Object-side stochasticization

# Object-side stochasticization and Giry-like algebras beyond Meas ## Trigger *Source: ChatGPT discussion while reading nLab Markov category page, Giry monad page, and ACT2026 Session B materials.* Ryuya's observation: > If Markov categories arise because we start from deterministic objects such as `Set`/`Meas` and stochasticize only the morphisms, then there should also be a suitable stochasticization of objects. The Eilenberg--Moore category of the Giry monad should be a candidate for that. Immediate correction by Ryuya: > Characterizing Giry-algebras inside `Meas` is probably the wrong direction because `Meas` itself is a bad base category. Instead, use a better generalized-space setting, e.g. Lawvere-style codiscrete/cohesive topoi or the topos of light condensed sets, and then axiomatize the categorical structure of "Giry-algebras" in the same spirit that Markov categories axiomatize stochastic maps. Further refinement by Ryuya: > Like Johnstone's topological topos, maybe we should consider a topos generated by standard Borel spaces. This resembles the earlier idea of a topos generated by abstract hash values, and also resembles the synthetic differential geometry pattern: choose the right generating test objects, then build a topos/internal language around them. This should be remembered as a research seed, not as a settled claim. ## Revised core idea The safe slogan is no longer simply: ```text Giry EM algebras = stochasticized objects. ``` Rather: ```text Markov categories axiomatize stochastic morphisms without committing to Meas. We want an analogous axiomatics for stochasticized objects / barycentric objects, preferably internal to a good topos or generalized space category. ``` There are two canonical monadic presentations: ```text Kleisli: same objects, stochastic/effectful morphisms C_T(X,Y) = C(X, T Y) Eilenberg--Moore: changed objects, deterministic morphisms preserving T-structure object = (A, a: T A -> A) ``` But `Meas^Giry` may be the wrong literal target. The intended object-side structure is more abstract: ```text an object A equipped with coherent barycenter / averaging / integration operations, formulated in a base category whose spaces are better behaved than Meas. ``` ## Conceptual distinction to investigate - `Kl(T)` stochasticizes morphisms. - `C^T` stochasticizes objects. - Markov categories abstract away from the bad concrete category of measurable spaces by axiomatizing copy/delete/normalization/conditioning behavior. - The desired new project is to abstract away from `Meas^Giry` similarly: axiomatize the object-level structure of probability absorption / barycenters / integration. Possible slogan: ```text Markov category : stochastic maps, axiomatically. Giry-like algebra category : stochastic objects, axiomatically. ``` ## Lawvere question Question to investigate: > Did Lawvere already formulate an object-side, topos-based axiomatics of probabilistic/barycentric objects, perhaps in codiscrete/cohesive/topos-of-continuum work? First-pass status: - Lawvere did initiate the categorical study of probabilistic mappings / Markov kernels in 1962. - Lawvere also pushed topos-theoretic foundations for continuum physics and argued that a topos should be viewed as an algebra of continuous set-valued functions on a generalized space, rather than naively as the generalized space itself. - I have not yet found evidence that Lawvere explicitly gave the desired modern axiomatics: "Giry-algebra-like stochastic objects inside a good topos, analogous to Markov categories for stochastic morphisms." - The closest Lawvere-adjacent line may be algebraic theories/toposes and integration as an infinitary or sheaf/topos-internal algebraic theory. ## Candidate construction: standard-Borel-generated topos This is the new proposed direction. Analogy: ```text Johnstone topological topos: choose topological/interval-like generating data build a topos in which realization/topological structure has an internal language SDG / Cahiers-style toposes: choose infinitesimal test objects build a topos in which infinitesimal calculus is internal Proposed Borel/probability topos: choose standard Borel spaces / countably generated measurable tests build a topos in which measurable/probabilistic structure is internal ``` The point is not to turn `StdBorel` itself into a topos. It is to use standard Borel spaces as a site/generating class, analogous to using infinitesimal algebras or Cartesian spaces in SDG, or topological generators in Johnstone-style topological topos constructions. Possible site sketches: ```text BorelTest = small skeleton of standard Borel spaces E_Bor = Sh(BorelTest, J_Bor) ``` Key question: what is the right coverage `J_Bor`? Candidates to test: 1. coverage generated by countable measurable partitions; 2. coverage generated by Borel surjections/quotients; 3. coverage generated by standard probability kernels or disintegration-friendly maps; 4. coverage generated by countably separating families of Borel maps to `[0,1]` or `2^N`; 5. a regular/canonical topology induced by the embedding into a better topological category such as Polish/QCB/light-condensed. Expected benefit: - `Standard Borel` is much better behaved than arbitrary `Meas`: it is countably generated/separated, closed under countable products and coproducts in good cases, and regular conditional probabilities exist in standard Borel settings. - It may supply the correct test-object category for probability, just as infinitesimal objects supply the correct test category for SDG. Potential danger: - All uncountable standard Borel spaces are Borel-isomorphic, so naive object geometry