cong I$9 admits strict Kolmogorov products, so does $\discard_Y \circ f = \discard_X$0 (with span projections), but more generally there is always a lax Kolmogorov product (Mohammed, 5 Sep 2025).

5. Principal Examples and Operations

Key examples that instantiate the construction include:

• Standard Borel Spaces and Markov Kernels: $\discard_Y \circ f = \discard_X$1, $\discard_Y \circ f = \discard_X$2 has objects as standard Borel spaces, morphisms as partial Markov kernels given by measurable $\discard_Y \circ f = \discard_X$3 and Markov kernel $\discard_Y \circ f = \discard_X$4.
• Finitely Supported Distributions: $\discard_Y \circ f = \discard_X$5 as the Kleisli category of finite-valued distribution monad, with $\discard_Y \circ f = \discard_X$6 corresponding to partially defined finite-distribution kernels.
• Multivalued Maps: $\discard_Y \circ f = \discard_X$7 (or $\discard_Y \circ f = \discard_X$8), $\discard_Y \circ f = \discard_X$9 describe “secure” relations as spans of total relations.

All properties transferred by the general structure theorems—monoidal CD-structure, restriction poset, representability, conditionals, Kolmogorov products, and idempotents—hold in these cases (Mohammed, 5 Sep 2025).

A further extension applies to partial algebras for probability monads. If $f: X \to Y$0 is representable (with distribution monad $f: X \to Y$1), then $f: X \to Y$2 is also representable, and the monad on the copyable subcategory extends to partial algebras (i.e., spans $f: X \to Y$3 satisfying the unit and associativity up to the domain). On $f: X \to Y$4 this recovers the situation where integration of real-valued functions is defined only for those probability measures with finite expectation.

6. Comparisons and Generalizations

This framework generalizes classical categories of partial morphisms, such as $f: X \to Y$5-categories, dominical categories, and restriction categories, to a broader setting accommodating nondeterminism and general noncartesian monoidal structures. For comparison, the Cartier–Dieudonné (CD) category of $f: X \to Y$6-kernels in the context of Dieudonné theory and $f: X \to Y$7-divisible groups is an instance of a CD-category in the sense of $f: X \to Y$8-vector spaces equipped with additional semilinear operators, but it lacks the probabilistic and partial-structural flavors that arise through the partial kernel construction (Ziegler, 2015).

A notable extension is the poset-enrichment of each hom-set in $f: X \to Y$9, compatible with the restriction structure, allowing fine-grained comparison of partial morphisms. The structure theorems prove that every morphism is quasi-total, every essential property transfers, and new forms of universal constructions (e.g., lax Kolmogorov products) provide expressive tools for partial probabilistic reasoning.

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References

• "Partializations of Markov categories" (Mohammed, 5 Sep 2025)
• "$f$0-kernels occurring in an isogeny class of $f$1-divisible groups" (Ziegler, 2015)