Partializations of Markov categories

Areeb Shah Mohammed

Partializations of Markov categories

T0 review · 0 major / 6 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read For any 'partializable' Markov category, the category of spans with deterministic monomorphisms as inclusions is a CD category of partial stochastic maps in which every map is quasi-total, carries the copy–discard structure, and inherits re

desk verdict Solid and useful extension of partial-map categories to non-cartesian Markov categories; the BorelStoch example checks out, but expect compressed proofs and some reliance on companion preprints. read the letter →

arxiv 2509.05094 v1 pith:54HIVFTJ submitted 2025-09-05 math.CT cs.LOmath.PR

Areeb Shah Mohammed This is my paper

classification math.CTcs.LOmath.PR MSC 18M0518A3260A0560B05

keywords MarkovcategoriespartialmapsrestrictionCDquasi-totalmorphismscategoricalprobabilityKolmogorovproductsalgebras

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Markov categories model stochastic maps, but many natural operations—averaging, taking expectations, empirical distributions—are only defined on part of their input. This paper shows that a large class of Markov categories can be systematically extended to categories of partial stochastic maps: a "partializable" Markov category (positive, with deterministic monomorphisms stable under pullback and tensoring) yields a CD category whose morphisms are spans with a deterministic subobject as domain, composed by pullback. The construction succeeds where earlier frameworks for partial maps stopped: it carries a genuinely non-cartesian monoidal product, so genuine randomness is compatible with partiality. Every morphism is quasi-total—its domain of definition is deterministic—and the category is positive quasi-Markov, hence a restriction category whose partial order is exactly "is a restriction of". The paper proves that representability, conditionals, idempotent splitting, and Kolmogorov products all transfer, supplies partial algebras (the mean on the nonnegative reals for the Giry monad), and introduces lax Kolmogorov products to keep infinite tensor products functorial on partial maps.

What carries the argument

The central object is the partialization Partial(C), built as a category of spans whose wrong-way legs are deterministic monomorphisms of C. The argument is carried by two interacting mechanisms: the domain endomorphism dom(f) defined from the composite of deletion with f (and its pre-composition with the copy map), and the quasi-totality equation f∘dom(f) = f. Because C is positive, quasi-totality upgrades to a full restriction structure, identifying the span (D, i, i) with the CD-theoretic domain and making the restriction partial order coincide with the span order. Pullback-stability of deterministic monomorphisms is what makes composition of spans well defined; in the flagship example th

What would settle it

In BorelStoch, compose the fair-coin state ∗ → {H,T} with the partial kernel f defined only on {H} that returns a. In Partial(BorelStoch) the composite must have empty domain—the coin lands in the complement with probability 1/2, and the composite is defined only where the first map lands surely in the second's domain—whereas the competing sub-stochastic composition returns probability 1/2. Computing this one composite distinguishes the two and tests the quasi-totality claim, since a composite that assigned positive probability to a would violate the defining equation of a partialization.

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Extended reading notes

Core claim

On the paper's own terms: if C is a partializable Markov category—meaning C is positive, pullbacks of deterministic monomorphisms exist and are deterministic, and deterministic monomorphisms close under tensoring—then the span construction Partial(C) is a CD category of partial kernels that extends C fully faithfully. Its morphisms are equivalence classes of spans X ← D → Y with D ↣ X a deterministic monomorphism; composition is by pullback and tensoring is componentwise. The decisive technical fact is that every morphism of Partial(C) is quasi-total (it absorbs its own domain), and that the CD-theoretic domain dom(u) coincides with the span-theoretic domain (D, i, i); combined with positivi

Load-bearing premise

The whole construction depends on deterministic monomorphisms being closed under pullback and tensoring: if a pullback of a deterministic subobject along an arbitrary stochastic map does not exist or is not deterministic, the span composition is undefined and Partial(C) is not a category; in the flagship example this is guaranteed by the descriptive set theory fact that every measurable injection between standard Borel spaces is a Borel isomorphism onto its image.

