{"id":"c6ecedb9-4460-4ca9-8c07-4293edfe50ad","arxiv_id":"2509.05094","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Given a partializable Markov category C, the span-based category Partial(C) is a positive quasi-Markov CD category that inherits representability, conditionals, Kolmogorov products, and idempotent splittings.","lead":"This paper constructs categories of partially defined stochastic maps, called partializations, from Markov categories that satisfy a few extra conditions. The main example gives partially defined Markov kernels between standard Borel spaces, and the construction preserves structural properties used in categorical probability.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"BorelStoch pullback verification is the load-bearing step; the flagship example depends on Lusin–Souslin and the measurability of S={x:f(T|x)=1}.","rationale":"The reader identified the BorelStoch pullback stability as the weakest assumption, and I agree that it is the most load-bearing point in terms of the paper's primary example and the plausibility of the general definition. However, the verification in Example 3.6 is mathematically correct: the Lusin–Souslin theorem and the measure-theoretic argument for the universal property are standard and apply to standard Borel spaces. The abstract conditional theorem is proven through general categorical arguments that do not hide additional assumptions beyond Definition 3.4. I found no internal inconsistency or gap in the proofs of the main transfer theorems (representability, conditionals, idempotent splitting, Kolmogorov products). Thus the concern does not warrant changing the accept verdict; it is a note on the example's reliance on a nontrivial external theorem rather than a flaw in the paper's argument.","tokens_in":44320,"tokens_out":27263,"duration_ms":295638,"concrete_test":"Formalize Example 3.6 in a proof assistant (e.g., Lean or Coq) by proving that for any Markov kernel f:X→Y and Borel T⊆Y, the set S={x:f(T|x)=1} is Borel, that f|_S:S→T is a Markov kernel, and that the resulting square is a pullback in BorelStoch; specifically, verify the universal property for an arbitrary kernel h:W→X with f∘h factoring through T.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central theorem is conditional on the definition of partializable Markov category (Definition 3.4), whose key nontrivial axiom is that pullbacks of deterministic monomorphisms exist and are deterministic. The flagship example BorelStoch (Example 3.6) is where this axiom is least automatable: it relies on the descriptive set theory fact that every measurable injection between standard Borel spaces is a Borel isomorphism onto its image, and then constructs the pullback as S={x:f(T|x)=1}. If S were not Borel, or if the restricted kernel f|_S were not a Markov kernel, or if the universal property failed, Partial(BorelStoch) would not be a category and the paper's main advertised application would collapse. The proof of the universal property also uses the measure-theoretic fact that a nonnegative integral vanishes iff its integrand vanishes almost everywhere, a property specific to standard Borel spaces. While the abstract theorem would survive for the other semiring examples, the paper's motivation and intuitive content are largely carried by BorelStoch, making this verification load-bearing for the paper as presented.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":44600,"tokens_out":30883,"duration_ms":329906,"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.","major_comments":[],"minor_comments":[{"comment":"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.","section":"§3.2, Example 3.6"},{"comment":"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.","section":"§2.2.3, Definition 2.3"},{"comment":"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.","section":"§3.3, Proposition 3.24"},{"comment":"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.","section":"§5, Definition 5.2"},{"comment":"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.","section":"§4.1.1, Proposition 4.8"},{"comment":"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.","section":"General notation"}],"recommendation":"minor_revision","confidential_remarks":"The paper is a companion to [FGL+25] and relies on it for Definition 2.11 and Proposition 2.12. It would be advisable for the editor to confirm that the companion is available and in a comparable state. The BorelStoch pullback verification flagged by the skeptic is, in my reading, correct; it is only presented too tersely. The manuscript is long but well organized; no concerns about novelty or scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhat 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.\n\nThe 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.\n\nOverall, 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.\n\nRecommendation: send it out for peer review; the referee should ask for clarification of the compressed proofs rather than a reworking of the main ideas.","headline":"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.","tokens_in":45014,"tokens_out":3824,"would_cite":true,"duration_ms":36783,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["18M05","18A32","60A05","60B05"],"pacs":[],"model":"deepseek-v4-flash","headline":"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","keywords":["Markov categories","partial maps","restriction categories","CD categories","quasi-total morphisms","categorical probability","Kolmogorov products","partial algebras"],"falsifier":"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.","tokens_in":44256,"feed_emoji":"🎲","tokens_out":11415,"duration_ms":102321,"temperature":0.7,"pith_summary":"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.","feed_headline":"Partial stochastic maps form a category that keeps its randomness","feed_subtitle":"The span recipe that builds partial functions from sets now works for Markov kernels.","key_machinery":"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","core_discovery":"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","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the stable-system-of-monomorphisms span construction, the associated restriction structure on span categories, and the restriction-category axioms used throughout.","marker":"[CL02]"},{"why":"Furnishes Markov categories, the flagship BorelStoch example, determinism, and positivity of the base categories.","marker":"[Fri20]"},{"why":"Defines the infinite tensor and strict Kolmogorov products that Section 5 extends to lax Kolmogorov products.","marker":"[FR20]"},{"why":"Provides Corollary 15.2, the measurable-injection fact that makes deterministic subobjects of a standard Borel space exactly its measurable subsets, grounding Example 3.6.","marker":"[Kec95]"},{"why":"Source of the quasi-totality definition and of the comparison categories BorelStoch≤1 and Kl(D≤1), which the paper distinguishes from its partializations.","marker":"[DLR23]"},{"why":"Defines positive quasi-Markov categories and their poset enrichment, conditions and language reused for Proposition 3.1 and the representability transfer.","marker":"[FGL+25]"},{"why":"Gives the CD-theoretic domain definition and shows strict oplax cartesian categories are quasi-Markov, the bridge Proposition 3.1 generalizes.","marker":"[FGTC23]"},{"why":"Proves positive strict oplax cartesian categories are restriction categories; Proposition 3.1 extends this beyond the copyable/oplax setting.","marker":"[CGT25b]"},{"why":"Establishes representability and determinism in Kleisli categories of entire-semiring distribution monads, feeding Examples 3.9 and 3.12 and the representability transfer.","marker":"[FGPR23]"}],"fun_headline_variants":["Partial stochastic maps form a category that keeps its randomness","Span construction builds CD categories from partializable Markov categories","Partial Markov kernels: a categorical construction preserving key properties","From partializable Markov categories to CD categories of partial kernels","Quasi-total partial morphisms: a natural construction for Markov categories"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Partial stochastic maps form a category that keeps its randomness","Span construction builds CD categories from partializable Markov categories","Partial Markov kernels: a categorical construction preserving key properties","From partializable Markov categories to CD categories of partial kernels","Quasi-total partial morphisms: a natural construction for Markov categories"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000498,"raw_usage":{"total_tokens":2250,"prompt_tokens":691,"completion_tokens":1559,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":435,"completion_tokens_details":{"reasoning_tokens":1478}},"tokens_in":435,"tokens_out":1559,"duration_ms":12239,"temperature":1.0,"reasoning_tokens":1478,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T05:36:35.858244+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}