Sample-Smooth Spaces: A Convenient Category for Differentiable Probabilistic Programming
Abstract
We introduce the category SSS of sample-smooth spaces over a mixed site. The test objects are the products Ωn := ℝn × 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 Ωn → 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 ℝn; and it has mixed morphisms ω↦ (W(ω),Φ(ω)), which turn measurability of a smooth family from an axiom into a consequence. 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∞ 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 Πdashv ♭ dashv natural dashv ♯ dashv Λ, making SSS cohesive over quasi-universal spaces. The point is the probability monad. Defining the plots of P(𝒳) as push-forwards of 𝒳-plots at every test object, P is an unconditional strong commutative affine monad on all of 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.
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BibTeX
@misc{forre2026samplesmooth,
title = {{Sample-Smooth Spaces: A Convenient Category for Differentiable Probabilistic Programming}},
author = {Patrick Forré},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26270},
eprint = {2609.26270},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26270},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Sample-Smooth Spaces: A Convenient Category for Differentiable Probabilistic Programming
AU - Patrick Forré
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.26270
UR - https://arxiv.org/abs/2609.26270
AB - We introduce the category SSS of sample-smooth spaces over a mixed site. The test objects are the products Ω_n := ℝ^n × 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 → 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 ℝ^n; and it has mixed morphisms ω↦ (W(ω),Φ(ω)), which turn measurability of a smooth family from an axiom into a consequence. 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^∞ 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 Πdashv ♭ dashv natural dashv ♯ dashv Λ, making SSS cohesive over quasi-universal spaces. The point is the probability monad. Defining the plots of P(𝒳) as push-forwards of 𝒳-plots at every test object, P is an unconditional strong commutative affine monad on all of 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.
ER -