collapses. The topology/coverage/probability doctrine must retain more than bare Borel isomorphism type. - A mere sheaf topos on `StdBorel` may internalize measurable sets but not automatically probability, kernels, integration, or barycenters. - Need to decide whether probability is a monad, a doctrine, a valuation object, or an algebraic/geometric theory internal to the generated topos. ## Link to abstract hash-value generated topos Ryuya's earlier "abstract hash value generated topos" idea seems structurally similar: ```text choose a class of observable/test values H build a topos generated by H-tests study objects through their H-valued observations ``` Standard Borel spaces could play the role of a continuous/probabilistic analogue of hash values: ```text hash/test value idea: generated by abstract observation tokens standard Borel idea: generated by measurable observation spaces SDG idea: generated by infinitesimal probes ``` The common pattern is: ```text not: start with a bad concrete category and characterize its algebras but: choose the right probes, generate a topos, then axiomatize the intended internal structure. ``` ## Prior research located in first pass ### 1. Markov categories and Kleisli probability monads - Tobias Fritz, "A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics" (2019/2020). - Develops Markov categories as a synthetic probability framework. - Treats conditioning, disintegration, conditional independence, sufficient statistics abstractly. - arXiv: https://arxiv.org/abs/1908.07021 - Tobias Fritz, Tomas Gonda, Paolo Perrone, Eigil Fjeldgren Rischel, "Representable Markov Categories and Comparison of Statistical Experiments in Categorical Probability" (2020). - Introduces representable Markov categories, where one can talk internally about spaces of distributions. - Explicitly relates Markov categories and Kleisli categories of probability monads. - arXiv: https://arxiv.org/abs/2010.07416 - Sean Moss and Paolo Perrone, "Probability monads with submonads of deterministic states" (LICS 2022 extended version). - Directly relevant to the pure/deterministic tension. - In Markov categories, deterministic morphisms are defined by copying/discarding equations; in monadic semantics, pure maps live in the base category. - Studies sober objects and sobrification for probability monads, including the Giry monad. - arXiv: https://arxiv.org/abs/2204.07003 - Eigil Fjeldgren Rischel, "The Universal Property of Measure-Theoretic Probability" (2025). - Gives a universal property of `BorelStoch`, the Markov category of standard Borel spaces and Markov kernels. - Important for any standard-Borel-generated direction. - arXiv: https://arxiv.org/abs/2512.15485 ### 2. Giry Eilenberg--Moore algebras as a warning case, not the endpoint - Ernst-Erich Doberkat, "Eilenberg-Moore algebras for stochastic relations" (Information and Computation, 2006). - Earlier work on EM algebras for stochastic/probabilistic relations. - Need to inspect details and limitations, especially Polish/continuity assumptions. - Kirk Sturtz, "The factorization of the Giry monad" (2017). - Claims/factors Giry monad through convex measurable spaces. - arXiv: https://arxiv.org/abs/1707.00488 - Tomas Crhak, "On functors from category of Giry algebras to category of convex spaces" (2018). - Important warning: refutes a too-strong equivalence claim between convex spaces and Giry EM algebras. - Shows no such simple equivalence exists in the asserted form. - arXiv: https://arxiv.org/abs/1804.01345 - Tomas Crhak, "A note on sigma-algebras on sets of affine and measurable maps to the unit interval" (2018). - Gives counterexamples concerning sigma-algebras used in Sturtz's earlier proof. - arXiv: https://arxiv.org/abs/1803.07956 - Kirk Sturtz, "Characterizing Giry-algebras as coseparable super convex spaces" (2019). - Withdrawn. The arXiv page says there is a fundamental error related to the claim that the category of Giry algebras has a coseparator. - Keep as a warning not to overidentify Giry algebras with convenient convex spaces. - arXiv: https://arxiv.org/abs/1907.03209 ### 3. Better base categories / topological and topos-like directions - Peter Johnstone, "Aspects of Topology" / topological topos line. - Johnstone's topological topos was designed to present geometric realization as a geometric morphism between toposes. - This is the main analogy for a standard-Borel-generated topos. - Jean Goubault-Larrecq and Xiaodong Jia, "Algebras of the extended probabilistic powerdomain monad" (2019). - EM algebras of valuation/probabilistic powerdomain monads on `TOP_0` are characterized as locally convex sober topological cones; algebra maps are continuous linear maps in key cases. - Very relevant as a topological analogue of "objects that absorb probability by barycentres". - arXiv: https://arxiv.org/abs/1903.07472 - Tobias Fritz and Paolo Perrone, "A Probability Monad as the Colimit of Spaces of Finite Samples" (2017). - Builds a probability monad on complete metric spaces via a colimit of finite samples. - Develops integration and measures-on-measures without ordinary measure theory. - Algebras are equivalent to closed convex subsets of Banach spaces with short affine maps. - arXiv: https://arxiv.org/abs/1712.05363 - Peter Kristel and Benedikt Peterseim, "A Topologically Enriched Probability Monad on the Cartesian Closed Category of CGWH Spaces" (2024). - Constructs a Riesz probability monad on CGWH spaces, extending Radon and Giry and enriched topologically. - Restriction to QCB spaces is strongly affine, making independence/determinism interact well. - arXiv: https://arxiv.org/abs/2404.08430 - Ruben Van