Editorial extensions

If this is right

In BorelStoch, Dist, SetMulti, and their finite variants, the construction yields concrete categories of partial Markov kernels, partial discrete kernels, and partial multivalued maps; composition is 'secure': a composite is defined only where the first map lands surely inside the second's domain, not merely possibly (Warning 3.13).
In a representable partializable Markov category, Partial(C) is representable and the distribution monad extends to the copyable subcategory; over the nonnegative reals in Partial(BorelStoch), the mean defines a partial algebra whose domain is exactly the distributions with finite expectation (Propositions 4.2, 4.7, 4.8), while the analogous construction on all of the reals fails (Warning 4.9).
Whenever C has conditionals, so does Partial(C), with the conditional defined on the largest possible domain; Partial(C) is therefore a partial Markov category in the sense of the cited literature whenever C has conditionals (Proposition 4.10, Corollary 4.11).
Idempotents of Partial(C) are idempotents on their domains; they split, and are static/strong/balanced, exactly when their domain idempotents are. Hence in Partial(BorelStoch) every idempotent splits and is balanced.
A K-indexed family of partial maps induces an infinite tensor computed componentwise; the strict Kolmogorov product in C becomes both a strict and a lax Kolmogorov product in Partial(C), restoring functoriality of the infinite tensor that the strict universal property alone loses.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The "sure landing" composition suggests a spectrum of partial-composition doctrines between 'defined with probability 1' (Partial(C)) and 'defined with positive probability' (sub-stochastic categories); one could interpolate by replacing the pullback set {x : f(T|x)=1} with a threshold {x : f(T|x) > 1−ε}, producing a family of categories parametrized by ε and revealing which probabilistic properti
Since positivity is only shown to be sufficient for the restriction structure, a natural next step—left open by the paper—is to characterize quasi-Markov categories whose domain preorder is a poset enrichment; such a characterization would say exactly which 'positive-like' axioms are needed for partiality without randomness.
The partial-algebra framework is a template for other partially defined probabilistic operations: applying the same construction to higher moments or to other monads would yield partial algebras whose domains are finiteness or integrability conditions, bringing categorical formulations of laws of large numbers closer to the analytic statements.
Because Partial(C) is constructed precisely so that quasi-total morphisms close under composition, it is a candidate ambient category for formalizing partially defined stochastic constructions such as empirical sampling, where the infinite-sample map is defined only on convergent sequences.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper is a solid extension of partial-map theory to non-cartesian Markov categories. The main construction—Partial(C) from a partializable Markov category—is new and useful, and the transfer theorems (representability, conditionals, idempotent splitting, Kolmogorov products) appear to hold. I checked the BorelStoch example carefully, since the stress-test note flagged it as load-bearing. The concern does not land: S={x:f(T|x)=1} is Borel by the measurability of f(T|.), and the argument that f h factoring through T forces h to land in S uses the standard fact that a nonnegative integral vanishes iff the integrand vanishes a.e., which is valid in standard Borel spaces. The restricted kernel f|_S is well-defined and Markov. So the flagship example is fine.

What the paper does well: it gives a uniform span-based construction that avoids the cartesian restriction of earlier p-categories and restriction categories. The result that positive quasi-Markov categories are restriction categories (Prop 3.1) is a nice bridge. The introduction of lax Kolmogorov products is a genuine attempt to retain functoriality of infinite tensor in the presence of partial maps, and the proofs that strict and lax products agree in Partial(C) are convincing. The examples are well chosen and worked out in detail.

The soft spots: several proofs are compressed to string-diagram assertions, especially the naturality checks in Prop 3.17 and the restriction-category axioms in Prop 3.1. A referee will want more detail there. The paper also leans on companion preprints [FGL+25] for some background results (e.g., Prop 2.12), which is acceptable but means the reader must trust work in progress. The lax Kolmogorov product definition is somewhat unusual, with the 'greatest morphism' universal property; it works, but it would help to see a worked example that is not just the induced strict product.

Overall, this is a competent and useful contribution to categorical probability. It deserves a serious referee and will probably be accepted with revisions. I would bring it to a reading group focused on either partial map categories or categorical probability, and I would cite it in future work on partial stochastic maps.

Recommendation: send it out for peer review; the referee should ask for clarification of the compressed proofs rather than a reworking of the main ideas.

Referee Report

0 major / 6 minor

Summary. The paper introduces the notion of a partializable Markov category and constructs, for every such category C, a category Partial(C) of spans whose left legs are deterministic monomorphisms. The main theorem is that Partial(C) is a positive quasi-Markov CD category whose hom-sets carry a restriction structure and a poset enrichment, with total maps recovering C. The paper further proves transfer theorems: representability, conditionals, splitting/static/strong/balanced idempotents, and strict Kolmogorov products all pass from C to Partial(C). A new notion of lax Kolmogorov product is introduced to make infinite tensor products functorial on partial maps, and partial algebras for the distribution monad are defined. The flagship examples are BorelStoch, Dist, SetMulti, and Kleisli categories for semiring-valued distributions; the paper works out the expectation partial algebra on R≥0 in detail.

Significance. If the results are correct, this is a substantial and useful contribution to categorical probability: it supplies a systematic, non-cartesian framework for partially defined stochastic maps, connecting the older theory of p-categories/restriction categories to modern Markov categories. The construction is parameter-free and the main preservation theorems are broadly applicable. The paper also gives a concrete, non-trivial partial algebra—expectation on R≥0—and introduces a genuinely new notion of lax Kolmogorov product. The strongest strengths are the detailed checks of the semiring-valued and multivalued examples and the theorem that positivity transfers to the partialization. The proofs are often string-diagrammatic and terse, but the central claims are credible.

minor comments (6)

