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

arXiv: https://arxiv.org/abs/2609.26270 · HTML: https://mlsanity.com/paper/2609.26270/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 32 references that are in the map.
- First posted on arXiv 14 Aug 2026.

## Builds on (most cited first, 15 of 15)

- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Categorical Reparameterization with Gumbel-Softmax](https://mlsanity.com/paper/1611.01144/) (2016)
- [The Concrete Distribution: A Continuous Relaxation of Discrete Random Variables](https://mlsanity.com/paper/1611.00712/) (2016)
- [Monte Carlo Gradient Estimation in Machine Learning](https://mlsanity.com/paper/1906.10652/) (2019)
- [Implicit Reparameterization Gradients](https://mlsanity.com/paper/1805.08498/) (2018)
- [A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics](https://mlsanity.com/paper/1908.07021/) (2019)
- [Stochastic Backpropagation and Approximate Inference in Deep Generative Models](https://mlsanity.com/paper/1401.4082/) (2014)
- [A Convenient Category for Higher-Order Probability Theory](https://mlsanity.com/paper/1701.02547/) (2017)
- [Correctness of Automatic Differentiation via Diffeologies and Categorical Gluing](https://mlsanity.com/paper/2001.02209/) (2020)
- [ADEV: Sound Automatic Differentiation of Expected Values of Probabilistic Programs](https://mlsanity.com/paper/2212.06386/) (2022)
- [A Domain Theory for Statistical Probabilistic Programming](https://mlsanity.com/paper/1811.04196/) (2018)
- [Smoothness Analysis for Probabilistic Programs with Application to Optimised Variational Inference](https://mlsanity.com/paper/2208.10530/) (2022)
- [ωPAP Spaces: Reasoning Denotationally About Higher-Order, Recursive Probabilistic and Differentiable Programs](https://mlsanity.com/paper/2302.10636/) (2023)
- [Monads need not be endofunctors](https://mlsanity.com/paper/1412.7148/) (2014)
- [Concrete categories and higher-order recursion (With applications including probability, differentiability, and full abstraction)](https://mlsanity.com/paper/2205.15917/) (2022)

## Abstract (arXiv)

We introduce the category SSS of sample-smooth spaces over a mixed site. The test objects are the products Omegan := ℝn × 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 Omegan → 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 ℝn; and it has mixed morphisms ω mapsto (W(ω),Phi(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 Pi dashv flat dashv natural dashv sharp dashv Lambda, making SSS cohesive over quasi-universal spaces. The point is the probability monad. Defining the plots of mathsfP(X) as push-forwards of X-plots at every test object, mathsfP 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.