Belle, "Probability monads as codensity monads" (2021). - Constructs probability monads from categorical/codensity data over countable distributions and uses integral representation theorems. - Relevant to rebuilding probability without committing to raw Meas. - arXiv: https://arxiv.org/abs/2111.01250 - Zev Shirazi, "Commutativity and liftings of codensity monads of probability measures" (2024). - Studies commutativity, affineness, liftings of probability monads, and exact pointwise monoidality. - Notes a specific obstruction for the Giry monad: probability bimeasures need not extend to measures, while standard Borel restrictions behave better. - arXiv: https://arxiv.org/abs/2405.12917 - Felix Cherubini, Thierry Coquand, Freek Geerligs, Hugo Moeneclaey, "A Foundation for Synthetic Stone Duality" (2024). - Uses the higher topos corresponding to light condensed sets, with HoTT axioms, to do synthetic topology. - Not probability-specific, but important as a candidate base topos direction. - arXiv: https://arxiv.org/abs/2412.03203 ### 4. Lawvere-theory / sheaf-topos algebra of integration - Boaz Haberman, "Algebraic theories and commutativity in a sheaf topos" (2018). - Defines `C`-ary Lawvere theories for a site of definition of a Grothendieck topos. - Categories of models form stacks over the topos and are complete/cocomplete in the internal sense. - Gives a convenient category of linear spaces generated by the theory of Lebesgue integration. - This looks closer to Ryuya's desired "axiomatize Giry-algebra-like structure in a topos" than direct `Meas^Giry` characterizations. - arXiv: https://arxiv.org/abs/1803.09378 - Tom Leinster, "A categorical derivation of Lebesgue integration" (2020). - Characterizes `L^p` spaces and integration via universal properties. - Not directly a topos/Markov-category axiomatics, but useful for treating integration as universal structure rather than as sigma-algebra technology. - arXiv: https://arxiv.org/abs/2011.00412 ## Research question Can we formulate an axiomatic theory of stochasticized objects in a good generalized-space environment? Possible formulation: ```text Input: a cohesive/codiscrete/light-condensed/topos-like category H a probability/integration doctrine P on H Output: an axiomatic category of P-algebras / barycentric objects / integration objects analogous to Markov categories, but object-sided rather than morphism-sided. ``` Desired properties: 1. Do not begin with `Meas`. 2. Treat barycenters/integration as primitive universal/categorical structure. 3. Recover classical Giry/Radon/Kantorovich examples by realization functors. 4. Interact with Markov categories via a Kleisli/EM or representability bridge. 5. Ideally support an internal language in a topos or cohesive/light-condensed setting. 6. Consider a `StdBorel`-generated topos as an intermediate candidate between raw `Meas` and light-condensed/cohesive worlds. ## Possible connection to Ryuya's existing themes - Local state classifiers: compare state classifiers with barycenter/integration classifiers. Both may be classifier objects for a fibration/doctrine of local observations. - Topoi of automata / dynamical systems: ask whether probability/possibility doctrines in presheaf or sheaf topoi yield stochasticized automata/state objects. - Quotient/hyperconnected themes: object-side stochasticization may be a quotient/completion process over deterministic state spaces. - Abstract hash-value generated topos: compare hash/test-value generation with standard-Borel/measurable-test generation. - ACT2026 Session B: - Talk 1: normalized stochastic kernels expose limits of Kleisli-style stochastic morphisms under normalization. - Talk 2: definable Markov categories restrict which stochastic morphisms/conditionals remain tame. - Talk 3: possibilistic belief objects may be an object-side classifier for updating, analogous in spirit to EM/power-object structure. ## Immediate next actions 1. Treat `Meas^Giry` as a warning/example, not as the foundation. 2. Search Lawvere archives for continuum physics, codiscrete/cohesive topos, integration, and probability references. 3. Read Haberman 2018 before Giry-algebra characterization papers; it is closer to a sheaf/topos algebraic-theory route. 4. Read Fritz--Perrone 2017 and Kristel--Peterseim 2024 for non-Meas probability monads with better algebra categories. 5. Read Cherubini--Coquand--Geerligs--Moeneclaey 2024 for light-condensed internal topology; then ask what probability/integration doctrine should live there. 6. Formulate an abstract definition candidate: "barycentric Markov object category" or "probability-algebra doctrine". 7. Compare with representable Markov categories: are distribution objects `P X` free stochasticized objects, or do they live in a separate EM-like world? 8. Make a toy site from a small skeleton of standard Borel spaces and test candidate coverages: countable partitions, Borel quotients, and observation maps into `[0,1]` or `2^N`. 9. Check whether `BorelStoch`'s universal property can be restated as a universal property of a generated topos plus an internal probability doctrine. ## Warning Do not state naively that `GiryAlg` is simply `Convex spaces`. The literature contains corrections and subtleties, and one Sturtz characterization paper was withdrawn. The safe conceptual claim is: ```text Giry EM algebras are spaces equipped with a coherent barycenter/expectation operation for probability measures, but Meas may be the wrong base category for the theory Ryuya wants. ``` The exact equivalence with convex/superconvex/measurable structures depends on separation, measurability, set-theoretic hypotheses, and the chosen base category.