[§3.2, Example 3.6] The pullback verification for BorelStoch is the load-bearing example, and the underlying argument is correct, but as written it does not explicitly state that the restricted kernel f|_S defines a Markov kernel S→T nor that the square commutes; it only draws the necessary condition and proves the universal property. Please spell out these two checks. Also briefly justify that deterministic monomorphisms are exactly measurable subset inclusions, citing Lusin–Souslin as done.
[§2.2.3, Definition 2.3] The displayed restriction axioms, especially R.3 and R.4, have notational inconsistencies (e.g. R.4 appears as "¯g f=f g f", which does not type correctly). Please restate the axioms in the standard Cockett–Lack form, or explain the notation for domains so that Proposition 3.1's verification is readable.
[§3.3, Proposition 3.24] The transfer of positivity from C to Partial(C) is proved by a large string-diagram calculation. The final reduction to “applying positivity of C to the deterministic gh” is terse; a short prose explanation of which equation is being used and how the pullback legs are eliminated would materially improve the proof.
[§5, Definition 5.2] The definition of lax Kolmogorov product is phrased in terms of a “greatest morphism g” satisfying π_F g ≤ f_F. Please clarify the orientation of the poset and why this universal property is the correct lax-limit notion, especially because in a poset-enriched setting the direction of the 2-cells is easy to reverse by accident.
[§4.1.1, Proposition 4.8] The notation BorelMeas is used in the proof ("a span in BorelMeas ∼= BorelStochdet") without being introduced earlier in the paper. Please define it as the category of standard Borel spaces and measurable maps, or replace it by BorelStochdet throughout.
[General notation] The text mixes "Cdet", "C_det", "Partial(C) cop", and "Partial(C)_cop". A consistent notation for the deterministic/copyable subcategory would help, since the paper repeatedly navigates between these subcategories.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction; minor overlapping-author background citations do not carry the central derivation.

full rationale

The central chain is self-contained: Partial(C) is defined directly as a span category over deterministic monomorphisms (Definition 3.5), and the main results — quasi-Markov structure, positivity, representability, conditionals, idempotent splitting, and Kolmogorov/lax products — are proved from the three partializability axioms (Definition 3.4) plus the CD/span machinery. No parameter is fitted and no derived quantity is renamed as a prediction. The flagship BorelStoch example (Example 3.6) is conditional on external descriptive set theory (Kechris Corollary 15.2) and standard measure-theoretic arguments; that makes it a load-bearing example, but not a circular one. The only overlapping-author citations are background definitions and Proposition 2.12 from the companion preprint [FGL+25]. These are not load-bearing: Proposition 3.1 independently proves that positive quasi-Markov categories are restriction categories, and Corollary 3.20 independently establishes the equivalence of the restriction order with the span order. No uniqueness theorem is imported from the authors, no ansatz is smuggled via citation, and no equation reduces to its own input by construction. The residual risk is correctness of cited background facts and stability of the BorelStoch pullback verification, not circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces new categorical definitions, including partializable Markov categories, the partialization Partial(C), lax Kolmogorov products, and partial algebras, but these are constructions proven from assumptions, not empirical entities with independent falsifiable consequences. There are no free parameters fitted to data. The main axioms are the defining conditions of the framework and the external Kechris theorem for the flagship example.

assumptions (5)

standard math CD category and Markov category axioms (copy, delete, naturality, totality) as in Definition 2.4.
Background framework for all statements; cited to Cho-Jacobs and Fritz.
domain assumption A partializable Markov category is positive, pullbacks of deterministic monomorphisms exist and are deterministic, and deterministic monomorphisms are closed under tensor (Definition 3.4).
The defining hypothesis of the main construction; if these fail, Partial(C) is not a span category in the intended sense.
domain assumption Every measurable injection between standard Borel spaces is a Borel isomorphism onto its image (Kechris 15.2).
Used in Example 3.6 to identify deterministic subobjects in BorelStoch with measurable subsets and to construct pullbacks.
domain assumption Representability data, distribution objects and sampling maps, exists when transfer of representability is claimed (Definition 2.13).
Needed for Proposition 4.2 and partial algebras; only assumed in the relevant sections.
domain assumption K-sized Kolmogorov products exist when transfer of infinite tensor products is claimed (Definition 2.16).
Needed for Section 5; the paper proves the induced products in Partial(C) from this.

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Cite this review

Pith. "Pith review of Partializations of Markov categories." pith.science (2026). https://pith.science/paper/54HIVFTJ

@misc{pith2026250905094,
  author       = {Pith},
  title        = {Pith review of: Partializations of Markov categories},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/54HIVFTJ}},
  note         = {Machine review of arXiv:2509.05094}
}

read the original abstract

The present work develops a construction of a CD category of partial kernels from a particular type of Markov category called a partializable Markov category. These are a generalization of earlier models of categories of partial morphisms such as p-categories, dominical categories, restriction categories, etc. to a non-deterministic/non-cartesian setting. Here all morphisms are quasi-total, with a natural poset enrichment corresponding to one morphism being a restriction of the other. Furthermore, various properties important to categorical probability are preserved, such as positivity, representability, conditionals, Kolmogorov products, and splittings of idempotents. We additionally discuss an alternative notion of Kolmogorov product suitable for partial maps, as well as partial algebras for probability monads. The primary example is that of the partialization of the category of standard Borel spaces and Markov kernels. Other examples include variants where the distributions are finitely supported, or where one considers multivalued maps instead.

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

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