{
  "abstract": "We introduce the category $\\mathbf{SSS}$ of sample-smooth spaces over a mixed site. The test objects are the products $Ω_n := \\mathbb{R}^n \\times Ω$ of a Cartesian space with the universal Hilbert cube $Ω$ carrying all universally measurable sets, and a space is a set with a family of admissible plots $Ω_n \\to \\mathcal{X}$ closed under precomposition. Smoothness and measurability are then not two structures glued along an axiom, but one structure over one site. The site has finite non-empty products, because $Ω$ absorbs its own square; its Karoubi envelope contains every $\\mathbb{R}^n$; and it has mixed morphisms $ω\\mapsto (W(ω),Φ(ω))$, which turn measurability of a smooth family from an axiom into a consequence. $\\mathbf{SSS}$ is a concrete quasitopos: complete, cocomplete, cartesian closed and locally cartesian closed, with a classifier for embeddings. Morphisms of Cartesian spaces are exactly the $C^\\infty$ maps and manifolds embed full and faithfully, both without Boman's theorem. Every object has tangent and cotangent spaces, every morphism a differential. The modalities sit in an adjoint string $Π\\dashv \\flat \\dashv \\natural \\dashv \\sharp \\dashv Λ$, making $\\mathbf{SSS}$ cohesive over quasi-universal spaces. The point is the probability monad. Defining the plots of $\\mathsf{P}(\\mathcal{X})$ as push-forwards of $\\mathcal{X}$-plots at every test object, $\\mathsf{P}$ is an unconditional strong commutative affine monad on all of $\\mathbf{SSS}$ -- functor, unit, product of kernels, multiplication and the monad laws are each one line of seed splitting -- and its Kleisli category, of differentiable simulators, is a Markov category. The reparametrisation trick holds by construction: every Kleisli morphism is plot-wise a sampler, stably under composition. A reflection theorem locates the whole gain in a single plot family.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.26270",
  "author_total": 1,
  "authors": [
    {
      "name": "Patrick Forré"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bz81mgvwew89h.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.26270",
  "fulltext_url": null,
  "id": "bz81mgvwew89h",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.26270",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
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      "2610|Homotopy and Cohomology in Algebraic Topology"
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  "openalex": "W7214110538",
  "pdf_url": "https://arxiv.org/pdf/2609.26270",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bz81mgvwew89h.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.26270",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.26270"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Sample-Smooth Spaces: A Convenient Category for Differentiable Probabilistic Programming",
  "title_text": "Sample-Smooth Spaces: A Convenient Category for Differentiable Probabilistic Programming",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/bz81mgvwew89h",
  "venue": "arXiv",
  "versions": []
}