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This paper proposes a novel categorical framework, $\mathbf{SSS}$, that unifies smoothness and measurability into a single structure to formalize differentiable probabilistic programming. By defining the probability monad within this quasitopos, the authors establish a rigorous foundation where reparametrization tricks hold by construction, enabling stable composition of differentiable simulators. This theoretical advance offers a unified mathematical language for integrating continuous dynamics with stochastic sampling in machine learning models.
Sample-Smooth Spaces: A Convenient Category for Differentiable Probabilistic Programming
arXiv:2609.26270v1 Announce Type: cross Abstract: We introduce the category $\mathbf{SSS}$ of sample-smooth spaces over a mixed site. The test objects are the products $\Omega_n := \mathbb{R}^n \times \Omega$ of a Cartesian space with the universal Hilbert cube $\Omega$ carrying all universally measurable sets, and a space is a set with a family of admissible plots $\Omega_n \to \mathcal{X}$ 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 $\Omega$ absorbs its own square; its Karoubi envelope contains every $\mathbb{R}^n$; and it has mixed morphisms $\omega \mapsto (W(\omega),\Phi(\omega))$, 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 $\Pi \dashv \flat \dashv \natural \dashv \sharp \dashv \Lambda$, 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:We introduce the category $\mathbf{SSS}$ of sample-smooth spaces over a mixed site. The test objects are the products $\Omega_n := \mathbb{R}^n \times \Omega$ of a Cartesian space with the universal Hilbert cube $\Omega$ carrying all universally measurable sets, and a space is a set with a family of admissible plots $\Omega_n \to \mathcal{X}$ 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 $\Omega$ absorbs its own square; its Karoubi envelope contains every $\mathbb{R}^n$; and it has mixed morphisms $\omega \mapsto (W(\omega),\Phi(\omega))$, 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 $\Pi \dashv \flat \dashv \natural \dashv \sharp \dashv \Lambda$, 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.
Category Theory (math.CT); Machine Learning (cs.LG); Logic in Computer Science (cs.LO); Programming Languages (cs.PL); Probability (math.PR)
arXiv:2609.26270 [math.CT]
(or arXiv:2609.26270v1 [math.CT] for this version)
https://doi.org/10.48550/arXiv.2609.26270 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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