Sample-Smooth Spaces: A Convenient Category for Differentiable Probabilistic Programming

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 $\Pi ⊣♭⊣♮⊣♯⊣\Lambda$, 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.